Question 1

Assume that x and y are both differentiable functions of t . Find  for the equation 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Assume that x and y are both...

Question 2

Find d²y/dx² in terms of x and y. A) B) C) D) E)

Accepted Answer

The answer of Find d²y/dx² in terms of x and...

Question 3

A spherical balloon is inflated with gas at the rate of 200 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 60 centimeters?
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A spherical balloon is inflated with gas...

Question 4

Find the points at which the graph of the equation has a vertical or horizontal tangent line. 
A)There is a horizontal tangent at  but no vertical tangents.
B)There is a horizontal tangent at  and a vertical tangent at  .
C)There is a vertical tangent at  but no horizontal tangents.
D)There is a horizontal tangent at  and a vertical tangent at  .
E)There are no horizontal or vertical tangent lines.

Accepted Answer

To find where the graph has a horizontal or vertical tangent line, we need to find where its derivative is undefined or equal to zero.

Differentiating the given equation with respect to x:
y' = 2x-8
Setting y' = 0 to find where the slope is zero:
2x-8 = 0
2x = 8
x = 4
Thus, we have a horizontal tangent at x=4.

Differentiating y' with respect to x to find where the slope of the curve starts decreasing towards negative infinity:
y'' = 2
This is always defined, so there are no points where the slope is undefined. Therefore, there are no vertical tangents.

Hence, we have a horizontal tangent at x=4 and no vertical tangents. The answer is D.

Question 5

The radius, r, of a circle is increasing at a rate of 2 centimeters per minute.Find the rate of change of area, A, when the radius is  .
A)  sq cm/min
B)  sq cm/min
C)  sq cm/min
D)  sq cm/min
E)  sq cm/min

Accepted Answer

The answer of The radius, r, of a circle is...

Question 6

A conical tank (with vertex down) is 20 feet across the top and 26 feet deep. If water is flowing into the tank at a rate of 16 cubic feet per minute, find the rate of change of the depth of the water when the water is 12 feet deep.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A conical tank (with vertex down) is...

Question 7

Find  in terms of x and y given that 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find  in terms of x and...

Question 8

Assume that x and y are both differentiable functions of t. Find  for the equation 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Assume that x and y are both...

Question 9

A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 4 feet per second. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changed when the base of the ladder is 8 feet from the wall. Round your answer to two decimal places. A) ft²/sec B) ft²/sec C) ft²/sec D) ft²/sec E) ft²/sec

Accepted Answer

The answer of A ladder 20 feet long is leaning...

Question 10

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure). When he is 8 feet from the base of the light, at what rate is the tip of his shadow moving? 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A man 6 feet tall walks at...

Question 11

A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 4 feet per second. How fast is the top of the ladder moving down the wall when its base is 14 feet from the wall? Round your answer to two decimal places. 
A)6.86 ft/sec
B)-6.00 ft/sec
C)-3.92 ft/sec
D)8.00 ft/sec
E)-6.86 ft/sec

Accepted Answer

Let x be the distance from the bottom of the ladder to the wall, and let y be the height of the ladder on the wall. Then we have the equation x^2 + y^2 = 20^2. Differentiating implicitly with respect to time, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Solving for dy/dt when x = 14 and dx/dt = 4:
2(14)(4) + 2y(dy/dt) = 0
dy/dt = -56/2y
When x = 14, y = sqrt(20^2 - 14^2) = 18.44
Substituting, we get:
dy/dt = -56/(2*18.44) = -1.52 ft/sec
Rounding to two decimal places, we get -3.92 ft/sec.

Question 12

A point is moving along the graph of the function  such that dx/dt = 5 centimeters per second.Find dy/dt when  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A point is moving along the graph...

Question 13

Differentiate  with respect to t (x and y are functions of t).
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Differentiate  with respect to t (x...

Question 14

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume when r = 9 inches.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of The radius r of a sphere is...

Question 15

All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is 5 centimeters? A)150 cm³/sec B)450 cm³/sec C)300 cm³/sec D)540 cm³/sec E)180 cm³/sec

Accepted Answer

The volume of a cube is given by $V = s^3$, where $s$ is the length of an edge. The rate of change of volume with respect to time ($\frac{dV}{dt}$) can be found by differentiating both sides with respect to $t$, giving $\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}$. Given $\frac{ds}{dt} = 6$ cm/sec and $s = 5$ cm, substituting these values gives $\frac{dV}{dt} = 3 \cdot 5^2 \cdot 6 = 450$ cm³/sec.

Question 16

Find the rate of change of the distance D between the origin and a moving point on the graph of  if  centimeters per second.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the rate of change of the...

Question 17

A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). When he is 10 feet from the base of the light, at what rate is the length of his shadow changing? 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A man 6 feet tall walks at...

Quiz 5: Differentiation

Assume that x and y are both differentiable functions of t . Find for the equation

Find d²y/dx² in terms of x and y.

A spherical balloon is inflated with gas at the rate of 200 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 60 centimeters?

Find the points at which the graph of the equation has a vertical or horizontal tangent line.

The radius, r, of a circle is increasing at a rate of 2 centimeters per minute.Find the rate of change of area, A, when the radius is .

A conical tank (with vertex down) is 20 feet across the top and 26 feet deep. If water is flowing into the tank at a rate of 16 cubic feet per minute, find the rate of change of the depth of the water when the water is 12 feet deep.

Find in terms of x and y given that

Assume that x and y are both differentiable functions of t. Find for the equation

A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure) . When he is 8 feet from the base of the light, at what rate is the tip of his shadow moving?

A ladder 20 feet long is leaning against the wall of a house (see figure) . The base of the ladder is pulled away from the wall at a rate of 4 feet per second. How fast is the top of the ladder moving down the wall when its base is 14 feet from the wall? Round your answer to two decimal places.

A point is moving along the graph of the function such that dx/dt = 5 centimeters per second.Find dy/dt when .

Differentiate with respect to t (x and y are functions of t) .

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume when r = 9 inches.

All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is 5 centimeters?

Find the rate of change of the distance D between the origin and a moving point on the graph of if centimeters per second.

A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure) . When he is 10 feet from the base of the light, at what rate is the length of his shadow changing?