Question 1

Find the indicated derivative.
-$\frac { \mathrm { dt } } { \mathrm { dx } }$ if $\mathrm { t } = \frac { \mathrm { x } } { 8 \mathrm { x } - 2 }$

Accepted Answer

A)  $- \frac { 2 } { ( 8 x - 2 ) ^ { 2 } }$ 
B)  $\frac { 16 x - 2 } { ( 8 x - 2 ) ^ { 2 } }$ 
C)  $- \frac { 2 x } { ( 8 x - 2 ) ^ { 2 } }$ 
D)  $- \frac { 2 } { 8 x - 2 }$ 
A)  $- \frac { 2 } { ( 8 x - 2 ) ^ { 2 } }$ 
B)  $\frac { 16 x - 2 } { ( 8 x - 2 ) ^ { 2 } }$ 
C)  $- \frac { 2 x } { ( 8 x - 2 ) ^ { 2 } }$ 
D)  $- \frac { 2 } { 8 x - 2 }$

Question 2

Find the slope of the curve at the indicated point.
-The equation for free fall at the surface of Planet $X$ is s $= 12.05 t ^ { 2 } \mathrm {~m}$ with $\mathrm { t }$ in seconds. Assume a rock is dropped from the top of a $600 \mathrm {~m}$ cliff. Find the speed of the rock at $t = 2 \mathrm { sec }$.

Accepted Answer

A)  $49.2 \mathrm {~m} / \mathrm { sec }$ 
B)  $24.1 \mathrm {~m} / \mathrm { sec }$ 
C)  $48.2 \mathrm {~m} / \mathrm { sec }$ 
D)  $23.1 \mathrm {~m} / \mathrm { sec }$ 
A)  $49.2 \mathrm {~m} / \mathrm { sec }$ 
B)  $24.1 \mathrm {~m} / \mathrm { sec }$ 
C)  $48.2 \mathrm {~m} / \mathrm { sec }$ 
D)  $23.1 \mathrm {~m} / \mathrm { sec }$

Question 3

Find the derivative of the function.
-$$y = \frac { x ^ { 3 } } { x - 1 }$$

Accepted Answer

A)  $y ^ { \prime } = \frac { 2 x ^ { 3 } - 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 2 x ^ { 3 } + 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { - 2 x ^ { 3 } - 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
D)  $y ^ { \prime } = \frac { 2 x ^ { 3 } + 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
A)  $y ^ { \prime } = \frac { 2 x ^ { 3 } - 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 2 x ^ { 3 } + 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { - 2 x ^ { 3 } - 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$ 
D)  $y ^ { \prime } = \frac { 2 x ^ { 3 } + 3 x ^ { 2 } } { ( x - 1 ) ^ { 2 } }$

Question 4

Find the derivative.
-$$y = 2 x ^ { 2 } e ^ { - x }$$

Accepted Answer

A)  $2 x e ^ { x } ( 2 - x )$ 
B)  $2 x e ^ { - x } ( x + 2 )$ 
C)  $4 \times e ^ { - x } ( 1 - x )$ 
D)  $2 \times \mathrm { e } ^ { - \mathrm { x } } ( 2 - \mathrm { x } )$ 
A)  $2 x e ^ { x } ( 2 - x )$ 
B)  $2 x e ^ { - x } ( x + 2 )$ 
C)  $4 \times e ^ { - x } ( 1 - x )$ 
D)  $2 \times \mathrm { e } ^ { - \mathrm { x } } ( 2 - \mathrm { x } )$

Question 5

Calculate the derivative of the function. Then find the value of the derivative as specified.
-$$f ( x ) = x ^ { 2 } + 7 x - 2 ; f ^ { \prime } ( 0 )$$

