Question 1

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 3 t + 2$ . Find the value of t at which the particle reverses its direction.

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B) 0 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E) 1
F)2
G) $\frac { 3 } { 4 }$
H)
$\frac { 4 } { 3 }$ 
A)  $\frac { 1 } { 2 }$ 
B) 0 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 2 } { 3 }$ 
E) 1
F)2
G) $\frac { 3 } { 4 }$
H)
$\frac { 4 } { 3 }$

Question 2

f and g are functions whose graphs are shown below. Let $F ( x ) = f ( g ( x ) ) , G ( x ) = g ( f ( x ) )$ and $H ( x ) = f ( f ( x ) )$ Find each derivative, if it exists. If it does not exist, explain.   (a) $F ^ { \prime } ( 2 )$ (b) $F ^ { \prime } ( 6 )$ (c) $F ^ { \prime } ( 8 )$ (d) $G ^ { \prime } ( 2 )$ (e) $G ^ { \prime } ( 4 )$ (f) $G ^ { \prime } ( 10 )$ (g) $H ^ { \prime } ( 4 )$ (h) $H ^ { \prime } ( 6 )$ (i) $H ^ { \prime } ( 10 )$

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) @#IMG-DLM& (d) @#IMG-DLM& undefined

Question 3

Find the exact value of $\tan \left( \cos ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)$ .

Accepted Answer

A) 1 
B)  $\sqrt { 3 }$ 
C)  $\sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E) 0
F) $\frac { \pi } { 4 }$ 
G) $\frac { \pi } { 3 }$
H) $\frac { \pi } { 6 }$ 
A) 1 
B)  $\sqrt { 3 }$ 
C)  $\sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E) 0
F) $\frac { \pi } { 4 }$ 
G) $\frac { \pi } { 3 }$
H) $\frac { \pi } { 6 }$

Question 4

Find the y-intercept of the tangent line to the curve $y = \sqrt { x + 3 }$ at the point (1, 2).

Accepted Answer

A)  $\frac { 3 } { 4 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$ 
F)1
G) $\frac { 3 } { 2 }$ 
H) $\frac { 5 } { 4 }$ 
A)  $\frac { 3 } { 4 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D) 2 
E)  $\frac { 1 } { 2 }$ 
F)1
G) $\frac { 3 } { 2 }$ 
H) $\frac { 5 } { 4 }$

Question 5

Find $\frac { d y } { d x } .$ (a) $y = \sqrt { 2 x - x ^ { 2 } }$ (b) $y = \left( x ^ { 2 } + 1 \right) ^ { 50 }$ (c) $y = \sin \left( \frac { 1 } { x } \right)$ (d) $y = \tan ( 3 x )$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 6

Let $y = f ( x )$ If $x y ^ { 3 } + x y = 6 \text { and } f ( 3 ) = 1 , \text { find } f ^ { \prime } ( 3 )$

Accepted Answer

A) 0 
B) 1 
C) 2 
D)  $\frac { 1 } { 3 }$ 
E)  $- 4$
F) $\frac { 1 } { 5 }$
G) $- \frac { 1 } { 6 }$
H)8 
A) 0 
B) 1 
C) 2 
D)  $\frac { 1 } { 3 }$ 
E)  $- 4$
F) $\frac { 1 } { 5 }$
G) $- \frac { 1 } { 6 }$
H)8

Question 7

Find an equation of the tangent line to the curve $y = \left( x ^ { 2 } + 1 \right) ^ { 3 }$ at the point (1, 8).

Accepted Answer

The slope is 24, so

Question 8

Find the domain of the function $f ( x ) = \sin ^ { - 1 } ( 3 - 2 x )$ .

