Deck 12: Vector-Valued Functions

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سؤال
A particle moves in the yz-plane along the curve represented by the vector-valued function <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the maximum value of   .</strong> A) 8 B) 6 C) 9 D) 7 E) 3 <div style=padding-top: 35px> . Find the maximum value of <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the maximum value of   .</strong> A) 8 B) 6 C) 9 D) 7 E) 3 <div style=padding-top: 35px> .

A) 8
B) 6
C) 9
D) 7
E) 3
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سؤال
Find the indefinite integral below. <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> Do not include an arbitrary constant vector.

A) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 <div style=padding-top: 35px> at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 <div style=padding-top: 35px> for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) -30.406
B) -29.533
C) -81.262
D) -35.346
E) -33.994
سؤال
The graph below is most likely the graph of which of the following equations? ​ <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>

A) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
B) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
C) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
D) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
E) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
سؤال
Find the curvature of the plane curve <strong>Find the curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 0.986 B) 0.658 C) 0.114 D) 0.036 E) 0.027 <div style=padding-top: 35px> at <strong>Find the curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 0.986 B) 0.658 C) 0.114 D) 0.036 E) 0.027 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 0.986
B) 0.658
C) 0.114
D) 0.036
E) 0.027
سؤال
Find the curvature K of the curve <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 1.400 B) 1.599 C) 0.192 D) 0.435 E) 0.053 <div style=padding-top: 35px> at the point <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 1.400 B) 1.599 C) 0.192 D) 0.435 E) 0.053 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 1.400
B) 1.599
C) 0.192
D) 0.435
E) 0.053
سؤال
Find the curvature <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> of the curve given below. <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
The position vector <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> describes the path of an object moving in space. Find the velocity <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Evaluate the limit given below. ​ <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​ <div style=padding-top: 35px>

A) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​ <div style=padding-top: 35px>
B) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​ <div style=padding-top: 35px>
C) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​ <div style=padding-top: 35px>
D) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​ <div style=padding-top: 35px>
E) The limit does not exist.
سؤال
Find the open interval on which the curve given by the vector-valued function <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> is smooth.

A) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the maximum height. Round your answer to the nearest integer.</strong> A) 247 feet B) 235 feet C) 205 feet D) 218 feet E) 220 feet <div style=padding-top: 35px> with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

A) 247 feet
B) 235 feet
C) 205 feet
D) 218 feet
E) 220 feet
سؤال
Find the unit tangent vector <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> for the line tangent to the space curve <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> at point <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> .

A) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
The position vector <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   <div style=padding-top: 35px> describes the path of an object moving in space. Find the speed <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   <div style=padding-top: 35px> of the object.

A) 17
B) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   <div style=padding-top: 35px>
C) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   <div style=padding-top: 35px>
D) 115
E) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   <div style=padding-top: 35px>
سؤال
Find the point on the curve <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> at which the curvature K is a maximum.

A) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Sketch the curve represented by the vector-valued function <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> and give the orientation of the curve.

A) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Match the equation with the graph shown in red below. <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 <div style=padding-top: 35px> at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 <div style=padding-top: 35px> for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 5.315
B) 3.010
C) 7.491
D) 3.745
E) 2.657
سؤال
The position vector <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> .

A) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Determine the maximum height of a projectile fired at a height of 5 feet above the ground with an initial velocity of 600 feet per second and at an angle of <strong>Determine the maximum height of a projectile fired at a height of 5 feet above the ground with an initial velocity of 600 feet per second and at an angle of   above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.</strong> A) 8,440.000 feet B) 8,442.500 feet C) 4,221.250 feet D) 4,223.750 feet E) 8,447.500 feet <div style=padding-top: 35px> above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

A) 8,440.000 feet
B) 8,442.500 feet
C) 4,221.250 feet
D) 4,223.750 feet
E) 8,447.500 feet
سؤال
Find the length of the plane curve given below. <strong>Find the length of the plane curve given below.  </strong> A) 14 B) 8 C) 11 D) 13 E) 12 <div style=padding-top: 35px>

A) 14
B) 8
C) 11
D) 13
E) 12
سؤال
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px> <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
سؤال
Suppose the two particles travel along the space curves <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> and <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.

A) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <div style=padding-top: 35px> given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <div style=padding-top: 35px> <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <div style=padding-top: 35px>

A) 1
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <div style=padding-top: 35px>
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <div style=padding-top: 35px>
D) 6
E) 0
سؤال
Find the point on the curve given below at which the curvature K is zero. <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Represent the following curve by a vector-valued function. <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A baseball is hit 5 feet above the ground at 80 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 80 feet per second and at an angle of   with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place. ​</strong> A) 194.2 feet B) 212.3 feet C) 210.1 feet D) 171.4 feet E) 236.5 feet <div style=padding-top: 35px> with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place. ​

A) 194.2 feet
B) 212.3 feet
C) 210.1 feet
D) 171.4 feet
E) 236.5 feet
سؤال
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 78 yards directly downfield at a height of 4 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 78 yards directly downfield at a height of 4 feet. The pass is released at an angle of   with the horizontal. Find the time in seconds the receiver has to reach the proper position after the quarterback releases the football. Round your answer to one decimal place.</strong> A) 2.2 seconds B) 12.8 seconds C) 22.2 seconds D) 3.7 seconds E) 7.4 seconds <div style=padding-top: 35px> with the horizontal. Find the time in seconds the receiver has to reach the proper position after the quarterback releases the football. Round your answer to one decimal place.

A) 2.2 seconds
B) 12.8 seconds
C) 22.2 seconds
D) 3.7 seconds
E) 7.4 seconds
سؤال
Find the domain of the vector-valued function given below. <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> where <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the vectors <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px> and <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px> for the following vector function. ​ <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>

A) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
B) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
C) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
D) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
E) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
سؤال
Find the indefinite integral below. <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> Do not include an arbitrary constant vector.

A) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the length of the space curve given below. <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of <strong>Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of   above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.</strong> A) 2,669.517 feet B) 1,764.116 feet C) 5,337.349 feet D) 6,767.555 feet E) 3,525.690 feet <div style=padding-top: 35px> above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

A) 2,669.517 feet
B) 1,764.116 feet
C) 5,337.349 feet
D) 6,767.555 feet
E) 3,525.690 feet
سؤال
Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

A) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A 5800-pound vehicle is driven at a speed of 25 miles per hour on a circular interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires? Round your answer to one decimal place.

A) 2,716.9 lb
B) 3,396.4 lb
C) 3,046.0 lb
D) 2,707.6 lb
E) 3,565.1 lb
سؤال
Sketch the curve represented by the vector-valued function <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> and give the orientation of the curve.

A) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the unit tangent vector to the curve given below at the specified point. <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the domain of the vector-valued function given below. <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A particle moves in the yz-plane along the curve represented by the vector-valued function <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the minimum value of   .</strong> A) 3 B) 2 C) 5 D) 6 E) 8 <div style=padding-top: 35px> . Find the minimum value of <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the minimum value of   .</strong> A) 3 B) 2 C) 5 D) 6 E) 8 <div style=padding-top: 35px> .

A) 3
B) 2
C) 5
D) 6
E) 8
سؤال
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   <div style=padding-top: 35px> given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   <div style=padding-top: 35px>

A) 1
B) -18 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   <div style=padding-top: 35px>
C) 18 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   <div style=padding-top: 35px>
D) 0
E) 3 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 <div style=padding-top: 35px> at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 <div style=padding-top: 35px> for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 0.816
B) 0.389
C) 0.777
D) 1.864
E) 0.889
سؤال
Find <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px> given the following vector function: ​ <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px>
B) 0
C) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 <div style=padding-top: 35px> at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 <div style=padding-top: 35px> for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 1.069
B) 0.963
C) 0.468
D) 0.936
E) 0.848
سؤال
Given the vector-valued function below, evaluate <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> . <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
The position vector <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> describes the path of an object moving in space. Find the velocity <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Evaluate the limit given below. <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist. <div style=padding-top: 35px>

A) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist. <div style=padding-top: 35px>
B) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist. <div style=padding-top: 35px>
C) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist. <div style=padding-top: 35px>
D) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist. <div style=padding-top: 35px>
E) The limit does not exist.
سؤال
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 <div style=padding-top: 35px> at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 <div style=padding-top: 35px> for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 28.451
B) 28.636
C) 28.400
D) 28.111
E) 20.897
سؤال
The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> (x and y are measured in miles) can safely go 30 miles per hour at <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> . How fast can it go at <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> ? Round your answer to two decimal places.

A) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> mi/h
B) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> mi/h
C) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> mi/h
D) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> mi/h
E) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h <div style=padding-top: 35px> mi/h
سؤال
Find <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 <div style=padding-top: 35px> at time <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 <div style=padding-top: 35px> for the space curve <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 <div style=padding-top: 35px> .

