Question 1

Find the slope of the curve at the indicated point.
-$$y = x ^ { 2 } + 10 x , x = - 4$$

Accepted Answer

A)  $m = 6$ 
B)  $m = - 24$ 
C)  $m = - 8$ 
D)  $m = 2$ 
A)  $m = 6$ 
B)  $m = - 24$ 
C)  $m = - 8$ 
D)  $m = 2$

Question 2

Find the second derivative.
-$$\mathrm { s } = \frac { 17 \mathrm { t } ^ { 3 } } { 3 } + 17$$

Accepted Answer

A)  $17 t ^ { 2 }$ 
B)  $34 t + 17$ 
C)  $34 \mathrm { t }$ 
D)  $17 \mathrm { t }$ 
A)  $17 t ^ { 2 }$ 
B)  $34 t + 17$ 
C)  $34 \mathrm { t }$ 
D)  $17 \mathrm { t }$

Question 3

Graph the equation and its tangent
-$$\text { Graph } y = 5 x ^ { 2 } \text { and the tangent to the curve at the point whose } x \text {-coordinate is } 1 \text {. }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 4

Find the slope of the curve at the indicated point.
-Assume that a watermelon dropped from a tall building falls $y = 16 \mathrm { t } ^ { 2 } \mathrm { ft }$ in $\mathrm { t }$ sec. Find the watermelon's average speed during the first $4 \sec$ of fall and the speed at the instant $t = 4 \mathrm { sec }$.

Accepted Answer

A)  $128 \mathrm { ft } / \mathrm { sec } ; 65 \mathrm { ft } / \mathrm { sec }$ 
B)  $32 \mathrm { ft } / \mathrm { sec } ; 64 \mathrm { ft } / \mathrm { sec }$ 
C)  $64 \mathrm { ft } / \mathrm { sec } ; 128 \mathrm { ft } / \mathrm { sec }$ 
D)  $65 \mathrm { ft } / \mathrm { sec } ; 130 \mathrm { ft } / \mathrm { sec }$ 
A)  $128 \mathrm { ft } / \mathrm { sec } ; 65 \mathrm { ft } / \mathrm { sec }$ 
B)  $32 \mathrm { ft } / \mathrm { sec } ; 64 \mathrm { ft } / \mathrm { sec }$ 
C)  $64 \mathrm { ft } / \mathrm { sec } ; 128 \mathrm { ft } / \mathrm { sec }$ 
D)  $65 \mathrm { ft } / \mathrm { sec } ; 130 \mathrm { ft } / \mathrm { sec }$

Question 5

Find the derivative of the function.
-$$s = \frac { t ^ { 7 } + 3 t + 6 } { t ^ { 2 } }$$

Accepted Answer

A)  $\frac { \mathrm { ds } } { \mathrm { dt } } = \mathrm { t } ^ { 4 } - \frac { 3 } { \mathrm { t } ^ { 2 } } - \frac { 6 } { \mathrm { t } ^ { 3 } }$ 
B)  $\frac { \mathrm { ds } } { \mathrm { dt } } = 12 \mathrm { t } ^ { 9 } + 8 \mathrm { t } ^ { 2 } + 12 \mathrm { t }$ 
C)  $\frac { d s } { d t } = 5 t ^ { 4 } - \frac { 3 } { t ^ { 2 } } - \frac { 12 } { t ^ { 3 } }$ 
D)  $\frac { d s } { d t } = 5 t ^ { 4 } + \frac { 3 } { t ^ { 2 } } + \frac { 12 } { t ^ { 3 } }$ 
A)  $\frac { \mathrm { ds } } { \mathrm { dt } } = \mathrm { t } ^ { 4 } - \frac { 3 } { \mathrm { t } ^ { 2 } } - \frac { 6 } { \mathrm { t } ^ { 3 } }$ 
B)  $\frac { \mathrm { ds } } { \mathrm { dt } } = 12 \mathrm { t } ^ { 9 } + 8 \mathrm { t } ^ { 2 } + 12 \mathrm { t }$ 
C)  $\frac { d s } { d t } = 5 t ^ { 4 } - \frac { 3 } { t ^ { 2 } } - \frac { 12 } { t ^ { 3 } }$ 
D)  $\frac { d s } { d t } = 5 t ^ { 4 } + \frac { 3 } { t ^ { 2 } } + \frac { 12 } { t ^ { 3 } }$

Question 6

Find the derivatives of all orders of the function.
-$$y = \frac { 8 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 13$$

Accepted Answer

The answer of Find the derivatives of all orders of...