Accepted Answer

A)  $[ 0,1 ]$ 
B)  $[ 1,2 ]$ 
C)  $[ 0,2 ]$ 
D)  $[ 0,3 ]$ 
E)  $[ 1,3 ]$ 
F) $[ 2,3 ]$ 
G) $[ - 3 , - 1 ]$ 
H) $[ - 3 , - 2 ]$ 
A)  $[ 0,1 ]$ 
B)  $[ 1,2 ]$ 
C)  $[ 0,2 ]$ 
D)  $[ 0,3 ]$ 
E)  $[ 1,3 ]$ 
F) $[ 2,3 ]$ 
G) $[ - 3 , - 1 ]$ 
H) $[ - 3 , - 2 ]$

Question 9

If $f ( x ) = - \frac { 1 } { x ^ { 2 } }$ , find $f ^ { \prime } ( 1 )$ .

Accepted Answer

A)  $- \frac { 1 } { 3 }$ 
B)  $- 2$ 
C)  $\frac { 1 } { 3 }$ 
D)  $- \frac { 1 } { 4 }$ 
E)  $\frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 2 }$

G)2

H) $\frac { 1 } { 4 }$ 
A)  $- \frac { 1 } { 3 }$ 
B)  $- 2$ 
C)  $\frac { 1 } { 3 }$ 
D)  $- \frac { 1 } { 4 }$ 
E)  $\frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 2 }$

G)2

H) $\frac { 1 } { 4 }$

Question 10

A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?

Accepted Answer

A) $\pi$ 
B) 2$\pi$ 
C) $\frac { \pi } { 3 }$ 
D) 6$\pi$ 
E) 3$\pi$
F) $\frac { \pi } { 6 }$
G)$\frac { \pi } { 12 }$ 
H) $\frac { \pi } { 2 }$ 
A) $\pi$ 
B) 2$\pi$ 
C) $\frac { \pi } { 3 }$ 
D) 6$\pi$ 
E) 3$\pi$
F) $\frac { \pi } { 6 }$
G)$\frac { \pi } { 12 }$ 
H) $\frac { \pi } { 2 }$

Question 11

The position function for a particle is $s ( t ) = 16 t ^ { 2 } + 48 t + 100$ , where s is measured in feet and t is measured in seconds.(a) Find the velocity at t = 2.(b) When does the velocity equal zero?

Accepted Answer

(a) -16 ft

Question 12

Suppose that $w = u \circ v$ and $u ( 0 ) = 1 , v ( 0 ) = 2 , u ^ { \prime } ( 0 ) = 3 , u ^ { \prime } ( 2 ) = 4 , v ^ { \prime } ( 0 ) = 5$ and $v ^ { \prime } ( 2 ) = 6$ Find $w ^ { \prime } ( 0 )$

Accepted Answer

A) 5 
B) 10 
C) 15 
D) 20 
E) 25
F)30
G)35
H)40 
A) 5 
B) 10 
C) 15 
D) 20 
E) 25
F)30
G)35
H)40

Question 13

The angular displacement $\theta$ of a simple pendulum is given by $\theta = \theta _ { 0 } \sin ( \omega t + \phi )$ where $\theta _ { 0 }$ is the angular amplitude, $\omega$ the angular frequency and $\theta$ a phase constant depending on initial conditions. If we are given that $\omega$ = 10 and $\phi = \frac { \pi } { 2 }$ 
, find the angular velocity $\frac { d \theta } { d t }$ when $\theta = \frac { \theta _ { 0 } } { 2 }$ .

Accepted Answer

@#LAT-DLM& ; when @#LAT-DLM& ,

Question 14

If $f ( x ) = \cos ^ { 2 } ( 2 x ) , \text { find } f ^ { \prime } \left( \frac { \pi } { 3 } \right)$

Accepted Answer

A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 4 }$ 
C) 1 
D)  $- 1$ 
E)  $- \sqrt { 3 }$
F) $- \frac { \sqrt { 3 } } { 2 }$ 
G) $\frac { \sqrt { 3 } } { 2 }$
H) $- \frac { \sqrt { 3 } } { 4 }$ 
A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 4 }$ 
C) 1 
D)  $- 1$ 
E)  $- \sqrt { 3 }$
F) $- \frac { \sqrt { 3 } } { 2 }$ 
G) $\frac { \sqrt { 3 } } { 2 }$
H) $- \frac { \sqrt { 3 } } { 4 }$

Question 15

Find $\frac { d y } { d x }$ if $x \sin y + \cos 2 y + x e ^ { x ^ { 4 } } = 4$ .