Accepted Answer

A)  $f ^ { \prime } ( x ) = 2 x + 7 ; f ^ { \prime } ( 0 ) = 7$ 
B)  $f ^ { \prime } ( x ) = x + 7 ; f ^ { \prime } ( 0 ) = 7$ 
C)  $f ^ { \prime } ( x ) = 2 x ; f ^ { \prime } ( 0 ) = 0$ 
D)  $f ^ { \prime } ( x ) = 2 x - 2 ; f ^ { \prime } ( 0 ) = - 2$ 
A)  $f ^ { \prime } ( x ) = 2 x + 7 ; f ^ { \prime } ( 0 ) = 7$ 
B)  $f ^ { \prime } ( x ) = x + 7 ; f ^ { \prime } ( 0 ) = 7$ 
C)  $f ^ { \prime } ( x ) = 2 x ; f ^ { \prime } ( 0 ) = 0$ 
D)  $f ^ { \prime } ( x ) = 2 x - 2 ; f ^ { \prime } ( 0 ) = - 2$

Question 6

Find the derivative.
-$$s = 2 t ^ { 2 } + 5 t - 8$$

Accepted Answer

A)  $4 t + 5$ 
B)  $4 t ^ { 2 } + 5$ 
C)  $2 t ^ { 2 } + 5$ 
D)  $2 t + 5$ 
A)  $4 t + 5$ 
B)  $4 t ^ { 2 } + 5$ 
C)  $2 t ^ { 2 } + 5$ 
D)  $2 t + 5$

Question 7

Find the second derivative.
-$$y = \frac { 11 x ^ { 3 } } { 6 } - 9$$

Accepted Answer

A)  $\frac { 11 } { 2 } \mathrm { x } ^ { 2 }$ 
B)  $11 x - 9$ 
C)  $11 \mathrm { x }$ 
D)  $\frac { 11 } { 6 } x$ 
A)  $\frac { 11 } { 2 } \mathrm { x } ^ { 2 }$ 
B)  $11 x - 9$ 
C)  $11 \mathrm { x }$ 
D)  $\frac { 11 } { 6 } x$

Question 8

Estimate the slope of the curve at the indicated point.
-

Accepted Answer

A)  $- 2$ 
B)  $\frac { 1 } { 20 }$ 
C)  2 
D)  $- \frac { 1 } { 20 }$ 
A)  $- 2$ 
B)  $\frac { 1 } { 20 }$ 
C)  2 
D)  $- \frac { 1 } { 20 }$

Question 9

Find the second derivative.
-$$y = 4 x ^ { 2 } + 9 x + 5 x ^ { - 3 }$$

Accepted Answer

A)  $8 + 60 x ^ { - 1 }$ 
B)  $8 + 60 x ^ { - 5 }$ 
C)  $8 x + 9 - 15 x ^ { - 4 }$ 
D)  $8 - 60 x ^ { - 5 }$ 
A)  $8 + 60 x ^ { - 1 }$ 
B)  $8 + 60 x ^ { - 5 }$ 
C)  $8 x + 9 - 15 x ^ { - 4 }$ 
D)  $8 - 60 x ^ { - 5 }$

Question 10

Given the graph of f, find any values of x at which f ' is not defined.
-

Accepted Answer

A)  x = -3, 0, 3 
B)  x = -2, 0, 2 
C)  x = -2, 2 
D)  x = -3, 3 
A)  x = -3, 0, 3 
B)  x = -2, 0, 2 
C)  x = -2, 2 
D)  x = -3, 3

Question 11

Find the derivative of the function.
-$$y = \frac { ( x + 4 ) ( x + 1 ) } { ( x - 4 ) ( x - 1 ) }$$

Accepted Answer

A)  $y ^ { \prime } = \frac { - x ^ { 2 } + 8 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 10 x ^ { 2 } + 40 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { 10 x ^ { 2 } - 40 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
D)  $\mathrm { y } ^ { \prime } = \frac { 10 \mathrm { x } - 40 } { ( \mathrm { x } - 4 ) ^ { 2 } ( \mathrm { x } - 1 ) ^ { 2 } }$ 
A)  $y ^ { \prime } = \frac { - x ^ { 2 } + 8 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 10 x ^ { 2 } + 40 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { 10 x ^ { 2 } - 40 } { ( x - 4 ) ^ { 2 } ( x - 1 ) ^ { 2 } }$ 
D)  $\mathrm { y } ^ { \prime } = \frac { 10 \mathrm { x } - 40 } { ( \mathrm { x } - 4 ) ^ { 2 } ( \mathrm { x } - 1 ) ^ { 2 } }$

