Question 1

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

Accepted Answer

A)  $g ^ { \prime } ( x ) = - \frac { 2 } { x ^ { 2 } } ; g ^ { \prime } ( - 2 ) = - \frac { 1 } { 2 }$ 
B)  $g ^ { \prime } ( x ) = \frac { 2 } { x ^ { 2 } } ; g ^ { \prime } ( - 2 ) = \frac { 1 } { 2 }$ 
C)  $g ^ { \prime } ( x ) = - 2 x ^ { 2 } ; g ^ { \prime } ( - 2 ) = - 8$ 
D)  $g ^ { \prime } ( x ) = - 2 ; g ^ { \prime } ( - 2 ) = - 2$ 
A)  $g ^ { \prime } ( x ) = - \frac { 2 } { x ^ { 2 } } ; g ^ { \prime } ( - 2 ) = - \frac { 1 } { 2 }$ 
B)  $g ^ { \prime } ( x ) = \frac { 2 } { x ^ { 2 } } ; g ^ { \prime } ( - 2 ) = \frac { 1 } { 2 }$ 
C)  $g ^ { \prime } ( x ) = - 2 x ^ { 2 } ; g ^ { \prime } ( - 2 ) = - 8$ 
D)  $g ^ { \prime } ( x ) = - 2 ; g ^ { \prime } ( - 2 ) = - 2$

Question 2

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

Accepted Answer

A)  x = 5 
B)  x = 2 
C)  x = 2, 5 
D)  Defined for all values of x 
A)  x = 5 
B)  x = 2 
C)  x = 2, 5 
D)  Defined for all values of x

Question 3

Find the derivative.
-$$w = z ^ { - 3 } - \frac { 1 } { z }$$

Accepted Answer

A)  $3 z ^ { - 4 } - \frac { 1 } { z ^ { 2 } }$ 
B)  $- 3 z ^ { - 4 } - \frac { 1 } { z ^ { 2 } }$ 
C)  $- 3 z ^ { - 4 } + \frac { 1 } { z ^ { 2 } }$ 
D)  $z ^ { - 4 } + \frac { 1 } { z ^ { 2 } }$ 
A)  $3 z ^ { - 4 } - \frac { 1 } { z ^ { 2 } }$ 
B)  $- 3 z ^ { - 4 } - \frac { 1 } { z ^ { 2 } }$ 
C)  $- 3 z ^ { - 4 } + \frac { 1 } { z ^ { 2 } }$ 
D)  $z ^ { - 4 } + \frac { 1 } { z ^ { 2 } }$

Question 4

Provide an appropriate response.
-$\text { Does the curve } y = \sqrt { x } \text { ever have a negative slope? If so, where? Give reasons for your answer. }$

Accepted Answer

The curve @#LAT-DLM& never has a negative slope. T

Question 5

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable?
-$x = 0$

Accepted Answer

A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable 
A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable

Question 6

Provide an appropriate response.
-$$\text { Does the graph of } f ( x ) = \left\{ \begin{array} { l l } 
x \cos \frac { 1 } { x ^ { \prime } } & x \neq 0 \
0 , & x = 0
\end{array} \right. \text { have a tangent at the origin? Give reasons for your answer. }$$

Accepted Answer

The function f is no

Question 7

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable?
-$$x = - 1$$

Accepted Answer

A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable 
A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable

Question 8

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
-  $ y=x^{2} \quad y=\sqrt{x} $