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 16

Suppose that $g ( x )$ is a differentiable function. Find $f ^ { \prime } ( x )$ for each of the following, in terms of $g ( x )$ and $g ^ { \prime } ( x )$ .(a) $f ( x ) = e ^ { x } g ( x )$ (b) $f ( x ) = \frac { 3 x - 1 } { g ( x ) }$ (c) $f ( x ) = g ( x ) \left( 3 x ^ { 2 } - 8 \right)$ (d) $f ( x ) = \sqrt { x } ( g ( x ) + 4 )$

Accepted Answer

The answer of Suppose that $g ( x )$ is...

Question 17

Let $x ( t ) = 200 \left( 1 - e ^ { - \frac { 3 } { 200 } t } \right)$ be the amount of salt (in kg) in a tank after time t minutes. Find:
(a) How much salt is in the tank after 1 hour?
(b) Find the rate of change of salt after 1 hour?

Accepted Answer

The answer of Let \(x ( t ) = 200...

Question 18

Find the slope of the tangent to the curve $x = 1 - 2 \sin t , y = 3 \cos t$ when $t = \frac { \pi } { 3 }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 2 }$ 
C)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$ 
E)  $- \frac { \sqrt { 3 } } { 2 }$
F) $- \frac { 2 \sqrt { 3 } } { 2 }$
G) $- \frac { 3 \sqrt { 3 } } { 2 }$

H) $- \frac { \sqrt { 3 } } { 3 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 2 }$ 
C)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$ 
E)  $- \frac { \sqrt { 3 } } { 2 }$
F) $- \frac { 2 \sqrt { 3 } } { 2 }$
G) $- \frac { 3 \sqrt { 3 } } { 2 }$

H) $- \frac { \sqrt { 3 } } { 3 }$

Question 19

Find the y-intercept of the tangent line to the curve $y = x \sin ( 2 x )$ at the point ( $\frac { \pi } { 2 }$ , 0).

Accepted Answer

The answer of Find the y-intercept of the tangent line...

Question 20

If $\sqrt { x } + \sqrt { y } = 3$ find the value of $\frac { d y } { d x }$ at the point (4, 1).

Accepted Answer

A)  $- 3$ 
B) 0 
C) 2 
D)  $- 1$ 
E) 1
F)3
G) $- 2$
H)4 
A)  $- 3$ 
B) 0 
C) 2 
D)  $- 1$ 
E) 1
F)3
G) $- 2$
H)4

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 3 t + 2$ . Find the value of t at which the particle reverses its direction.

Find the exact value of $\tan \left( \cos ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)$ .

Find the y-intercept of the tangent line to the curve $y = \sqrt { x + 3 }$ at the point (1, 2).

Find $\frac { d y } { d x } .$ (a) $y = \sqrt { 2 x - x ^ { 2 } }$ (b) $y = \left( x ^ { 2 } + 1 \right) ^ { 50 }$ (c) $y = \sin \left( \frac { 1 } { x } \right)$ (d) $y = \tan ( 3 x )$

Let $y = f ( x )$ If $x y ^ { 3 } + x y = 6 \text { and } f ( 3 ) = 1 , \text { find } f ^ { \prime } ( 3 )$

Find an equation of the tangent line to the curve $y = \left( x ^ { 2 } + 1 \right) ^ { 3 }$ at the point (1, 8).

Find the domain of the function $f ( x ) = \sin ^ { - 1 } ( 3 - 2 x )$ .

If $f ( x ) = - \frac { 1 } { x ^ { 2 } }$ , find $f ^ { \prime } ( 1 )$ .

