Question 1

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

Accepted Answer

A)  $11 \pi$ 
B)  $36 \pi$ 
C)  $8 \pi$ 
D)  $6 \pi$ 
E)  $12 \pi$ 
A)  $11 \pi$ 
B)  $36 \pi$ 
C)  $8 \pi$ 
D)  $6 \pi$ 
E)  $12 \pi$

Question 2

Find the derivative of the function. $f(x)$ = sinh 4x

Accepted Answer

A)  -4 cosh 4x 
B)  4 sinh 4x 
C)  -sinh 4x 
D)  4 cosh 4x 
A)  -4 cosh 4x 
B)  4 sinh 4x 
C)  -sinh 4x 
D)  4 cosh 4x

Question 3

Find the derivative of the function. $y=8 \cos ^{-1}\left(\sin ^{-1} t\right)$

Accepted Answer

A)  $y^{\prime}=-\frac{8}{\sqrt{1-\left(\sin ^{-1}(t)\right)^{2}}}$ 
B)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)\left(1-\sin ^{-1}(t)\right)}}$ 
C)  $y^{\prime}=-\frac{8}{\sqrt{\left(1+t^{2}\right)\left(1+\left(\sin ^{-1}(t)\right)^{2}\right)}}$ 
D)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\sin ^{-1}(t)\right)^{2}\right)}}$ 
E)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)}}$ 
A)  $y^{\prime}=-\frac{8}{\sqrt{1-\left(\sin ^{-1}(t)\right)^{2}}}$ 
B)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)\left(1-\sin ^{-1}(t)\right)}}$ 
C)  $y^{\prime}=-\frac{8}{\sqrt{\left(1+t^{2}\right)\left(1+\left(\sin ^{-1}(t)\right)^{2}\right)}}$ 
D)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\sin ^{-1}(t)\right)^{2}\right)}}$ 
E)  $y^{\prime}=-\frac{8}{\sqrt{\left(1-t^{2}\right)}}$

Question 4

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[3]{\frac{x^{8}+1}{x-1}}$

Accepted Answer

A)  $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$ 
B)  $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$ 
C)  $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)(x-1)}$ 
D)  $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)(x-1)}$ 
A)  $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$ 
B)  $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$ 
C)  $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)(x-1)}$ 
D)  $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)(x-1)}$

Question 5

Find the rate of change of y with respect to x at the given values of x and y. $2 x y^{2}-5 x^{2} y-375=0$ ; x = 3, y = -5

Accepted Answer

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

Question 6

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=t^{10} e^{-t}$ ; t = 1

Accepted Answer

The answer of s(t) is the position of a body...

Question 7

If $y=3 x^{2}+6 x$ and $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ when $x=2$ .

Accepted Answer

A)  60 
B)  $210$ 
C)  300 
D)  260 
E)  None of these 
A)  60 
B)  $210$ 
C)  300 
D)  260 
E)  None of these

Question 8

Two cars start moving from the same point. One travels south at $50$ mi/h and the other travels west at $40$ mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Accepted Answer

A)  $55.42 ~\mathrm{mi} / \mathrm{h}$ 
B)  $76.43 ~\mathrm{mi} / \mathrm{h}$ 
C)  $81.38 ~\mathrm{mi} / \mathrm{h}$ 
D)  $65.49 ~\mathrm{mi} / \mathrm{h}$ 
E)  $64.03~ \mathrm{mi} / \mathrm{h}$ 
A)  $55.42 ~\mathrm{mi} / \mathrm{h}$ 
B)  $76.43 ~\mathrm{mi} / \mathrm{h}$ 
C)  $81.38 ~\mathrm{mi} / \mathrm{h}$ 
D)  $65.49 ~\mathrm{mi} / \mathrm{h}$ 
E)  $64.03~ \mathrm{mi} / \mathrm{h}$

Question 9

Find the derivative of the function. $h(t)=\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}$

Accepted Answer

A)  $3 t^{2}-3+8 e^{t}$ 
B)  $3 t^{2}-3+e^{t}$ 
C)  $\frac{3}{8}$ $t^{2}-$ $\frac{3}{8}$ $+e^{t}$ 
D)  $9 t^{8}-21 t^{6}+e^{t}$ 
A)  $3 t^{2}-3+8 e^{t}$ 
B)  $3 t^{2}-3+e^{t}$ 
C)  $\frac{3}{8}$ $t^{2}-$ $\frac{3}{8}$ $+e^{t}$ 
D)  $9 t^{8}-21 t^{6}+e^{t}$

Question 10

If $f(x)=6 \cos x+\sin ^{2} x$ , find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Accepted Answer

A)  $f^{\prime \prime}(x)=-6 \cos (2 x)+2 \cos (x)$ 
B)  $f^{\prime}(x)=-6 \sin (x)+\sin (2 x)$ 
C)  $f^{\prime}(x)=-6 \sin (2 x)+\sin (x)$ 
D)  $f^{\prime \prime}(x)=-6 \cos (x)+2 \cos (2 x)$ 
E)  $f^{\prime \prime}(x)=-2 \cos (2 x)+6 \cos (x)$ 
A)  $f^{\prime \prime}(x)=-6 \cos (2 x)+2 \cos (x)$ 
B)  $f^{\prime}(x)=-6 \sin (x)+\sin (2 x)$ 
C)  $f^{\prime}(x)=-6 \sin (2 x)+\sin (x)$ 
D)  $f^{\prime \prime}(x)=-6 \cos (x)+2 \cos (2 x)$ 
E)  $f^{\prime \prime}(x)=-2 \cos (2 x)+6 \cos (x)$

Question 11

Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$

Accepted Answer

The answer of Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$...

