Question 1

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

Accepted Answer

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

Question 2

Find the derivative.
-$$y = 14 - 12 x ^ { 2 }$$

Accepted Answer

A)  -24 
B)  14 - 24x 
C)  14 - 12x 
D)  -24x 
A)  -24 
B)  14 - 24x 
C)  14 - 12x 
D)  -24x

Question 3

Find the derivative of the function.
-$$y = \frac { 3 - 2 x ^ { 3 } + x ^ { 6 } } { x ^ { 9 } }$$

Accepted Answer

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

Question 4

Find the slope of the curve at the indicated point.
-The power $\mathrm { P }$ (in $\mathrm { W }$ ) generated by a particular windmill is given by $\mathrm { P } = 0.015 \mathrm {~V} ^ { 3 }$ where $\mathrm { V }$ is the velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is $9.0 \mathrm { mph }$.

Accepted Answer

A)  8.1 W/mph 
B)  3.6 W/mph 
C)  0.4 W/mph 
D)  21.9 W/mph 
A)  8.1 W/mph 
B)  3.6 W/mph 
C)  0.4 W/mph 
D)  21.9 W/mph

Question 5

Find the derivatives of all orders of the function.
-$$y = \frac { x ^ { 7 } } { 15,120 }$$

Accepted Answer

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

Question 6

Provide an appropriate response.
-Find the slope of the tangent to the curve $y = \sqrt { x }$ at the point where $x = 16$.

Accepted Answer

A)  $\mathrm { m } = - \frac { 1 } { 8 }$ 
B)  $\mathrm { m } = \frac { 1 } { 8 }$ 
C)  $m = \frac { 1 } { 20 }$ 
D)  $m = - \frac { 1 } { 20 }$ 
A)  $\mathrm { m } = - \frac { 1 } { 8 }$ 
B)  $\mathrm { m } = \frac { 1 } { 8 }$ 
C)  $m = \frac { 1 } { 20 }$ 
D)  $m = - \frac { 1 } { 20 }$

Question 7

Find the slope of the curve at the indicated point.
-$$y = \frac { 6 } { 5 + x } , x = 9$$

Accepted Answer

A)  $m = \frac { 3 } { 7 }$ 
B)  $m = - \frac { 3 } { 98 }$ 
C)  $\mathrm { m } = \frac { 3 } { 98 }$ 
D)  $m = - \frac { 3 } { 7 }$ 
A)  $m = \frac { 3 } { 7 }$ 
B)  $m = - \frac { 3 } { 98 }$ 
C)  $\mathrm { m } = \frac { 3 } { 98 }$ 
D)  $m = - \frac { 3 } { 7 }$

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.
-  $$y = x \quad y = 2 x$$

Accepted Answer

A)  Since $\lim _ { x \rightarrow0 ^ { + } } f ^ { \prime } ( x ) = - 2$ while $\lim _ { x \rightarrow 0 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = 0$. 
B)  Since $\lim _ { x \rightarrow0 ^ { + } } f ^ { \prime } ( x ) = 2$ while $\lim _ { x \rightarrow - ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is not differentiable at $x = 0$. 
C)  Since $\lim _ { x \rightarrow 0^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )$ is not differentiable at $x = 0$. 
D)  Since $\lim _ { x \rightarrow0^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is differentiable at $x = 0$. 
A)  Since $\lim _ { x \rightarrow0 ^ { + } } f ^ { \prime } ( x ) = - 2$ while $\lim _ { x \rightarrow 0 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = 0$. 
B)  Since $\lim _ { x \rightarrow0 ^ { + } } f ^ { \prime } ( x ) = 2$ while $\lim _ { x \rightarrow - ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is not differentiable at $x = 0$. 
C)  Since $\lim _ { x \rightarrow 0^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )$ is not differentiable at $x = 0$. 
D)  Since $\lim _ { x \rightarrow0^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is differentiable at $x = 0$.