A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?

The position function for a particle is $s ( t ) = 16 t ^ { 2 } + 48 t + 100$ , where s is measured in feet and t is measured in seconds.(a) Find the velocity at t = 2.(b) When does the velocity equal zero?

Suppose that $w = u \circ v$ and $u ( 0 ) = 1 , v ( 0 ) = 2 , u ^ { \prime } ( 0 ) = 3 , u ^ { \prime } ( 2 ) = 4 , v ^ { \prime } ( 0 ) = 5$ and $v ^ { \prime } ( 2 ) = 6$ Find $w ^ { \prime } ( 0 )$

If $f ( x ) = \cos ^ { 2 } ( 2 x ) , \text { find } f ^ { \prime } \left( \frac { \pi } { 3 } \right)$

Find $\frac { d y } { d x }$ if $x \sin y + \cos 2 y + x e ^ { x ^ { 4 } } = 4$ .

Let $x ( t ) = 200 \left( 1 - e ^ { - \frac { 3 } { 200 } t } \right)$ be the amount of salt (in kg) in a tank after time t minutes. Find: (a) How much salt is in the tank after 1 hour? (b) Find the rate of change of salt after 1 hour?

Find the slope of the tangent to the curve $x = 1 - 2 \sin t , y = 3 \cos t$ when $t = \frac { \pi } { 3 }$

Find the y-intercept of the tangent line to the curve $y = x \sin ( 2 x )$ at the point ( $\frac { \pi } { 2 }$ , 0).

If $\sqrt { x } + \sqrt { y } = 3$ find the value of $\frac { d y } { d x }$ at the point (4, 1).

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Exam 3: Differentiation Rules

A particle moves along a straight line with equation of motion s=t2−3t+2s = t ^ { 2 } - 3 t + 2s=t2−3t+2 . Find the value of t at which the particle reverses its direction.

Find the exact value of tan⁡(cos⁡−122)\tan \left( \cos ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)tan(cos−122​​) .

Find the y-intercept of the tangent line to the curve y=x+3y = \sqrt { x + 3 }y=x+3​ at the point (1, 2).

Find dydx.\frac { d y } { d x } .dxdy​. (a) y=2x−x2y = \sqrt { 2 x - x ^ { 2 } }y=2x−x2​ (b) y=(x2+1)50y = \left( x ^ { 2 } + 1 \right) ^ { 50 }y=(x2+1)50 (c) y=sin⁡(1x)y = \sin \left( \frac { 1 } { x } \right)y=sin(x1​) (d) y=tan⁡(3x)y = \tan ( 3 x )y=tan(3x)

Let y=f(x)y = f ( x )y=f(x) If xy3+xy=6 and f(3)=1, find f′(3)x y ^ { 3 } + x y = 6 \text { and } f ( 3 ) = 1 , \text { find } f ^ { \prime } ( 3 )xy3+xy=6 and f(3)=1, find f′(3)

Find an equation of the tangent line to the curve y=(x2+1)3y = \left( x ^ { 2 } + 1 \right) ^ { 3 }y=(x2+1)3 at the point (1, 8).

Find the domain of the function f(x)=sin⁡−1(3−2x)f ( x ) = \sin ^ { - 1 } ( 3 - 2 x )f(x)=sin−1(3−2x) .

If f(x)=−1x2f ( x ) = - \frac { 1 } { x ^ { 2 } }f(x)=−x21​ , find f′(1)f ^ { \prime } ( 1 )f′(1) .

A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?

The position function for a particle is s(t)=16t2+48t+100s ( t ) = 16 t ^ { 2 } + 48 t + 100s(t)=16t2+48t+100 , where s is measured in feet and t is measured in seconds.(a) Find the velocity at t = 2.(b) When does the velocity equal zero?