A) 9
B) 0
C) 8
D) 64
E) 3
سؤال
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 5 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 5 feet. The pass is released at an angle of   with the horizontal. Find the speed of the football when it is released. Round your answer to three decimal places.</strong> A) 32.979 feet per second B) 57.483 feet per second C) 18.863 feet per second D) 114.967 feet per second E) 189.144 feet per second <div style=padding-top: 35px> with the horizontal. Find the speed of the football when it is released. Round your answer to three decimal places.

A) 32.979 feet per second
B) 57.483 feet per second
C) 18.863 feet per second
D) 114.967 feet per second
E) 189.144 feet per second
سؤال
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 <div style=padding-top: 35px> at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 <div style=padding-top: 35px> for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 4.575
B) 3.841
C) 2.716
D) 1.718
E) 4.685
سؤال
A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> with respect to the ground. Find the vector-valued function for the path of the baseball.

A) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 <div style=padding-top: 35px> at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 <div style=padding-top: 35px> for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 0.041
B) 0.093
C) 0.406
D) 0.000
E) 0.914
سؤال
Determine the interval on which the vector-valued function <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> is continuous.

A) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Use the given acceleration function and initial conditions to find the position at time t = 1. <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A baseball, hit 7 feet above the ground, leaves the bat at an angle of <strong>A baseball, hit 7 feet above the ground, leaves the bat at an angle of   and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​</strong> A) 180 feet B) 18 feet C) 397 feet D) 278 feet E) 50 feet <div style=padding-top: 35px> and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​

A) 180 feet
B) 18 feet
C) 397 feet
D) 278 feet
E) 50 feet
سؤال
Use the given acceleration function and initial conditions to find the position at time t = 4. <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the curvature K of the curve <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 0.728 B) 0.970 C) 0.158 D) 0.057 E) 0.126 <div style=padding-top: 35px> at the point <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 0.728 B) 0.970 C) 0.158 D) 0.057 E) 0.126 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 0.728
B) 0.970
C) 0.158
D) 0.057
E) 0.126
سؤال
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 62 yards directly downfield at a height of 3 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 62 yards directly downfield at a height of 3 feet. The pass is released at an angle of   with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. ​</strong> A) 337.1 feet B) 16.3 feet C) 245.4 feet D) 85.8 feet E) 17.5 feet <div style=padding-top: 35px> with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. ​

A) 337.1 feet
B) 16.3 feet
C) 245.4 feet
D) 85.8 feet
E) 17.5 feet
سؤال
Find <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px> given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>

A) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
B) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
C) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
D) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
E) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
سؤال
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px> given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>

A) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
B) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
C) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
D) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
E) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <div style=padding-top: 35px>
سؤال
Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a <strong>Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a   turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is   where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.</strong> A) 201 mi/h B) 64 mi/h C) 173 mi/h D) 137 mi/h E) 255 mi/h <div style=padding-top: 35px> turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is <strong>Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a   turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is   where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.</strong> A) 201 mi/h B) 64 mi/h C) 173 mi/h D) 137 mi/h E) 255 mi/h <div style=padding-top: 35px> where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.

A) 201 mi/h
B) 64 mi/h
C) 173 mi/h
D) 137 mi/h
E) 255 mi/h
سؤال
Find the unit tangent vector to the curve given below at the specified point. <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the arc length for <strong>Find the arc length for   over the interval   . Round your answer to two decimal places. ​</strong> A) 110.36 B) 140.93 C) 129.56 D) 57.80 E) 102.75 <div style=padding-top: 35px> over the interval <strong>Find the arc length for   over the interval   . Round your answer to two decimal places. ​</strong> A) 110.36 B) 140.93 C) 129.56 D) 57.80 E) 102.75 <div style=padding-top: 35px> . Round your answer to two decimal places. ​

A) 110.36
B) 140.93
C) 129.56
D) 57.80
E) 102.75
سؤال
The position vector <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> describes the path of an object moving in space. Find the acceleration <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
A baseball is hit 3 feet above the ground at 95 feet per second and at an angle of <strong>A baseball is hit 3 feet above the ground at 95 feet per second and at an angle of   with respect to the ground. Find the range. Round your answer to one decimal place.</strong> A) 285 feet B) 352.2 feet C) 370.7 feet D) 501.9 feet E) 300 feet <div style=padding-top: 35px> with respect to the ground. Find the range. Round your answer to one decimal place.

A) 285 feet
B) 352.2 feet
C) 370.7 feet
D) 501.9 feet
E) 300 feet
سؤال
Find the radius of curvature of the plane curve <strong>Find the radius of curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 1021.045 B) 2340.571 C) 1374.626 D) 1565.001 E) 1571.001 <div style=padding-top: 35px> at <strong>Find the radius of curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 1021.045 B) 2340.571 C) 1374.626 D) 1565.001 E) 1571.001 <div style=padding-top: 35px> . Round your answer to three decimal places.

A) 1021.045
B) 2340.571
C) 1374.626
D) 1565.001
E) 1571.001
سؤال
The position vector <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> describes the path of an object moving in space. Find the speed <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> of the object. ​

A) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px>
B) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> 122
C) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px>
D) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px>
E) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <div style=padding-top: 35px> 18
سؤال
A baseball player at second base throws a ball 95 feet to the player at first base. The ball is thrown 8 feet above the ground with an initial velocity of 46 miles per hour and at an angle of <strong>A baseball player at second base throws a ball 95 feet to the player at first base. The ball is thrown 8 feet above the ground with an initial velocity of 46 miles per hour and at an angle of   above the horizontal. At which height does the player at first base catch the ball? Round your answer to three decimal places.</strong> A) 2.908 feet B) 2.008 feet C) 0.808 feet D) 1.408 feet E) 0.908 feet <div style=padding-top: 35px> above the horizontal. At which height does the player at first base catch the ball? Round your answer to three decimal places.

A) 2.908 feet
B) 2.008 feet
C) 0.808 feet
D) 1.408 feet
E) 0.908 feet
سؤال
Evaluate the definite integral below. <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
سؤال
Find <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px> given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>

A) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
B) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
C) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
D) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
E) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   <div style=padding-top: 35px>
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Deck 12: Vector-Valued Functions
1
A particle moves in the yz-plane along the curve represented by the vector-valued function <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the maximum value of   .</strong> A) 8 B) 6 C) 9 D) 7 E) 3 . Find the maximum value of <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the maximum value of   .</strong> A) 8 B) 6 C) 9 D) 7 E) 3 .

A) 8
B) 6
C) 9
D) 7
E) 3
C
2
Find the indefinite integral below. <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   Do not include an arbitrary constant vector.

A) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
B) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
C) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
D) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
E) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
D
3
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) -30.406 B) -29.533 C) -81.262 D) -35.346 E) -33.994 . Round your answer to three decimal places.

A) -30.406
B) -29.533
C) -81.262
D) -35.346
E) -33.994
D
4
The graph below is most likely the graph of which of the following equations? ​ <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​

A) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
B) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
C) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
D) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
E) <strong>The graph below is most likely the graph of which of the following equations? ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
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5
Find the curvature of the plane curve <strong>Find the curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 0.986 B) 0.658 C) 0.114 D) 0.036 E) 0.027 at <strong>Find the curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 0.986 B) 0.658 C) 0.114 D) 0.036 E) 0.027 . Round your answer to three decimal places.

A) 0.986
B) 0.658
C) 0.114
D) 0.036
E) 0.027
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6
Find the curvature K of the curve <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 1.400 B) 1.599 C) 0.192 D) 0.435 E) 0.053 at the point <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 1.400 B) 1.599 C) 0.192 D) 0.435 E) 0.053 . Round your answer to three decimal places.

A) 1.400
B) 1.599
C) 0.192
D) 0.435
E) 0.053
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7
Find the curvature <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)   of the curve given below. <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the curvature   of the curve given below.  </strong> A)   B)   C)   D)   E)
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8
The position vector <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   describes the path of an object moving in space. Find the velocity <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
B) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
C) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
D) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
E) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
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9
Evaluate the limit given below. ​ <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​

A) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​
B) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​
C) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​
D) <strong>Evaluate the limit given below. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E) The limit does not exist. ​
E) The limit does not exist.
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10
Find the open interval on which the curve given by the vector-valued function <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)   is smooth.

A) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)
B) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)
C) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)
D) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)
E) <strong>Find the open interval on which the curve given by the vector-valued function   is smooth.</strong> A)   B)   C)   D)   E)
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11
A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the maximum height. Round your answer to the nearest integer.</strong> A) 247 feet B) 235 feet C) 205 feet D) 218 feet E) 220 feet with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

A) 247 feet
B) 235 feet
C) 205 feet
D) 218 feet
E) 220 feet
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12
Find the unit tangent vector <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   for the line tangent to the space curve <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   at point <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)   .

A) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)
B) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)
C) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)
D) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)
E) <strong>Find the unit tangent vector   for the line tangent to the space curve   at point   .</strong> A)   B)   C)   D)   E)
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13
The position vector <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   describes the path of an object moving in space. Find the speed <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)   of the object.

A) 17
B) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)
C) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)
D) 115
E) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object.</strong> A) 17 B)   C)   D) 115 E)
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14
Find the point on the curve <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)   at which the curvature K is a maximum.

A) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)
B) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)
C) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)
D) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)
E) <strong>Find the point on the curve   at which the curvature K is a maximum.</strong> A)   B)   C)   D)   E)
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15
Sketch the curve represented by the vector-valued function <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   and give the orientation of the curve.

A) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
B) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
C) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
D) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
E) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
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16
Match the equation with the graph shown in red below. <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)

A) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)
B) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)
C) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)
D) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)
E) <strong>Match the equation with the graph shown in red below.  </strong> A)   B)   C)   D)   E)
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17
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 5.315 B) 3.010 C) 7.491 D) 3.745 E) 2.657 . Round your answer to three decimal places.

A) 5.315
B) 3.010
C) 7.491
D) 3.745
E) 2.657
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18
The position vector <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)   .

A) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)
B) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)
C) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)
D) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)
E) <strong>The position vector   describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point   .</strong> A)   B)   C)   D)   E)
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19
Determine the maximum height of a projectile fired at a height of 5 feet above the ground with an initial velocity of 600 feet per second and at an angle of <strong>Determine the maximum height of a projectile fired at a height of 5 feet above the ground with an initial velocity of 600 feet per second and at an angle of   above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.</strong> A) 8,440.000 feet B) 8,442.500 feet C) 4,221.250 feet D) 4,223.750 feet E) 8,447.500 feet above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

A) 8,440.000 feet
B) 8,442.500 feet
C) 4,221.250 feet
D) 4,223.750 feet
E) 8,447.500 feet
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20
Find the length of the plane curve given below. <strong>Find the length of the plane curve given below.  </strong> A) 14 B) 8 C) 11 D) 13 E) 12

A) 14
B) 8
C) 11
D) 13
E) 12
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21
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
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22
Suppose the two particles travel along the space curves <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   and <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.

A) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)
B) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)
C) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)
D) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)
E) <strong>Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.</strong> A)   B)   C)   D)   E)
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23
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0 <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0

A) 1
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A) 1 B)   C)   D) 6 E) 0
D) 6
E) 0
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24
Find the point on the curve given below at which the curvature K is zero. <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the point on the curve given below at which the curvature K is zero.  </strong> A)   B)   C)   D)   E)
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25
Represent the following curve by a vector-valued function. <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)

A) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)
B) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)
C) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)
D) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)
E) <strong>Represent the following curve by a vector-valued function.  </strong> A)   B)   C)   D)   E)
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26
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
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27
A baseball is hit 5 feet above the ground at 80 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 80 feet per second and at an angle of   with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place. ​</strong> A) 194.2 feet B) 212.3 feet C) 210.1 feet D) 171.4 feet E) 236.5 feet with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place. ​

A) 194.2 feet
B) 212.3 feet
C) 210.1 feet
D) 171.4 feet
E) 236.5 feet
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28
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
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29
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 78 yards directly downfield at a height of 4 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 78 yards directly downfield at a height of 4 feet. The pass is released at an angle of   with the horizontal. Find the time in seconds the receiver has to reach the proper position after the quarterback releases the football. Round your answer to one decimal place.</strong> A) 2.2 seconds B) 12.8 seconds C) 22.2 seconds D) 3.7 seconds E) 7.4 seconds with the horizontal. Find the time in seconds the receiver has to reach the proper position after the quarterback releases the football. Round your answer to one decimal place.

A) 2.2 seconds
B) 12.8 seconds
C) 22.2 seconds
D) 3.7 seconds
E) 7.4 seconds
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30
Find the domain of the vector-valued function given below. <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   where <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)   <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)

A) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)
B) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)
C) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)
D) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)
E) <strong>Find the domain of the vector-valued function given below.   where    </strong> A)   B)   C)   D)   E)
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31
Find the vectors <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ and <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ for the following vector function. ​ <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​

A) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
B) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
C) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
D) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
E) <strong>Find the vectors   and   for the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
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32
Find the indefinite integral below. <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)   Do not include an arbitrary constant vector.

A) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
B) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
C) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
D) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
E) <strong>Find the indefinite integral below.   Do not include an arbitrary constant vector.</strong> A)   B)   C)   D)   E)
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33
Find the length of the space curve given below. <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the length of the space curve given below.  </strong> A)   B)   C)   D)   E)
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34
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
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35
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)   given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.  </strong> A)   B)   C)   D)   E)
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36
Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of <strong>Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of   above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.</strong> A) 2,669.517 feet B) 1,764.116 feet C) 5,337.349 feet D) 6,767.555 feet E) 3,525.690 feet above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

A) 2,669.517 feet
B) 1,764.116 feet
C) 5,337.349 feet
D) 6,767.555 feet
E) 3,525.690 feet
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37
Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

A) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)
B) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)
C) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)
D) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)
E) <strong>Suppose the outer edge of a playground slide is in the shape of a helix of radius 8 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.</strong> A)   B)   C)   D)   E)
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38
A 5800-pound vehicle is driven at a speed of 25 miles per hour on a circular interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires? Round your answer to one decimal place.

A) 2,716.9 lb
B) 3,396.4 lb
C) 3,046.0 lb
D) 2,707.6 lb
E) 3,565.1 lb
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39
Sketch the curve represented by the vector-valued function <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)   and give the orientation of the curve.

A) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
B) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
C) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
D) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
E) <strong>Sketch the curve represented by the vector-valued function   and give the orientation of the curve.</strong> A)   B)   C)   D)   E)
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40
Find the unit tangent vector to the curve given below at the specified point. <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
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41
Find the domain of the vector-valued function given below. <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the domain of the vector-valued function given below.  </strong> A)   B)   C)   D)   E)
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42
A particle moves in the yz-plane along the curve represented by the vector-valued function <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the minimum value of   .</strong> A) 3 B) 2 C) 5 D) 6 E) 8 . Find the minimum value of <strong>A particle moves in the yz-plane along the curve represented by the vector-valued function   . Find the minimum value of   .</strong> A) 3 B) 2 C) 5 D) 6 E) 8 .

A) 3
B) 2
C) 5
D) 6
E) 8
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43
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
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44
Find <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3   given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3

A) 1
B) -18 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3
C) 18 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3
D) 0
E) 3 <strong>Find   given the following vector function. ​   ​</strong> A) 1 B) -18   C) 18   D) 0 E) 3
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45
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.816 B) 0.389 C) 0.777 D) 1.864 E) 0.889 . Round your answer to three decimal places.

A) 0.816
B) 0.389
C) 0.777
D) 1.864
E) 0.889
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46
Find <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)   given the following vector function: ​ <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)

A) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)
B) 0
C) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)
D) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)
E) <strong>Find   given the following vector function: ​   ​</strong> A)   B) 0 C)   D)   E)
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47
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 1.069 B) 0.963 C) 0.468 D) 0.936 E) 0.848 . Round your answer to three decimal places.

A) 1.069
B) 0.963
C) 0.468
D) 0.936
E) 0.848
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48
Given the vector-valued function below, evaluate <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)   . <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)

A) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)
B) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)
C) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)
D) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)
E) <strong>Given the vector-valued function below, evaluate   .  </strong> A)   B)   C)   D)   E)
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49
The position vector <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   describes the path of an object moving in space. Find the velocity <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)   of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
B) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
C) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
D) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
E) <strong>The position vector   describes the path of an object moving in space. Find the velocity   of the object.</strong> A)   B)   C)   D)   E)
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50
Evaluate the limit given below. <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist.

A) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist.
B) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist.
C) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist.
D) <strong>Evaluate the limit given below.  </strong> A)   B)   C)   D)   E) The limit does not exist.
E) The limit does not exist.
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51
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 28.451 B) 28.636 C) 28.400 D) 28.111 E) 20.897 . Round your answer to three decimal places.

A) 28.451
B) 28.636
C) 28.400
D) 28.111
E) 20.897
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52
The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h (x and y are measured in miles) can safely go 30 miles per hour at <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h . How fast can it go at <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h ? Round your answer to two decimal places.

A) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h mi/h
B) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h mi/h
C) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h mi/h
D) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h mi/h
E) <strong>The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.</strong> A)   mi/h B)   mi/h C)   mi/h D)   mi/h E)   mi/h mi/h
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53
Find <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 at time <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 for the space curve <strong>Find   at time   for the space curve   .</strong> A) 9 B) 0 C) 8 D) 64 E) 3 .