Question 7

Find the indicated derivative.
-$$\frac { d y } { d x } \text { if } y = 3 x ^ { 3 }$$

Accepted Answer

A)  $9 x ^ { 3 }$ 
B)  $9 x ^ { 2 }$ 
C)  $3 x ^ { 2 }$ 
D)  $9 x$ 
A)  $9 x ^ { 3 }$ 
B)  $9 x ^ { 2 }$ 
C)  $3 x ^ { 2 }$ 
D)  $9 x$

Question 8

Find the second derivative of the function.
-$$p = \left( \frac { q + 3 } { q } \right) \left( \frac { q + 5 } { q ^ { 2 } } \right)$$

Accepted Answer

A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = \frac { 2 } { \mathrm { q } } + \frac { 48 } { \mathrm { q } ^ { 2 } } + \frac { 180 } { \mathrm { q } ^ { 3 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = \frac { 2 } { \mathrm { q } ^ { 3 } } + \frac { 48 } { \mathrm { q } ^ { 4 } } + \frac { 180 } { \mathrm { q } ^ { 5 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = - \frac { 2 } { \mathrm { q } ^ { 3 } } - \frac { 48 } { \mathrm { q } ^ { 4 } } - \frac { 180 } { \mathrm { q } ^ { 5 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = - \frac { 1 } { \mathrm { q } ^ { 2 } } - \frac { 16 } { \mathrm { q } ^ { 3 } } - \frac { 45 } { \mathrm { q } ^ { 4 } }$ 
A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = \frac { 2 } { \mathrm { q } } + \frac { 48 } { \mathrm { q } ^ { 2 } } + \frac { 180 } { \mathrm { q } ^ { 3 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = \frac { 2 } { \mathrm { q } ^ { 3 } } + \frac { 48 } { \mathrm { q } ^ { 4 } } + \frac { 180 } { \mathrm { q } ^ { 5 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = - \frac { 2 } { \mathrm { q } ^ { 3 } } - \frac { 48 } { \mathrm { q } ^ { 4 } } - \frac { 180 } { \mathrm { q } ^ { 5 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { p } } { \mathrm { dq } ^ { 2 } } = - \frac { 1 } { \mathrm { q } ^ { 2 } } - \frac { 16 } { \mathrm { q } ^ { 3 } } - \frac { 45 } { \mathrm { q } ^ { 4 } }$

Question 9

Find the derivative.
-$$y = x ^ { 1 / 5 }$$

Accepted Answer

A)  $- \frac { 1 } { 5 } x ^ { - 6 / 5 }$ 
B)  $\frac { 6 } { 5 } x ^ { 6 / 5 }$ 
C)  $\frac { 1 } { 5 } x ^ { - 4 / 5 }$ 
D)  $\frac { 1 } { 5 } x ^ { - 1 / 5 }$ 
A)  $- \frac { 1 } { 5 } x ^ { - 6 / 5 }$ 
B)  $\frac { 6 } { 5 } x ^ { 6 / 5 }$ 
C)  $\frac { 1 } { 5 } x ^ { - 4 / 5 }$ 
D)  $\frac { 1 } { 5 } x ^ { - 1 / 5 }$

Question 10

Find the second derivative of the function.
-$$y = \frac { x ^ { 4 } + 9 } { x ^ { 2 } }$$

Accepted Answer

A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 2 + \frac { 54 } { x ^ { 4 } }$ 
B)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 2 - \frac { 54 } { x ^ { 4 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = 2 \mathrm { x } - \frac { 18 } { \mathrm { x } ^ { 3 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } \mathrm { x } ^ { 2 } } = 1 + \frac { 54 } { \mathrm { x } ^ { 4 } }$ 
A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 2 + \frac { 54 } { x ^ { 4 } }$ 
B)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = 2 - \frac { 54 } { x ^ { 4 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = 2 \mathrm { x } - \frac { 18 } { \mathrm { x } ^ { 3 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } \mathrm { x } ^ { 2 } } = 1 + \frac { 54 } { \mathrm { x } ^ { 4 } }$

Question 11

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$
-$$f ( x ) = \frac { 5 } { x + 6 }$$

Accepted Answer

A)  $\frac { 5 } { ( x + 6 ) ^ { 2 } }$ 
B)  $- \frac { 5 } { ( x + 6 ) ^ { 2 } }$ 
C)  $- \frac { 5 } { ( x + 6 ) }$ 
D)  $- \frac { 5 } { x ^ { 2 } }$ 
A)  $\frac { 5 } { ( x + 6 ) ^ { 2 } }$ 
B)  $- \frac { 5 } { ( x + 6 ) ^ { 2 } }$ 
C)  $- \frac { 5 } { ( x + 6 ) }$ 
D)  $- \frac { 5 } { x ^ { 2 } }$

Question 12

Provide an appropriate response.
-Evaluate $\lim _ { x \rightarrow 1 } \frac { x ^ { 33 } - 1 } { x - 1 } =$ by first converting it to a derivative at a particular $x$-value.

Accepted Answer

A)  32 
B)  33 
C)  $\frac { 1 } { 33 }$ 
D)  34 
A)  32 
B)  33 
C)  $\frac { 1 } { 33 }$ 
D)  34

Question 13

Find the derivatives of all orders of the function.
-$$y = ( x + 1 ) \left( x ^ { 2 } - 4 x + 8 \right)$$

Accepted Answer

The answer of Find the derivatives of all orders of...

Question 14

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

Accepted Answer

A)  x = 0 
B)  x = -2, 0, 2 
C)  x = -2, 2 
D)  Defined for all values of x 
A)  x = 0 
B)  x = -2, 0, 2 
C)  x = -2, 2 
D)  Defined for all values of x

Question 15

Provide an appropriate response.
-Find an equation for the tangent to the curve $y = \frac { 8 x } { x ^ { 2 } + 1 }$ at the point $( 1,4 )$.

Accepted Answer

A)  $y = x + 4$ 
B)  $y = 0$ 
C)  $y = 4$ 
D)  $y = 4 x$ 
A)  $y = x + 4$ 
B)  $y = 0$ 
C)  $y = 4$ 
D)  $y = 4 x$

Question 16

Provide an appropriate response.
-$\text { What is the range of values of the slope of the curve } y = x ^ { 3 } + 5 x - 2 \text { ? }$

Accepted Answer

The answer of Provide an appropriate response.
-\(\text { What is...

Question 17

Find the derivative of the function.
-$$y = \frac { ( x - 10 ) \left( x ^ { 2 } + 4 x \right) } { x ^ { 3 } }$$

Accepted Answer

A)  $\frac { d y } { d x } = x - \frac { 80 } { x ^ { 2 } } - \frac { 80 } { x ^ { 3 } }$ 
B)  $\frac { d y } { d x } = \frac { 6 } { x ^ { 2 } } + \frac { 80 } { x ^ { 3 } }$ 
C)  $\frac { d y } { d x } = \frac { 14 } { x ^ { 2 } } - \frac { 80 } { x ^ { 3 } }$ 
D)  $\frac { \mathrm { dy } } { \mathrm { dx } } = 80 + \frac { 80 } { \mathrm { x } }$ 
A)  $\frac { d y } { d x } = x - \frac { 80 } { x ^ { 2 } } - \frac { 80 } { x ^ { 3 } }$ 
B)  $\frac { d y } { d x } = \frac { 6 } { x ^ { 2 } } + \frac { 80 } { x ^ { 3 } }$ 
C)  $\frac { d y } { d x } = \frac { 14 } { x ^ { 2 } } - \frac { 80 } { x ^ { 3 } }$ 
D)  $\frac { \mathrm { dy } } { \mathrm { dx } } = 80 + \frac { 80 } { \mathrm { x } }$

Question 18

Differentiate the function and find the slope of the tangent line at the given value of the independent variable.
-$$g ( x ) = \frac { 5 } { 8 + x } , x = 3$$

Accepted Answer

A)  $- \frac { 5 } { 11 }$ 
B)  $- \frac { 5 } { 121 }$ 
C)  $\frac { 5 } { 11 }$ 
D)  $\frac { 5 } { 121 }$ 
A)  $- \frac { 5 } { 11 }$ 
B)  $- \frac { 5 } { 121 }$ 
C)  $\frac { 5 } { 11 }$ 
D)  $\frac { 5 } { 121 }$

Question 19

The graph of a function is given. Choose the answer that represents the graph of its derivative
-

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 20

Find y'.
-$$y = \left( \frac { 1 } { x ^ { 2 } } + 5 \right) \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + 5 \right)$$