Question 12

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy6; $\left(\frac{\pi}{2}, 1\right)$

Accepted Answer

A)  x = $\frac{\pi}{2}$ 
B)  y = 6x + 1 
C)  y = x 
D)  y = 1 
A)  x = $\frac{\pi}{2}$ 
B)  y = 6x + 1 
C)  y = x 
D)  y = 1

Question 13

Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$

Accepted Answer

The answer of Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$...

Question 14

Differentiate the function. $g(t)=t^{5} \ln 9 t$

Accepted Answer

A)  $1+\frac{\ln 9 t}{9 t}$ 
B)  $\frac{5}{9} t^{3}$ 
C)  $t^{4}(1+5 \ln 9 t)$ 
D)  $t^{4}\left(\frac{1}{9}+5 \ln 9 t\right)$ 
A)  $1+\frac{\ln 9 t}{9 t}$ 
B)  $\frac{5}{9} t^{3}$ 
C)  $t^{4}(1+5 \ln 9 t)$ 
D)  $t^{4}\left(\frac{1}{9}+5 \ln 9 t\right)$

Question 15

If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as $V=5000\left(1-\frac{t}{40}\right)^{2}$ .
Find the rate at which water is draining from the tank after 9 minutes.

Accepted Answer

The answer of If a tank holds 5000 gallons of...

Question 16

Suppose that $F(x)=f(g(x))$ and $g(14)=2$ , $g^{\prime}(14)=5$ , $f^{\prime}(14)=15$ , and $f^{\prime}(2)=16$ . Find $F^{\prime}(14)$ .

Accepted Answer

A)  20 
B)  $80$ 
C)  24 
D)  140 
E)  17 
A)  20 
B)  $80$ 
C)  24 
D)  140 
E)  17

Question 17

Find all points at which the tangent line is horizontal on the graph of the function. $y(x)=6 \sin x+\sin ^{2} x$

Accepted Answer

The answer of Find all points at which the tangent...

Question 18

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028.
Find the velocity of the blood at radius r = $0.001$

Accepted Answer

The answer of Refer to the law of laminar flow....

Question 19

Calculate $y^{\prime}$ . $x e^{y}=y-8$

Accepted Answer

The answer of Calculate $y^{\prime}$ . $x e^{y}=y-8$...

Question 20

If $f(t)=\sqrt{9 t+1}$ , find $f^{\prime \prime}(4)$ .

Accepted Answer

A)  $-0.010$ 
B)  $0.015$ 
C)  $-0.033$ 
D)  $-0.22$ 
E)  $0.044$ 
A)  $-0.010$ 
B)  $0.015$ 
C)  $-0.033$ 
D)  $-0.22$ 
E)  $0.044$

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

Find the derivative of the function. $f(x)$ = sinh 4x

Find the derivative of the function. $y=8 \cos ^{-1}\left(\sin ^{-1} t\right)$

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[3]{\frac{x^{8}+1}{x-1}}$

Find the rate of change of y with respect to x at the given values of x and y. $2 x y^{2}-5 x^{2} y-375=0$ ; x = 3, y = -5

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=t^{10} e^{-t}$ ; t = 1

If $y=3 x^{2}+6 x$ and $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ when $x=2$ .

Two cars start moving from the same point. One travels south at $50$ mi/h and the other travels west at $40$ mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Find the derivative of the function. $h(t)=\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}$

If $f(x)=6 \cos x+\sin ^{2} x$ , find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁶; $\left(\frac{\pi}{2}, 1\right)$

Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$

Differentiate the function. $g(t)=t^{5} \ln 9 t$

If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as $V=5000\left(1-\frac{t}{40}\right)^{2}$ . Find the rate at which water is draining from the tank after 9 minutes.

Suppose that $F(x)=f(g(x))$ and $g(14)=2$ , $g^{\prime}(14)=5$ , $f^{\prime}(14)=15$ , and $f^{\prime}(2)=16$ . Find $F^{\prime}(14)$ .

Find all points at which the tangent line is horizontal on the graph of the function. $y(x)=6 \sin x+\sin ^{2} x$

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028. Find the velocity of the blood at radius r = $0.001$

Calculate $y^{\prime}$ . $x e^{y}=y-8$

If $f(t)=\sqrt{9 t+1}$ , find $f^{\prime \prime}(4)$ .