Question 12

Given the graph of f, find any values of x at which f ' is not defined.
-

Accepted Answer

A)  x = 0, 3 
B)  x = 3 
C)  x = 0 
D)  Defined for all values of x 
A)  x = 0, 3 
B)  x = 3 
C)  x = 0 
D)  Defined for all values of x

Question 13

Find y'.
-$$y = \left( \frac { 1 } { x } + 7 \right) \left( x - \frac { 1 } { x } + 7 \right)$$

Accepted Answer

A)  $- \frac { 1 } { x ^ { 3 } } - 7$ 
B)  $\frac { 1 } { x ^ { 3 } } + 7$ 
C)  $\frac { 2 } { x ^ { 3 } } + 7$ 
D)  $- \frac { 2 } { x ^ { 3 } } - 7$ 
A)  $- \frac { 1 } { x ^ { 3 } } - 7$ 
B)  $\frac { 1 } { x ^ { 3 } } + 7$ 
C)  $\frac { 2 } { x ^ { 3 } } + 7$ 
D)  $- \frac { 2 } { x ^ { 3 } } - 7$

Question 14

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative.
-$$u ( 1 ) = 4 , u ^ { \prime } ( 1 ) = - 7 , v ( 1 ) = 6 , v ^ { \prime } ( 1 ) = - 3 \text {. }$$
$\frac { \mathrm { d } } { \mathrm { dx } } \left( \frac { \mathrm { v } } { \mathrm { u } } \right)$ at $\mathrm { x } = 1$

Accepted Answer

A)  $- \frac { 27 } { 8 }$ 
B)  $- \frac { 15 } { 8 }$ 
C)  $\frac { 15 } { 8 }$ 
D)  $\frac { 15 } { 2 }$ 
A)  $- \frac { 27 } { 8 }$ 
B)  $- \frac { 15 } { 8 }$ 
C)  $\frac { 15 } { 8 }$ 
D)  $\frac { 15 } { 2 }$

Question 15

Provide an appropriate response.
-The curve $y = a x ^ { 2 } + b x + c$ passes through the point $( 2,32 )$ and is tangent to the line $y = 4 x$ at the origin. Find $a$, $b$, and $c$.

Accepted Answer

A)  a = 0, b = 6, c = 4 
B)  a = 4, b = 0, c = 6 
C)  a = 6, b = 4, c = 0 
D)  a = 7, b = 0, c = 0 
A)  a = 0, b = 6, c = 4 
B)  a = 4, b = 0, c = 6 
C)  a = 6, b = 4, c = 0 
D)  a = 7, b = 0, c = 0

Question 16

Calculate the derivative of the function. Then find the value of the derivative as specified.
-$$\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } \text { if } \mathrm { r } = \frac { 4 } { \sqrt { 12 - \theta } }$$

Accepted Answer

A)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { d } r } { \mathrm {~d} \theta } \mid \theta = 3 = - \frac { 2 } { 27 }$ 
B)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 2 } { 27 }$ 
C)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 4 } { 27 }$ 
D)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { dr } } { \mathrm { d } \theta } \mid \theta = 3 = - \frac { 4 } { 27 }$ 
A)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { d } r } { \mathrm {~d} \theta } \mid \theta = 3 = - \frac { 2 } { 27 }$ 
B)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 2 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 2 } { 27 }$ 
C)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } = \frac { 4 } { 27 }$ 
D)  $\frac { \mathrm { dr } } { \mathrm { d } \theta } = - \frac { 4 } { ( 12 - \theta ) ^ { 3 / 2 } } ; \frac { \mathrm { dr } } { \mathrm { d } \theta } \mid \theta = 3 = - \frac { 4 } { 27 }$

Question 17

Provide an appropriate response.
-Over what intervals of x-values, if any, does the function y = x55 decrease as x increases? For what values of x, if any, is y' negative? How are your answers related?