Accepted Answer

A)  Since $\lim _ { x \rightarrow1  ^ { + } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2$ while $\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 1 } { 2 } , \mathrm { f } ( \mathrm { x } )$ is not differentiable at $x = 1$. 
B)  Since $\lim _ { x \rightarrow 1^ { + } } \mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { 2 }$ while  $\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 1 , \mathrm { f } ( \mathrm { x } )$ is not differentiable at $x = 1$. 
C)  Since $\lim _ { x \rightarrow1 ^ { + } } f ^ { \prime } ( x ) = \frac { 1 } { 2 }$ while  $\lim _ { x \rightarrow 1 ^ { - } }f ^ { \prime } ( x ) = 2 , f ( x )$ is not differentiable at $x = 1$. 
D)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 2$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )$ is differentiable at $x = 1$. 
A)  Since $\lim _ { x \rightarrow1  ^ { + } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2$ while $\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 1 } { 2 } , \mathrm { f } ( \mathrm { x } )$ is not differentiable at $x = 1$. 
B)  Since $\lim _ { x \rightarrow 1^ { + } } \mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { 2 }$ while  $\lim _ { x \rightarrow 1 ^ { - } } \mathrm { f } ^ { \prime } ( \mathrm { x } ) = 1 , \mathrm { f } ( \mathrm { x } )$ is not differentiable at $x = 1$. 
C)  Since $\lim _ { x \rightarrow1 ^ { + } } f ^ { \prime } ( x ) = \frac { 1 } { 2 }$ while  $\lim _ { x \rightarrow 1 ^ { - } }f ^ { \prime } ( x ) = 2 , f ( x )$ is not differentiable at $x = 1$. 
D)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 2$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )$ is differentiable at $x = 1$.

Question 9

Provide an appropriate response.
-Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.

Accepted Answer

Yes, a tangent line to a graph

Question 10

Find the derivative of the function.
-$$p = \left( \frac { q ^ { 8 } + 4 } { 2 q } \right) \left( \frac { q ^ { 7 } + 6 } { q } \right)$$

Accepted Answer

A)  $\frac { \mathrm { dp } } { \mathrm { dq } } = \frac { 13 } { 2 } \mathrm { q } ^ { 12 } - \frac { 24 } { \mathrm { q } ^ { 3 } }$ 
B)  $\frac { d p } { d q } = \frac { 1 } { 2 } q ^ { 12 } + 2 q ^ { 4 } + 3 q ^ { 5 } + \frac { 24 } { q ^ { 3 } }$ 
C)  $\frac { d p } { d q } = \frac { 13 } { 2 } q ^ { 12 } + 10 q ^ { 4 } + 18 q ^ { 5 } - \frac { 24 } { q ^ { 3 } }$ 
D)  $\frac { d p } { d q } = \frac { 17 } { 2 } q ^ { 16 } + 18 q ^ { 8 } + 30 q ^ { 9 } - \frac { 24 } { q ^ { 3 } }$ 
A)  $\frac { \mathrm { dp } } { \mathrm { dq } } = \frac { 13 } { 2 } \mathrm { q } ^ { 12 } - \frac { 24 } { \mathrm { q } ^ { 3 } }$ 
B)  $\frac { d p } { d q } = \frac { 1 } { 2 } q ^ { 12 } + 2 q ^ { 4 } + 3 q ^ { 5 } + \frac { 24 } { q ^ { 3 } }$ 
C)  $\frac { d p } { d q } = \frac { 13 } { 2 } q ^ { 12 } + 10 q ^ { 4 } + 18 q ^ { 5 } - \frac { 24 } { q ^ { 3 } }$ 
D)  $\frac { d p } { d q } = \frac { 17 } { 2 } q ^ { 16 } + 18 q ^ { 8 } + 30 q ^ { 9 } - \frac { 24 } { q ^ { 3 } }$

Question 11

Find the second derivative of the function.
-$$y = \frac { ( x - 6 ) \left( x ^ { 2 } + 3 x \right) } { x ^ { 3 } }$$

Accepted Answer

A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 6 } { x } - \frac { 108 } { x ^ { 2 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = \frac { 3 } { x ^ { 2 } } + \frac { 36 } { x ^ { 3 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - \frac { 6 } { x ^ { 3 } } - \frac { 108 } { x ^ { 4 } }$ 
D)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 6 } { x ^ { 3 } } + \frac { 108 } { x ^ { 4 } }$ 
A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 6 } { x } - \frac { 108 } { x ^ { 2 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = \frac { 3 } { x ^ { 2 } } + \frac { 36 } { x ^ { 3 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = - \frac { 6 } { x ^ { 3 } } - \frac { 108 } { x ^ { 4 } }$ 
D)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 6 } { x ^ { 3 } } + \frac { 108 } { x ^ { 4 } }$

Question 12

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

Accepted Answer

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

Question 13

Find the second derivative of the function.
-$$y = \frac { 3 - 2 x ^ { 3 } + x ^ { 5 } } { x ^ { 9 } }$$