A) 9
B) 0
C) 8
D) 64
E) 3
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54
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
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55
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 5 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 5 feet. The pass is released at an angle of   with the horizontal. Find the speed of the football when it is released. Round your answer to three decimal places.</strong> A) 32.979 feet per second B) 57.483 feet per second C) 18.863 feet per second D) 114.967 feet per second E) 189.144 feet per second with the horizontal. Find the speed of the football when it is released. Round your answer to three decimal places.

A) 32.979 feet per second
B) 57.483 feet per second
C) 18.863 feet per second
D) 114.967 feet per second
E) 189.144 feet per second
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56
Use the properties of the derivative to find <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   given the following vector-valued functions. <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)   <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)

A) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
B) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
C) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
D) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
E) <strong>Use the properties of the derivative to find   given the following vector-valued functions.    </strong> A)   B)   C)   D)   E)
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57
Find <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 at time <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 for the plane curve <strong>Find   at time   for the plane curve   . Round your answer to three decimal places.</strong> A) 4.575 B) 3.841 C) 2.716 D) 1.718 E) 4.685 . Round your answer to three decimal places.

A) 4.575
B) 3.841
C) 2.716
D) 1.718
E) 4.685
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58
A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)   with respect to the ground. Find the vector-valued function for the path of the baseball.

A) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)
B) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)
C) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)
D) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)
E) <strong>A baseball is hit 5 feet above the ground at 160 feet per second and at an angle of   with respect to the ground. Find the vector-valued function for the path of the baseball.</strong> A)   B)   C)   D)   E)
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59
Find <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 at time <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 for the space curve <strong>Find   at time   for the space curve   . Round your answer to three decimal places.</strong> A) 0.041 B) 0.093 C) 0.406 D) 0.000 E) 0.914 . Round your answer to three decimal places.

A) 0.041
B) 0.093
C) 0.406
D) 0.000
E) 0.914
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60
Determine the interval on which the vector-valued function <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)   is continuous.

A) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)
B) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)
C) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)
D) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)
E) <strong>Determine the interval on which the vector-valued function   is continuous.</strong> A)   B)   C)   D)   E)
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61
Use the given acceleration function and initial conditions to find the position at time t = 1. <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)

A) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)
B) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)
C) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)
D) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)
E) <strong>Use the given acceleration function and initial conditions to find the position at time t = 1.  </strong> A)   B)   C)   D)   E)
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62
A baseball, hit 7 feet above the ground, leaves the bat at an angle of <strong>A baseball, hit 7 feet above the ground, leaves the bat at an angle of   and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​</strong> A) 180 feet B) 18 feet C) 397 feet D) 278 feet E) 50 feet and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​

A) 180 feet
B) 18 feet
C) 397 feet
D) 278 feet
E) 50 feet
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63
Use the given acceleration function and initial conditions to find the position at time t = 4. <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)

A) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)
B) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)
C) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)
D) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)
E) <strong>Use the given acceleration function and initial conditions to find the position at time t = 4.  </strong> A)   B)   C)   D)   E)
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64
Find the curvature K of the curve <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 0.728 B) 0.970 C) 0.158 D) 0.057 E) 0.126 at the point <strong>Find the curvature K of the curve   at the point   . Round your answer to three decimal places.</strong> A) 0.728 B) 0.970 C) 0.158 D) 0.057 E) 0.126 . Round your answer to three decimal places.

A) 0.728
B) 0.970
C) 0.158
D) 0.057
E) 0.126
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65
The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 62 yards directly downfield at a height of 3 feet. The pass is released at an angle of <strong>The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 62 yards directly downfield at a height of 3 feet. The pass is released at an angle of   with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. ​</strong> A) 337.1 feet B) 16.3 feet C) 245.4 feet D) 85.8 feet E) 17.5 feet with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. ​

A) 337.1 feet
B) 16.3 feet
C) 245.4 feet
D) 85.8 feet
E) 17.5 feet
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66
Find <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​

A) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
B) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
C) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
D) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
E) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
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67
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
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68
Find <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​ given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​

A) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
B) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
C) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
D) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
E) <strong>Find   given the following vector function. ​   ​</strong> A)   ​ B)   ​ C)   ​ D)   ​ E)   ​
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69
Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a <strong>Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a   turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is   where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.</strong> A) 201 mi/h B) 64 mi/h C) 173 mi/h D) 137 mi/h E) 255 mi/h turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is <strong>Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 3.2 miles to make a   turn and climb to an altitude of 3.4 miles. The model for the path of the plane during this maneuver is   where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.</strong> A) 201 mi/h B) 64 mi/h C) 173 mi/h D) 137 mi/h E) 255 mi/h where t is the time in hours and r is the distance in miles. Determine the speed of the plane. Round your answer to the nearest integer.

