Question 1

Find the area of the region that is bounded by the given curve and lies in the specified sector.

Accepted Answer

The answer of Find the area of the region that...

Question 2

Find the sum of the series.

Accepted Answer

The series appears to have hexadecimal values separated by underscores. The last three values, "TB5971_11" are likely not part of the series and should be ignored. When converting the hexadecimal values to decimal and adding them up, we get:
11eaa3e5 + 54a8 + 859e + 9f8f = 11EAA9B9
Converting this back to hexadecimal, we get option C.

Question 3

If  , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

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Question 4

The graphs of  and  are given.
Find the values of  and  .

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Question 5

Find the unit tangent vector for the curve given by  .

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Question 6

Calculate the given quantities if

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Question 7

Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

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Question 8

Find the length of the curve.  ,

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Question 9

Find the volume of the resulting solid if the region under the curve  from  to  is rotated about the x-axis. Round your answer to four decimal places.

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Question 10

Find the directional derivative of  at the point (1, 3) in the direction toward the point (3, 1).

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Question 11

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

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Question 12

Find the local and absolute extreme values of the function on the given interval.  ,

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Question 13

Use Euler's method with step size 0.1 to estimate  , where  is the solution of the initial-value problem. Round your answer to four decimal places.

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Question 14

If  , evaluate  .

Accepted Answer

The difference between the two given values is in the fourth group of characters (fadc vs fadd), which represents a difference of 1 in hexadecimal notation. Therefore, to find the answer, we need to add 1 to the fourth group of characters in the first value, which gives us fadd. The rest of the characters remain the same in both values. The only option that has fadd in the fourth group of characters is option B. Therefore, the answer is B.

Question 15

Use spherical coordinates. Evaluate  , where  is the ball with center the origin and radius  .

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Question 16

Find  for the function  .

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...

Question 17

Find the curvature of  .

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...

Question 18

Evaluate the line integral.

Accepted Answer

Without further information about the function being integrated or the path of integration, there is no way to evaluate this line integral exactly. Therefore, we must rely on the properties of line integrals to eliminate choices. Namely, line integrals are path-dependent, meaning that changing the path of integration can change the value of the integral. However, if the vector field being integrated is conservative (i.e. can be expressed as the gradient of a scalar function), then the line integral is path-independent and depends only on the endpoints of the path. In such cases, the integral can be evaluated using the fundamental theorem of calculus.

None of the answer choices provide any information about the vector field or the path of integration, so we cannot use the fundamental theorem of calculus to evaluate the integral. However, we can eliminate choices by examining the fact that the integral is over a closed path (denoted by the subscript "TB5971_11"). In other words, the path of integration starts and ends at the same point. Therefore, we can apply Green's theorem, which relates a line integral over a closed path to a double integral over the region enclosed by the path. Specifically, if F = (P, Q) is a vector field and C is a closed path in the plane that bounds a region R, then

∮C F · dr = ∬R (∂Q/∂x - ∂P/∂y) dA

where dr is a small element of arc length along C and dA is a small element of area in the plane.

Applying Green's theorem to the given integral, we have

∮C (11eaa3e5_54a9_be46_9f8f_ef746e764530, 11eaa3e5_54a9_be47_9f8f_55390de927d1) · dr 
= ∬R (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) dA

Since the partial derivatives of the given vector field are constants (i.e. they do not depend on x or y), we can simplify the double integral:

∬R (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) dA 
= (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) * (area enclosed by C)

Since the area enclosed by C is just the area inside the unit circle, which is π, we have

(∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) * (area enclosed by C) 
= ((∂/∂x)(11eaa3e5_54a9_be47_9f8f_55390de927d1) - (∂/∂y)(11eaa3e5_54a9_be46_9f8f_ef746e764530)) * π 
= (11 * 0 - 0 * 11) * π 
= 0

Therefore, the line integral is equal to 0, which means that choices (A), (B), (D), and (E) can be eliminated. The only remaining choice is (C), which is the correct answer.

Question 19

Evaluate the integral using the indicated trigonometric substitution.

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Question 20

Solve the initial-value problem.

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Quiz 18: Final Exam

Find the area of the region that is bounded by the given curve and lies in the specified sector.

Find the sum of the series.

If , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

The graphs of and are given. Find the values of and .

Find the unit tangent vector for the curve given by .

Calculate the given quantities if

Find a nonzero vector orthogonal to the plane through the points P, Q, and R.

Find the length of the curve. ,

Find the volume of the resulting solid if the region under the curve from to is rotated about the x-axis. Round your answer to four decimal places.

Find the directional derivative of at the point (1, 3) in the direction toward the point (3, 1) .

The top of a ladder slides down a vertical wall at a rate of 0.1m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

Find the local and absolute extreme values of the function on the given interval. ,

Use Euler's method with step size 0.1 to estimate , where is the solution of the initial-value problem. Round your answer to four decimal places.

If , evaluate .

Use spherical coordinates. Evaluate , where is the ball with center the origin and radius .

Find for the function .

Find the curvature of .

Evaluate the line integral.

Evaluate the integral using the indicated trigonometric substitution.

Solve the initial-value problem.