Accepted Answer

A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } \mathrm { x } ^ { 2 } } = \frac { 270 } { \mathrm { x } ^ { 12 } } + \frac { 84 } { \mathrm { x } ^ { 5 } } - \frac { 20 } { \mathrm { x } ^ { 3 } }$ 
B)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 27 } { x ^ { 10 } } + \frac { 12 } { x ^ { 7 } } - \frac { 4 } { x ^ { 5 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { 27 } { \mathrm { x } ^ { 11 } } + \frac { 12 } { \mathrm { x } ^ { 6 } } - \frac { 4 } { \mathrm { x } ^ { 4 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { d \mathrm { x } ^ { 2 } } = \frac { 270 } { \mathrm { x } ^ { 11 } } - \frac { 84 } { \mathrm { x } ^ { 8 } } + \frac { 20 } { \mathrm { x } ^ { 6 } }$ 
A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } \mathrm { x } ^ { 2 } } = \frac { 270 } { \mathrm { x } ^ { 12 } } + \frac { 84 } { \mathrm { x } ^ { 5 } } - \frac { 20 } { \mathrm { x } ^ { 3 } }$ 
B)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 27 } { x ^ { 10 } } + \frac { 12 } { x ^ { 7 } } - \frac { 4 } { x ^ { 5 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } } = \frac { 27 } { \mathrm { x } ^ { 11 } } + \frac { 12 } { \mathrm { x } ^ { 6 } } - \frac { 4 } { \mathrm { x } ^ { 4 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { d \mathrm { x } ^ { 2 } } = \frac { 270 } { \mathrm { x } ^ { 11 } } - \frac { 84 } { \mathrm { x } ^ { 8 } } + \frac { 20 } { \mathrm { x } ^ { 6 } }$

Question 14

Provide an appropriate response.
-A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engin defined by
$e = \frac { Q _ { h } - Q _ { c } } { Q _ { h } }$ where $\mathrm { Q } _ { \mathrm { h } }$ is the heat absorbed in one cycle and $\mathrm { Q } _ { \mathrm { C } }$, the heat released into a reservoir in one cycle, is a constant.
Find $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } ^ { 2 } }$.

Accepted Answer

A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { \mathrm { h } ^ { 2 } } } = \frac { - \mathrm { Q } _ { \mathrm { C } } } { 2 \mathrm { Q } _ { \mathrm { h } ^ { 2 } } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { h ^ { 2 } } } = \frac { \mathrm { Q } _ { \mathrm { c } } } { \mathrm { Qh } ^ { 3 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { h ^ { 2 } } } = \frac { \mathrm { Q } _ { \mathrm { c } } } { \mathrm { Qh } ^ { 2 } }$ 
D)  $\frac { d ^ { 2 } e } { d Q _ { h } ^ { 2 } } = \frac { - 2 Q _ { c } } { Q _ { h } ^ { 3 } }$ 
A)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { \mathrm { h } ^ { 2 } } } = \frac { - \mathrm { Q } _ { \mathrm { C } } } { 2 \mathrm { Q } _ { \mathrm { h } ^ { 2 } } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { h ^ { 2 } } } = \frac { \mathrm { Q } _ { \mathrm { c } } } { \mathrm { Qh } ^ { 3 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm { e } } { \mathrm { dQ } _ { h ^ { 2 } } } = \frac { \mathrm { Q } _ { \mathrm { c } } } { \mathrm { Qh } ^ { 2 } }$ 
D)  $\frac { d ^ { 2 } e } { d Q _ { h } ^ { 2 } } = \frac { - 2 Q _ { c } } { Q _ { h } ^ { 3 } }$

Question 15

Find the derivative of the function.
-$$f ( t ) = ( 6 - t ) \left( 6 + t ^ { 3 } \right) ^ { - 1 }$$