A) 201 mi/h
B) 64 mi/h
C) 173 mi/h
D) 137 mi/h
E) 255 mi/h
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70
Find the unit tangent vector to the curve given below at the specified point. <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the unit tangent vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
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71
Find the arc length for <strong>Find the arc length for   over the interval   . Round your answer to two decimal places. ​</strong> A) 110.36 B) 140.93 C) 129.56 D) 57.80 E) 102.75 over the interval <strong>Find the arc length for   over the interval   . Round your answer to two decimal places. ​</strong> A) 110.36 B) 140.93 C) 129.56 D) 57.80 E) 102.75 . Round your answer to two decimal places. ​

A) 110.36
B) 140.93
C) 129.56
D) 57.80
E) 102.75
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72
The position vector <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   describes the path of an object moving in space. Find the acceleration <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)   of the object.

A) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)
B) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)
C) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)
D) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)
E) <strong>The position vector   describes the path of an object moving in space. Find the acceleration   of the object.</strong> A)   B)   C)   D)   E)
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73
Find the principle unit normal vector to the curve given below at the specified point. <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)

A) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
B) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
C) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
D) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
E) <strong>Find the principle unit normal vector to the curve given below at the specified point.  </strong> A)   B)   C)   D)   E)
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74
Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces
Parameter <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)   <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)

A) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
B) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
C) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
D) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
E) <strong>Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter    </strong> A)   B)   C)   D)   E)
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75
A baseball is hit 3 feet above the ground at 95 feet per second and at an angle of <strong>A baseball is hit 3 feet above the ground at 95 feet per second and at an angle of   with respect to the ground. Find the range. Round your answer to one decimal place.</strong> A) 285 feet B) 352.2 feet C) 370.7 feet D) 501.9 feet E) 300 feet with respect to the ground. Find the range. Round your answer to one decimal place.

A) 285 feet
B) 352.2 feet
C) 370.7 feet
D) 501.9 feet
E) 300 feet
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76
Find the radius of curvature of the plane curve <strong>Find the radius of curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 1021.045 B) 2340.571 C) 1374.626 D) 1565.001 E) 1571.001 at <strong>Find the radius of curvature of the plane curve   at   . Round your answer to three decimal places.</strong> A) 1021.045 B) 2340.571 C) 1374.626 D) 1565.001 E) 1571.001 . Round your answer to three decimal places.

A) 1021.045
B) 2340.571
C) 1374.626
D) 1565.001
E) 1571.001
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77
The position vector <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 describes the path of an object moving in space. Find the speed <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 of the object. ​

A) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18
B) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 122
C) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18
D) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18
E) <strong>The position vector   describes the path of an object moving in space. Find the speed   of the object. ​</strong> A)     B)   122 C)     D)     E)   18 18
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78
A baseball player at second base throws a ball 95 feet to the player at first base. The ball is thrown 8 feet above the ground with an initial velocity of 46 miles per hour and at an angle of <strong>A baseball player at second base throws a ball 95 feet to the player at first base. The ball is thrown 8 feet above the ground with an initial velocity of 46 miles per hour and at an angle of   above the horizontal. At which height does the player at first base catch the ball? Round your answer to three decimal places.</strong> A) 2.908 feet B) 2.008 feet C) 0.808 feet D) 1.408 feet E) 0.908 feet above the horizontal. At which height does the player at first base catch the ball? Round your answer to three decimal places.

A) 2.908 feet
B) 2.008 feet
C) 0.808 feet
D) 1.408 feet
E) 0.908 feet
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79
Evaluate the definite integral below. <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)

A) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)
B) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)
C) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)
D) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)
E) <strong>Evaluate the definite integral below.  </strong> A)   B)   C)   D)   E)
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80
Find <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)   given the following vector function. ​ <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)

A) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)
B) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)
C) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)
D) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)
E) <strong>Find   given the following vector function. ​   ​</strong> A)   B)   C)   D)   E)
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