Question 9

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.
-$\begin{array} { l } 
u ( 1 ) = 5 , u ^ { \prime } ( 1 ) = - 7 , v ( 1 ) = 6 , v ^ { \prime } ( 1 ) = - 3 \
\frac { d } { d x } ( 2 u - 4 v ) \text { at } x = 1
\end{array}$

Accepted Answer

A)  - 26 
B)  34 
C)  - 2 
D)  - 14 
A)  - 26 
B)  34 
C)  - 2 
D)  - 14

Question 10

Provide an appropriate response.
-$$\text { Does the graph of } f ( x ) = \left\{ \begin{array} { l l } 
0 , & x > 0 \
5 , & x \leq 0
\end{array} \text { have a vertical tangent at the point } ( 0,5 ) ? \right. \text { Give reasons for your answer. }$$

Accepted Answer

The function is not differenti

Question 11

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$
-$f ( x ) = 4 x ^ { 2 } + 3 x + 5$

Accepted Answer

A)  $4 x + 3$ 
B)  $8 x ^ { 2 } + 3 x$ 
C)  $8 x + 3$ 
D)  $8 x$ 
A)  $4 x + 3$ 
B)  $8 x ^ { 2 } + 3 x$ 
C)  $8 x + 3$ 
D)  $8 x$

Question 12

Differentiate the function and find the slope of the tangent line at the given value of the independent variable.
-$$\mathrm { s } = 2 \mathrm { t } ^ { 4 } - 2 \mathrm { t } ^ { 3 } , \mathrm { t } = - 1$$

Accepted Answer

A)  14 
B)  $- 14$ 
C)  $- 2$ 
D)  2 
A)  14 
B)  $- 14$ 
C)  $- 2$ 
D)  2

Question 13

Find y'.
-$$y = \left( 5 x ^ { 3 } + 3 \right) \left( 2 x ^ { 7 } - 5 \right)$$

Accepted Answer

A)  $100 x ^ { 9 } + 42 x ^ { 6 } - 75 x$ 
B)  $20 x ^ { 9 } + 42 x ^ { 6 } - 75 x ^ { 2 }$ 
C)  $100 x ^ { 9 } + 42 x ^ { 6 } - 75 x ^ { 2 }$ 
D)  $20 x ^ { 9 } + 42 x ^ { 6 } - 75 x$ 
A)  $100 x ^ { 9 } + 42 x ^ { 6 } - 75 x$ 
B)  $20 x ^ { 9 } + 42 x ^ { 6 } - 75 x ^ { 2 }$ 
C)  $100 x ^ { 9 } + 42 x ^ { 6 } - 75 x ^ { 2 }$ 
D)  $20 x ^ { 9 } + 42 x ^ { 6 } - 75 x$

Question 14

Find the second derivative.
-$$y = 8 x ^ { 3 } - 4 x ^ { 2 } + 5 e ^ { x }$$

Accepted Answer

A)  $8 \mathrm { x } - 32 + 5 \mathrm { e } ^ { \mathrm { x } }$ 
B)  $32 x - 8 + 5 e ^ { x }$ 
C)  $48 x - 8 + 5 e ^ { x }$ 
D)  $8 x - 48 + 5 e ^ { x }$ 
A)  $8 \mathrm { x } - 32 + 5 \mathrm { e } ^ { \mathrm { x } }$ 
B)  $32 x - 8 + 5 e ^ { x }$ 
C)  $48 x - 8 + 5 e ^ { x }$ 
D)  $8 x - 48 + 5 e ^ { x }$

Question 15

Provide an appropriate response.
-Find all points $( x , y )$ on the graph of $y = \frac { x } { ( x - 5 ) }$ with tangent lines perpendicular to the line $y = 5 x - 4$.

Accepted Answer

A)  $( 10,2 )$ 
B)  $( 0,0 ) , ( 5,2 )$ 
C)  $( 0,0 )$ 
D)  $( 0,0 ) , ( 10,2 )$ 
A)  $( 10,2 )$ 
B)  $( 0,0 ) , ( 5,2 )$ 
C)  $( 0,0 )$ 
D)  $( 0,0 ) , ( 10,2 )$

Question 16

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
-  $$y = \frac { 1 } { x } \quad y = - 1$$

Accepted Answer

A)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 1$, $f ( x )$ is not differentiable at $x = - 1$. 
B)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = - 1$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0 , f ( x )$ is not differentiable at $x = - 1$. 
C)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = - 1$. 
D)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0$, $f ( x )$ is differentiable at $x = - 1$. 
A)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 1$, $f ( x )$ is not differentiable at $x = - 1$. 
B)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = - 1$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0 , f ( x )$ is not differentiable at $x = - 1$. 
C)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = - 1$. 
D)  Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0$, $f ( x )$ is differentiable at $x = - 1$.