Functions and Limits

Derivatives

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 3: Applications of Differentiation

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

Find the derivative of the function. f(x)f(x)f(x) = sinh 4x

Find the derivative of the function. y=8cos⁡−1(sin⁡−1t)y=8 \cos ^{-1}\left(\sin ^{-1} t\right)y=8cos−1(sin−1t)

Use logarithmic differentiation to find the derivative of the function. y=x8+1x−13y=\sqrt[3]{\frac{x^{8}+1}{x-1}}y=3x−1x8+1​​

Find the rate of change of y with respect to x at the given values of x and y. 2xy2−5x2y−375=02 x y^{2}-5 x^{2} y-375=02xy2−5x2y−375=0 ; x = 3, y = -5

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where t≥0t \geq 0t≥0 . Find the position, velocity, and speed of the body at the indicated time. s(t)=t10e−ts(t)=t^{10} e^{-t}s(t)=t10e−t ; t = 1

If y=3x2+6xy=3 x^{2}+6 xy=3x2+6x and dxdt=5\frac{d x}{d t}=5dtdx​=5 , find dydt\frac{d y}{d t}dtdy​ when x=2x=2x=2 .

Two cars start moving from the same point. One travels south at 505050 mi/h and the other travels west at 404040 mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Find the derivative of the function. h(t)=t9−3t7+8t6et8t6h(t)=\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}h(t)=8t6t9−3t7+8t6et​

If f(x)=6cos⁡x+sin⁡2xf(x)=6 \cos x+\sin ^{2} xf(x)=6cosx+sin2x , find f′(x)f^{\prime}(x)f′(x) and f′′(x)f^{\prime \prime}(x)f′′(x) .

Find f′(a)f^{\prime}(a)f′(a) . f(x)=2+x−5x5f(x)=2+x-5 x^{5}f(x)=2+x−5x5

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy6; (π2,1)\left(\frac{\pi}{2}, 1\right)(2π​,1)

Calculate y′y^{\prime}y′ . y=3xcos⁡xy=3 \sqrt{x} \cos \sqrt{x}y=3x​cosx​

Differentiate the function. g(t)=t5ln⁡9tg(t)=t^{5} \ln 9 tg(t)=t5ln9t

Suppose that F(x)=f(g(x))F(x)=f(g(x))F(x)=f(g(x)) and g(14)=2g(14)=2g(14)=2 , g′(14)=5g^{\prime}(14)=5g′(14)=5 , f′(14)=15f^{\prime}(14)=15f′(14)=15 , and f′(2)=16f^{\prime}(2)=16f′(2)=16 . Find F′(14)F^{\prime}(14)F′(14) .

Find all points at which the tangent line is horizontal on the graph of the function. y(x)=6sin⁡x+sin⁡2xy(x)=6 \sin x+\sin ^{2} xy(x)=6sinx+sin2x

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3,500 dynes /cm23,500 \text { dynes } / \mathrm{cm}^{2}3,500 dynes /cm2 and viscosity η\etaη =.028. Find the velocity of the blood at radius r = 0.0010.0010.001

Calculate y′y^{\prime}y′ . xey=y−8x e^{y}=y-8xey=y−8

If f(t)=9t+1f(t)=\sqrt{9 t+1}f(t)=9t+1​ , find f′′(4)f^{\prime \prime}(4)f′′(4) .

Functions and Limits

Derivatives

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the derivative of the function. $f(x)$ = sinh 4x

Find the derivative of the function. $y=8 \cos ^{-1}\left(\sin ^{-1} t\right)$

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[3]{\frac{x^{8}+1}{x-1}}$

Find the rate of change of y with respect to x at the given values of x and y. $2 x y^{2}-5 x^{2} y-375=0$ ; x = 3, y = -5

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s(t)=t^{10} e^{-t}$ ; t = 1

If $y=3 x^{2}+6 x$ and $\frac{d x}{d t}=5$ , find $\frac{d y}{d t}$ when $x=2$ .

Two cars start moving from the same point. One travels south at $50$ mi/h and the other travels west at $40$ mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

Find the derivative of the function. $h(t)=\frac{t^{9}-3 t^{7}+8 t^{6} e^{t}}{8 t^{6}}$

If $f(x)=6 \cos x+\sin ^{2} x$ , find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ .

Find $f^{\prime}(a)$ . $f(x)=2+x-5 x^{5}$

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁶; $\left(\frac{\pi}{2}, 1\right)$

Calculate $y^{\prime}$ . $y=3 \sqrt{x} \cos \sqrt{x}$

Differentiate the function. $g(t)=t^{5} \ln 9 t$

Suppose that $F(x)=f(g(x))$ and $g(14)=2$ , $g^{\prime}(14)=5$ , $f^{\prime}(14)=15$ , and $f^{\prime}(2)=16$ . Find $F^{\prime}(14)$ .

Find all points at which the tangent line is horizontal on the graph of the function. $y(x)=6 \sin x+\sin ^{2} x$

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028. Find the velocity of the blood at radius r = $0.001$

Calculate $y^{\prime}$ . $x e^{y}=y-8$

If $f(t)=\sqrt{9 t+1}$ , find $f^{\prime \prime}(4)$ .