Accepted Answer

The function @#LAT-DLM& decreases (as @#LAT-DLM& i

Question 18

Estimate the slope of the curve at the indicated point.
-

Accepted Answer

A)  Undefined 
B)  1 
C)  -1 
D)  0 
A)  Undefined 
B)  1 
C)  -1 
D)  0

Question 19

Given the graph of f, find any values of x at which f ' is not defined.
-

Accepted Answer

A)  x = -1, 1 
B)  x = -1, 0, 1 
C)  x = 0 
D)  Defined for all values of x 
A)  x = -1, 1 
B)  x = -1, 0, 1 
C)  x = 0 
D)  Defined for all values of x

Question 20

Find the derivative of the function.
-$$g ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }$$

Accepted Answer

A)  $g ^ { \prime } ( x ) = \frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
B)  $g ^ { \prime } ( x ) = \frac { x ^ { 4 } + 6 x ^ { 3 } + 5 x ^ { 2 } + 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
C)  $g ^ { \prime } ( x ) = \frac { 4 x ^ { 3 } + 18 x ^ { 2 } + 10 x + 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
D)  $g ^ { \prime } ( x ) = \frac { 6 x ^ { 2 } - 10 x - 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
A)  $g ^ { \prime } ( x ) = \frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
B)  $g ^ { \prime } ( x ) = \frac { x ^ { 4 } + 6 x ^ { 3 } + 5 x ^ { 2 } + 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
C)  $g ^ { \prime } ( x ) = \frac { 4 x ^ { 3 } + 18 x ^ { 2 } + 10 x + 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$ 
D)  $g ^ { \prime } ( x ) = \frac { 6 x ^ { 2 } - 10 x - 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$

Find the indicated derivative. - $\frac { \mathrm { dt } } { \mathrm { dx } }$ if $\mathrm { t } = \frac { \mathrm { x } } { 8 \mathrm { x } - 2 }$

Find the derivative of the function. - $y = \frac { x ^ { 3 } } { x - 1 }$

Find the derivative. - $y = 2 x ^ { 2 } e ^ { - x }$

Calculate the derivative of the function. Then find the value of the derivative as specified. - $f ( x ) = x ^ { 2 } + 7 x - 2 ; f ^ { \prime } ( 0 )$

Find the derivative. - $s = 2 t ^ { 2 } + 5 t - 8$

Find the second derivative. - $y = \frac { 11 x ^ { 3 } } { 6 } - 9$

Estimate the slope of the curve at the indicated point. -

Find the second derivative. - $y = 4 x ^ { 2 } + 9 x + 5 x ^ { - 3 }$

Given the graph of f, find any values of x at which f ' is not defined. -

Find the derivative of the function. - $y = \frac { ( x + 4 ) ( x + 1 ) } { ( x - 4 ) ( x - 1 ) }$

Given the graph of f, find any values of x at which f ' is not defined. -

Find y'. - $y = \left( \frac { 1 } { x } + 7 \right) \left( x - \frac { 1 } { x } + 7 \right)$

Provide an appropriate response. -The curve $y = a x ^ { 2 } + b x + c$ passes through the point $( 2,32 )$ and is tangent to the line $y = 4 x$ at the origin. Find $a$ , $b$ , and $c$ .

Calculate the derivative of the function. Then find the value of the derivative as specified. - $\left. \frac { \mathrm { dr } } { \mathrm { d } \theta } \right| _ { \theta = 3 } \text { if } \mathrm { r } = \frac { 4 } { \sqrt { 12 - \theta } }$

Provide an appropriate response. -Over what intervals of x-values, if any, does the function y = x55 decrease as x increases? For what values of x, if any, is y' negative? How are your answers related?