Accepted Answer

A)  $- \frac { 4 } { x ^ { 5 } } - 10 x$ 
B)  $- \frac { 1 } { x ^ { 5 } } + 10 x$ 
C)  $\frac { 4 } { x ^ { 3 } } + 10 x$ 
D)  $\frac { 4 } { \mathrm { x } ^ { 5 } } + 10 \mathrm { x }$ 
A)  $- \frac { 4 } { x ^ { 5 } } - 10 x$ 
B)  $- \frac { 1 } { x ^ { 5 } } + 10 x$ 
C)  $\frac { 4 } { x ^ { 3 } } + 10 x$ 
D)  $\frac { 4 } { \mathrm { x } ^ { 5 } } + 10 \mathrm { x }$

Find the slope of the curve at the indicated point. - $y = x ^ { 2 } + 10 x , x = - 4$

Find the second derivative. - $\mathrm { s } = \frac { 17 \mathrm { t } ^ { 3 } } { 3 } + 17$

Find the derivative of the function. - $s = \frac { t ^ { 7 } + 3 t + 6 } { t ^ { 2 } }$

Find the derivatives of all orders of the function. - $y = \frac { 8 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 13$

Find the indicated derivative. - $\frac { d y } { d x } \text { if } y = 3 x ^ { 3 }$

Find the second derivative of the function. - $p = \left( \frac { q + 3 } { q } \right) \left( \frac { q + 5 } { q ^ { 2 } } \right)$

Find the derivative. - $y = x ^ { 1 / 5 }$

Find the second derivative of the function. - $y = \frac { x ^ { 4 } + 9 } { x ^ { 2 } }$

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$ - $f ( x ) = \frac { 5 } { x + 6 }$

Provide an appropriate response. -Evaluate $\lim _ { x \rightarrow 1 } \frac { x ^ { 33 } - 1 } { x - 1 } =$ by first converting it to a derivative at a particular $x$ -value.

Find the derivatives of all orders of the function. - $y = ( x + 1 ) \left( x ^ { 2 } - 4 x + 8 \right)$

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

Provide an appropriate response. -Find an equation for the tangent to the curve $y = \frac { 8 x } { x ^ { 2 } + 1 }$ at the point $( 1,4 )$ .

Provide an appropriate response. - $\text { What is the range of values of the slope of the curve } y = x ^ { 3 } + 5 x - 2 \text { ? }$

Find the derivative of the function. - $y = \frac { ( x - 10 ) \left( x ^ { 2 } + 4 x \right) } { x ^ { 3 } }$

Differentiate the function and find the slope of the tangent line at the given value of the independent variable. - $g ( x ) = \frac { 5 } { 8 + x } , x = 3$

The graph of a function is given. Choose the answer that represents the graph of its derivative -

Find y'. - $y = \left( \frac { 1 } { x ^ { 2 } } + 5 \right) \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + 5 \right)$

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

Exam 3: Differentiation

Find the slope of the curve at the indicated point. - y=x2+10x,x=−4y = x ^ { 2 } + 10 x , x = - 4y=x2+10x,x=−4

Find the second derivative. - s=17t33+17\mathrm { s } = \frac { 17 \mathrm { t } ^ { 3 } } { 3 } + 17s=317t3​+17

Find the derivative of the function. - s=t7+3t+6t2s = \frac { t ^ { 7 } + 3 t + 6 } { t ^ { 2 } }s=t2t7+3t+6​

Find the derivatives of all orders of the function. - y=83x3+52x2−3x+13y = \frac { 8 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 13y=38​x3+25​x2−3x+13

Find the indicated derivative. - dydx if y=3x3\frac { d y } { d x } \text { if } y = 3 x ^ { 3 }dxdy​ if y=3x3

Find the second derivative of the function. - p=(q+3q)(q+5q2)p = \left( \frac { q + 3 } { q } \right) \left( \frac { q + 5 } { q ^ { 2 } } \right)p=(qq+3​)(q2q+5​)

Find the derivative. - y=x1/5y = x ^ { 1 / 5 }y=x1/5

Find the second derivative of the function. - y=x4+9x2y = \frac { x ^ { 4 } + 9 } { x ^ { 2 } }y=x2x4+9​

Provide an appropriate response. -Evaluate lim⁡x→1x33−1x−1=\lim _ { x \rightarrow 1 } \frac { x ^ { 33 } - 1 } { x - 1 } =limx→1​x−1x33−1​= by first converting it to a derivative at a particular xxx -value.

Find the derivatives of all orders of the function. - y=(x+1)(x2−4x+8)y = ( x + 1 ) \left( x ^ { 2 } - 4 x + 8 \right)y=(x+1)(x2−4x+8)