If f(x)=cos⁡2(2x), find f′(π3)f ( x ) = \cos ^ { 2 } ( 2 x ) , \text { find } f ^ { \prime } \left( \frac { \pi } { 3 } \right)f(x)=cos2(2x), find f′(3π​)

Find dydx\frac { d y } { d x }dxdy​ if xsin⁡y+cos⁡2y+xex4=4x \sin y + \cos 2 y + x e ^ { x ^ { 4 } } = 4xsiny+cos2y+xex4=4 .

Let x(t)=200(1−e−3200t)x ( t ) = 200 \left( 1 - e ^ { - \frac { 3 } { 200 } t } \right)x(t)=200(1−e−2003​t) be the amount of salt (in kg) in a tank after time t minutes. Find: (a) How much salt is in the tank after 1 hour? (b) Find the rate of change of salt after 1 hour?

Find the slope of the tangent to the curve x=1−2sin⁡t,y=3cos⁡tx = 1 - 2 \sin t , y = 3 \cos tx=1−2sint,y=3cost when t=π3t = \frac { \pi } { 3 }t=3π​

Find the y-intercept of the tangent line to the curve y=xsin⁡(2x)y = x \sin ( 2 x )y=xsin(2x) at the point ( π2\frac { \pi } { 2 }2π​ , 0).

If x+y=3\sqrt { x } + \sqrt { y } = 3x​+y​=3 find the value of dydx\frac { d y } { d x }dxdy​ at the point (4, 1).

Functions and Models

Limits and Derivatives

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Integrals

Applications of Integration

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Infinite Sequences and Series

Vectors and the Geometry of Space

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Multiple Integrals

Vector Calculus

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A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 3 t + 2$ . Find the value of t at which the particle reverses its direction.

Find the exact value of $\tan \left( \cos ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)$ .

Find the y-intercept of the tangent line to the curve $y = \sqrt { x + 3 }$ at the point (1, 2).

Find $\frac { d y } { d x } .$ (a) $y = \sqrt { 2 x - x ^ { 2 } }$ (b) $y = \left( x ^ { 2 } + 1 \right) ^ { 50 }$ (c) $y = \sin \left( \frac { 1 } { x } \right)$ (d) $y = \tan ( 3 x )$

Let $y = f ( x )$ If $x y ^ { 3 } + x y = 6 \text { and } f ( 3 ) = 1 , \text { find } f ^ { \prime } ( 3 )$

Find an equation of the tangent line to the curve $y = \left( x ^ { 2 } + 1 \right) ^ { 3 }$ at the point (1, 8).

Find the domain of the function $f ( x ) = \sin ^ { - 1 } ( 3 - 2 x )$ .

If $f ( x ) = - \frac { 1 } { x ^ { 2 } }$ , find $f ^ { \prime } ( 1 )$ .

The position function for a particle is $s ( t ) = 16 t ^ { 2 } + 48 t + 100$ , where s is measured in feet and t is measured in seconds.(a) Find the velocity at t = 2.(b) When does the velocity equal zero?

If $f ( x ) = \cos ^ { 2 } ( 2 x ) , \text { find } f ^ { \prime } \left( \frac { \pi } { 3 } \right)$

Find $\frac { d y } { d x }$ if $x \sin y + \cos 2 y + x e ^ { x ^ { 4 } } = 4$ .

Let $x ( t ) = 200 \left( 1 - e ^ { - \frac { 3 } { 200 } t } \right)$ be the amount of salt (in kg) in a tank after time t minutes. Find: (a) How much salt is in the tank after 1 hour? (b) Find the rate of change of salt after 1 hour?

Find the slope of the tangent to the curve $x = 1 - 2 \sin t , y = 3 \cos t$ when $t = \frac { \pi } { 3 }$

Find the y-intercept of the tangent line to the curve $y = x \sin ( 2 x )$ at the point ( $\frac { \pi } { 2 }$ , 0).

If $\sqrt { x } + \sqrt { y } = 3$ find the value of $\frac { d y } { d x }$ at the point (4, 1).