Question 17

Find the derivative.
-$$y = 7 x ^ { - 2 } - 5 x ^ { 3 } + 2 x$$

Accepted Answer

A)  $- 14 x ^ { - 1 } - 15 x ^ { 2 }$ 
B)  $- 14 x ^ { - 3 } - 15 x ^ { 2 }$ 
C)  $- 14 x ^ { - 3 } - 15 x ^ { 2 } + 2$ 
D)  $- 14 x ^ { - 1 } - 15 x ^ { 2 } + 2$ 
A)  $- 14 x ^ { - 1 } - 15 x ^ { 2 }$ 
B)  $- 14 x ^ { - 3 } - 15 x ^ { 2 }$ 
C)  $- 14 x ^ { - 3 } - 15 x ^ { 2 } + 2$ 
D)  $- 14 x ^ { - 1 } - 15 x ^ { 2 } + 2$

Question 18

Calculate the derivative of the function. Then find the value of the derivative as specified.
-$$f ( x ) = 5 x + 9 ; f ^ { \prime } ( 2 )$$

Accepted Answer

A)  $f ^ { \prime } ( x ) = 5 ; f ^ { \prime } ( 2 ) = 5$ 
B)  $f ^ { \prime } ( x ) = 9 ; f ^ { \prime } ( 2 ) = 9$ 
C)  $f ^ { \prime } ( x ) = 0 ; f ^ { \prime } ( 2 ) = 0$ 
D)  $f ^ { \prime } ( x ) = 5 x ; f ^ { \prime } ( 2 ) = 10$ 
A)  $f ^ { \prime } ( x ) = 5 ; f ^ { \prime } ( 2 ) = 5$ 
B)  $f ^ { \prime } ( x ) = 9 ; f ^ { \prime } ( 2 ) = 9$ 
C)  $f ^ { \prime } ( x ) = 0 ; f ^ { \prime } ( 2 ) = 0$ 
D)  $f ^ { \prime } ( x ) = 5 x ; f ^ { \prime } ( 2 ) = 10$

Question 19

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

Accepted Answer

A)  $\frac { d ^ { 2 } s } { d t ^ { 2 } } = 12 t ^ { 2 } + \frac { 4 } { t ^ { 3 } } + \frac { 36 } { t ^ { 4 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = 12 \mathrm { t } ^ { 4 } + \frac { 4 } { \mathrm { t } } + \frac { 36 } { \mathrm { t } ^ { 2 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { d \mathrm { t } ^ { 2 } } = 4 \mathrm { t } ^ { 2 } - \frac { 2 } { \mathrm { t } ^ { 3 } } - \frac { 12 } { \mathrm { t } ^ { 4 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = 4 \mathrm { t } ^ { 3 } - \frac { 2 } { \mathrm { t } ^ { 2 } } - \frac { 12 } { \mathrm { t } ^ { 3 } }$ 
A)  $\frac { d ^ { 2 } s } { d t ^ { 2 } } = 12 t ^ { 2 } + \frac { 4 } { t ^ { 3 } } + \frac { 36 } { t ^ { 4 } }$ 
B)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = 12 \mathrm { t } ^ { 4 } + \frac { 4 } { \mathrm { t } } + \frac { 36 } { \mathrm { t } ^ { 2 } }$ 
C)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { d \mathrm { t } ^ { 2 } } = 4 \mathrm { t } ^ { 2 } - \frac { 2 } { \mathrm { t } ^ { 3 } } - \frac { 12 } { \mathrm { t } ^ { 4 } }$ 
D)  $\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = 4 \mathrm { t } ^ { 3 } - \frac { 2 } { \mathrm { t } ^ { 2 } } - \frac { 12 } { \mathrm { t } ^ { 3 } }$

Question 20

Find the derivative.
-$$y = \frac { 1 } { 5 x ^ { 2 } } + \frac { 1 } { 9 x }$$

Accepted Answer

A)  $- \frac { 1 } { 5 x ^ { 3 } } + \frac { 1 } { 9 x ^ { 2 } }$ 
B)  $- \frac { 2 } { 5 x ^ { 3 } } - \frac { 1 } { 9 x ^ { 2 } }$ 
C)  $\frac { 2 } { 5 x ^ { 3 } } + \frac { 1 } { 9 x ^ { 2 } }$
$D ) = \frac { 2 } { 5 x } - \frac { 1 } { 9 x ^ { 2 } }$ 
A)  $- \frac { 1 } { 5 x ^ { 3 } } + \frac { 1 } { 9 x ^ { 2 } }$ 
B)  $- \frac { 2 } { 5 x ^ { 3 } } - \frac { 1 } { 9 x ^ { 2 } }$ 
C)  $\frac { 2 } { 5 x ^ { 3 } } + \frac { 1 } { 9 x ^ { 2 } }$
$D ) = \frac { 2 } { 5 x } - \frac { 1 } { 9 x ^ { 2 } }$

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

Find the derivative. - $y = 14 - 12 x ^ { 2 }$

Find the derivative of the function. - $y = \frac { 3 - 2 x ^ { 3 } + x ^ { 6 } } { x ^ { 9 } }$

Find the derivatives of all orders of the function. - $y = \frac { x ^ { 7 } } { 15,120 }$

Provide an appropriate response. -Find the slope of the tangent to the curve $y = \sqrt { x }$ at the point where $x = 16$ .

Find the slope of the curve at the indicated point. - $y = \frac { 6 } { 5 + x } , x = 9$

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)=5,(1)=-7,v(1)=6,(1)=-3 (2u-4v) at x=1

Provide an appropriate response. - $\text { Does the graph of } f ( x ) = \left\{ \begin{array} { l l } 0 , & x > 0 \\5 , & x \leq 0\end{array} \text { have a vertical tangent at the point } ( 0,5 ) ? \right. \text { Give reasons for your answer. }$

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$ - $f ( x ) = 4 x ^ { 2 } + 3 x + 5$

Differentiate the function and find the slope of the tangent line at the given value of the independent variable. - $\mathrm { s } = 2 \mathrm { t } ^ { 4 } - 2 \mathrm { t } ^ { 3 } , \mathrm { t } = - 1$

Find y'. - $y = \left( 5 x ^ { 3 } + 3 \right) \left( 2 x ^ { 7 } - 5 \right)$

Find the second derivative. - $y = 8 x ^ { 3 } - 4 x ^ { 2 } + 5 e ^ { x }$

Provide an appropriate response. -Find all points $( x , y )$ on the graph of $y = \frac { x } { ( x - 5 ) }$ with tangent lines perpendicular to the line $y = 5 x - 4$ .

Find the derivative. - $y = 7 x ^ { - 2 } - 5 x ^ { 3 } + 2 x$

Calculate the derivative of the function. Then find the value of the derivative as specified. - $f ( x ) = 5 x + 9 ; f ^ { \prime } ( 2 )$

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

Find the derivative. - $y = \frac { 1 } { 5 x ^ { 2 } } + \frac { 1 } { 9 x }$

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Exam 3: Differentiation

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

Find the derivative. - y=14−12x2y = 14 - 12 x ^ { 2 }y=14−12x2

Find the derivative of the function. - y=3−2x3+x6x9y = \frac { 3 - 2 x ^ { 3 } + x ^ { 6 } } { x ^ { 9 } }y=x93−2x3+x6​

Find the derivatives of all orders of the function. - y=x715,120y = \frac { x ^ { 7 } } { 15,120 }y=15,120x7​

Provide an appropriate response. -Find the slope of the tangent to the curve y=xy = \sqrt { x }y=x​ at the point where x=16x = 16x=16 .

Find the slope of the curve at the indicated point. - y=65+x,x=9y = \frac { 6 } { 5 + x } , x = 9y=5+x6​,x=9

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=xy=2xy = x \quad y = 2 xy=xy=2x