Accepted Answer

A)  $f ^ { \prime } ( t ) = \frac { - 4 t ^ { 3 } + 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
B)  $f ^ { \prime } ( t ) = \frac { 2 t ^ { 3 } - 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
C)  $f ^ { \prime } ( t ) = \frac { - 2 t ^ { 3 } + 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
D)  $f ^ { \prime } ( t ) = \frac { 2 t ^ { 3 } - 18 t ^ { 2 } - 6 } { 6 + t ^ { 3 } }$ 
A)  $f ^ { \prime } ( t ) = \frac { - 4 t ^ { 3 } + 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
B)  $f ^ { \prime } ( t ) = \frac { 2 t ^ { 3 } - 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
C)  $f ^ { \prime } ( t ) = \frac { - 2 t ^ { 3 } + 18 t ^ { 2 } - 6 } { \left( 6 + t ^ { 3 } \right) ^ { 2 } }$ 
D)  $f ^ { \prime } ( t ) = \frac { 2 t ^ { 3 } - 18 t ^ { 2 } - 6 } { 6 + t ^ { 3 } }$

Question 16

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable?
-$x = 0$

Accepted Answer

A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable 
A)  Differentiable 
B)  Continuous but not differentiable 
C)  Neither continuous nor differentiable

Question 17

Find the derivative of the function.
-$$y = \frac { x ^ { 2 } - 3 x + 2 } { x ^ { 7 } - 2 }$$

Accepted Answer

A)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 19 x ^ { 7 } - 14 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 13 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 14 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
D)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 14 x ^ { 6 } - 3 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
A)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 19 x ^ { 7 } - 14 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
B)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 13 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
C)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 14 x ^ { 6 } - 4 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$ 
D)  $y ^ { \prime } = \frac { - 5 x ^ { 8 } + 18 x ^ { 7 } - 14 x ^ { 6 } - 3 x + 6 } { \left( x ^ { 7 } - 2 \right) ^ { 2 } }$

Question 18

Determine if the piecewise defined function is differentiable at the origin.
-$$g ( x ) = \left\{ \begin{array} { l l l } 
x ^ { 7 / 8 } & \text { if } x \geq 0 & \
x ^ { 1 / 8 } & & \text { if } x < 0
\end{array} \right.$$

Accepted Answer

A)  Differentiable 
B)  Not differentiable 
A)  Differentiable 
B)  Not differentiable

Question 19

Graph the equation and its tangent
-$$\text { Graph } y = x ^ { 2 } - 2 x - 3 \text { and the tangent to the curve at the point whose } x \text {-coordinate is } 1 \text {. }$$

Accepted Answer

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

Question 20

Find y'.
-$$y = ( 4 x - 4 ) \left( 2 x ^ { 3 } - x ^ { 2 } + 1 \right)$$

Accepted Answer

A)  $8 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 4$ 
B)  $32 x ^ { 3 } - 36 x ^ { 2 } + 8 x + 4$ 
C)  $24 x ^ { 3 } + 36 x ^ { 2 } - 12 x + 4$ 
D)  $32 x ^ { 3 } - 12 x ^ { 2 } + 36 x + 4$ 
A)  $8 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 4$ 
B)  $32 x ^ { 3 } - 36 x ^ { 2 } + 8 x + 4$ 
C)  $24 x ^ { 3 } + 36 x ^ { 2 } - 12 x + 4$ 
D)  $32 x ^ { 3 } - 12 x ^ { 2 } + 36 x + 4$

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

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

Find the derivative. - $w = z ^ { - 3 } - \frac { 1 } { z }$

Provide an appropriate response. - $\text { Does the curve } y = \sqrt { x } \text { ever have a negative slope? If so, where? Give reasons for your answer. }$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = 0$

Provide an appropriate response. - $\text { Does the graph of } f ( x ) = \left\{ \begin{array} { l l } x \cos \frac { 1 } { x ^ { \prime } } & x \neq 0 \\0 , & x = 0\end{array} \right. \text { have a tangent at the origin? Give reasons for your answer. }$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = - 1$

Provide an appropriate response. -Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.

Find the derivative of the function. - $p = \left( \frac { q ^ { 8 } + 4 } { 2 q } \right) \left( \frac { q ^ { 7 } + 6 } { q } \right)$

Find the second derivative of the function. - $y = \frac { ( x - 6 ) \left( x ^ { 2 } + 3 x \right) } { x ^ { 3 } }$

Find the second derivative of the function. - $y = \frac { 3 - 2 x ^ { 3 } + x ^ { 5 } } { x ^ { 9 } }$

Find the derivative of the function. - $f ( t ) = ( 6 - t ) \left( 6 + t ^ { 3 } \right) ^ { - 1 }$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = 0$

Find the derivative of the function. - $y = \frac { x ^ { 2 } - 3 x + 2 } { x ^ { 7 } - 2 }$

Determine if the piecewise defined function is differentiable at the origin. - $g ( x ) = \left\{ \begin{array} { l l l } x ^ { 7 / 8 } & \text { if } x \geq 0 & \\x ^ { 1 / 8 } & & \text { if } x < 0\end{array} \right.$

Find y'. - $y = ( 4 x - 4 ) \left( 2 x ^ { 3 } - x ^ { 2 } + 1 \right)$

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Exam 3: Differentiation

Calculate the derivative of the function. Then find the value of the derivative as specified. - g(x)=−2x;g′(−2)g ( x ) = - \frac { 2 } { x } ; g ^ { \prime } ( - 2 )g(x)=−x2​;g′(−2)

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

Find the derivative. - w=z−3−1zw = z ^ { - 3 } - \frac { 1 } { z }w=z−3−z1​

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - x=0x = 0x=0

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - x=−1x = - 1x=−1

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. - y=x2y=x y=x^{2} \quad y=\sqrt{x} y=x2y=x​

Provide an appropriate response. -Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.

Find the derivative of the function. - p=(q8+42q)(q7+6q)p = \left( \frac { q ^ { 8 } + 4 } { 2 q } \right) \left( \frac { q ^ { 7 } + 6 } { q } \right)p=(2qq8+4​)(qq7+6​)

Find the second derivative of the function. - y=(x−6)(x2+3x)x3y = \frac { ( x - 6 ) \left( x ^ { 2 } + 3 x \right) } { x ^ { 3 } }y=x3(x−6)(x2+3x)​

Find the second derivative of the function. - y=3−2x3+x5x9y = \frac { 3 - 2 x ^ { 3 } + x ^ { 5 } } { x ^ { 9 } }y=x93−2x3+x5​

Find the derivative of the function. - f(t)=(6−t)(6+t3)−1f ( t ) = ( 6 - t ) \left( 6 + t ^ { 3 } \right) ^ { - 1 }f(t)=(6−t)(6+t3)−1

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - x=0x = 0x=0

Find the derivative of the function. - y=x2−3x+2x7−2y = \frac { x ^ { 2 } - 3 x + 2 } { x ^ { 7 } - 2 }y=x7−2x2−3x+2​

Determine if the piecewise defined function is differentiable at the origin. - g(x)={x7/8 if x≥0x1/8 if x<0g ( x ) = \left\{ \begin{array} { l l l } x ^ { 7 / 8 } & \text { if } x \geq 0 & \\x ^ { 1 / 8 } & & \text { if } x < 0\end{array} \right.g(x)={x7/8x1/8​ if x≥0​ if x<0​

Find y'. - y=(4x−4)(2x3−x2+1)y = ( 4 x - 4 ) \left( 2 x ^ { 3 } - x ^ { 2 } + 1 \right)y=(4x−4)(2x3−x2+1)

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

Find the derivative. - $w = z ^ { - 3 } - \frac { 1 } { z }$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = 0$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = - 1$

Find the derivative of the function. - $p = \left( \frac { q ^ { 8 } + 4 } { 2 q } \right) \left( \frac { q ^ { 7 } + 6 } { q } \right)$

Find the second derivative of the function. - $y = \frac { ( x - 6 ) \left( x ^ { 2 } + 3 x \right) } { x ^ { 3 } }$

Find the second derivative of the function. - $y = \frac { 3 - 2 x ^ { 3 } + x ^ { 5 } } { x ^ { 9 } }$

Find the derivative of the function. - $f ( t ) = ( 6 - t ) \left( 6 + t ^ { 3 } \right) ^ { - 1 }$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = 0$

Find the derivative of the function. - $y = \frac { x ^ { 2 } - 3 x + 2 } { x ^ { 7 } - 2 }$

Determine if the piecewise defined function is differentiable at the origin. - $g ( x ) = \left\{ \begin{array} { l l l } x ^ { 7 / 8 } & \text { if } x \geq 0 & \\x ^ { 1 / 8 } & & \text { if } x < 0\end{array} \right.$

Find y'. - $y = ( 4 x - 4 ) \left( 2 x ^ { 3 } - x ^ { 2 } + 1 \right)$