Estimate the slope of the curve at the indicated point. -

Given the graph of f, find any values of x at which f ' is not defined. -

Find the derivative of the function. - $g ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }$

Functions

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Exam 3: Differentiation

Find the indicated derivative. - dtdx\frac { \mathrm { dt } } { \mathrm { dx } }dxdt​ if t=x8x−2\mathrm { t } = \frac { \mathrm { x } } { 8 \mathrm { x } - 2 }t=8x−2x​

Find the derivative of the function. - y=x3x−1y = \frac { x ^ { 3 } } { x - 1 }y=x−1x3​

Find the derivative. - y=2x2e−xy = 2 x ^ { 2 } e ^ { - x }y=2x2e−x

Calculate the derivative of the function. Then find the value of the derivative as specified. - f(x)=x2+7x−2;f′(0)f ( x ) = x ^ { 2 } + 7 x - 2 ; f ^ { \prime } ( 0 )f(x)=x2+7x−2;f′(0)

Find the derivative. - s=2t2+5t−8s = 2 t ^ { 2 } + 5 t - 8s=2t2+5t−8

Find the second derivative. - y=11x36−9y = \frac { 11 x ^ { 3 } } { 6 } - 9y=611x3​−9

Estimate the slope of the curve at the indicated point. -

Find the second derivative. - y=4x2+9x+5x−3y = 4 x ^ { 2 } + 9 x + 5 x ^ { - 3 }y=4x2+9x+5x−3

Given the graph of f, find any values of x at which f ' is not defined. -

Find the derivative of the function. - y=(x+4)(x+1)(x−4)(x−1)y = \frac { ( x + 4 ) ( x + 1 ) } { ( x - 4 ) ( x - 1 ) }y=(x−4)(x−1)(x+4)(x+1)​

Given the graph of f, find any values of x at which f ' is not defined. -

Find y'. - y=(1x+7)(x−1x+7)y = \left( \frac { 1 } { x } + 7 \right) \left( x - \frac { 1 } { x } + 7 \right)y=(x1​+7)(x−x1​+7)

Provide an appropriate response. -The curve y=ax2+bx+cy = a x ^ { 2 } + b x + cy=ax2+bx+c passes through the point (2,32)( 2,32 )(2,32) and is tangent to the line y=4xy = 4 xy=4x at the origin. Find aaa , bbb , and ccc .

Provide an appropriate response. -Over what intervals of x-values, if any, does the function y = x55 decrease as x increases? For what values of x, if any, is y' negative? How are your answers related?

Estimate the slope of the curve at the indicated point. -

Given the graph of f, find any values of x at which f ' is not defined. -

Find the derivative of the function. - g(x)=x2+5x2+6xg ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }g(x)=x2+6xx2+5​

Functions

Limits and Derivatives

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Filters

Find the indicated derivative. - $\frac { \mathrm { dt } } { \mathrm { dx } }$ if $\mathrm { t } = \frac { \mathrm { x } } { 8 \mathrm { x } - 2 }$

Find the derivative of the function. - $y = \frac { x ^ { 3 } } { x - 1 }$

Find the derivative. - $y = 2 x ^ { 2 } e ^ { - x }$

Calculate the derivative of the function. Then find the value of the derivative as specified. - $f ( x ) = x ^ { 2 } + 7 x - 2 ; f ^ { \prime } ( 0 )$

Find the derivative. - $s = 2 t ^ { 2 } + 5 t - 8$

Find the second derivative. - $y = \frac { 11 x ^ { 3 } } { 6 } - 9$

Find the second derivative. - $y = 4 x ^ { 2 } + 9 x + 5 x ^ { - 3 }$

Find the derivative of the function. - $y = \frac { ( x + 4 ) ( x + 1 ) } { ( x - 4 ) ( x - 1 ) }$

Find y'. - $y = \left( \frac { 1 } { x } + 7 \right) \left( x - \frac { 1 } { x } + 7 \right)$

Provide an appropriate response. -The curve $y = a x ^ { 2 } + b x + c$ passes through the point $( 2,32 )$ and is tangent to the line $y = 4 x$ at the origin. Find $a$ , $b$ , and $c$ .

Find the derivative of the function. - $g ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }$