Question 1

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
​ $f ( x ) = \frac { 7 } { x }$

Accepted Answer

The answer of Compute the derivative function \(f ^ {...

Question 2

Calculate the limit algebraically.   $\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } - 5 x }$

Accepted Answer

A) -2.4 
B)  -0.4 
C)  10 
D)  -2 
E)  2.4 
A) -2.4 
B)  -0.4 
C)  10 
D)  -2 
E)  2.4

Question 3

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 6$ .
​ $f ( x ) = x ^ { 3 }$ ​
Please round the answer to the nearest whole number.

Accepted Answer

The answer of Estimate the slope of the tangent line...

Question 4

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ .
​ $f ( x ) = \frac { x ^ { 2 } - 5 x + 6 } { x - 3 }$ ; $a = 3$

Accepted Answer

The answer of Determine what, if any, value to assign...

Question 5

Use the graph to compute the following quantities.
​ $\lim_ { t \rightarrow -2 } f ( x )$ and $$\lim _ { x \rightarrow 2 } f ( x )$$ ​   ​
Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Use the graph to compute the following...

Question 6

The graph of a function f is given. Determine whether f is continuous on its domain. ​   ​

Accepted Answer

A) Continuous 
B)  Discontinuous 
A) Continuous 
B)  Discontinuous

Question 7

Compute $f ^ { \prime } ( 5 )$ .   $f ( x ) = \frac { 2.1 } { x }$

Accepted Answer

A)   $f ^ { \prime } ( 5 ) = - 0.42$ 
B)    $f ^ { \prime } ( 5 ) = - 0.084$ 
C)    $f ^ { \prime } ( 5 ) = 0.084$ 
D)    $f ^ { \prime } ( 5 ) = 1$ 
E)    $f ^ { \prime } ( 5 ) = 2.1$ 
A)   $f ^ { \prime } ( 5 ) = - 0.42$ 
B)    $f ^ { \prime } ( 5 ) = - 0.084$ 
C)    $f ^ { \prime } ( 5 ) = 0.084$ 
D)    $f ^ { \prime } ( 5 ) = 1$ 
E)    $f ^ { \prime } ( 5 ) = 2.1$

Question 8

Use a graph to determine whether the function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity.
​ $$h ( x ) = \left\{ \begin{array} { l l } 
\frac { | x | } { x } & \text { if } x \neq 0 \
0 & \text { if } x = 0
\end{array} \right.$$

Accepted Answer

The answer of Use a graph to determine whether the...

Question 9

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ .   $f ( x ) = x ^ { 2 } - 9$

Accepted Answer

A) 6 
B)  14 
C)  8 
D)  10 
E)  18 
A) 6 
B)  14 
C)  8 
D)  10 
E)  18

Question 10

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.  
  $f ( x ) = 8 - 5 x ^ { 3 }$

Accepted Answer

A)   $f ^ { \prime } ( x ) = 15 x ^ { 2 }$ 
B)    $f ^ { \prime } ( x ) = - 15 x$ 
C)    $f ^ { \prime } ( x ) = - 15 x ^ { 2 }$ 
D)    $f ^ { \prime } ( x ) = 8 - 15 x ^ { 2 }$ 
E)    $f ^ { \prime } ( x ) = 10 x ^ { 2 }$ 
A)   $f ^ { \prime } ( x ) = 15 x ^ { 2 }$ 
B)    $f ^ { \prime } ( x ) = - 15 x$ 
C)    $f ^ { \prime } ( x ) = - 15 x ^ { 2 }$ 
D)    $f ^ { \prime } ( x ) = 8 - 15 x ^ { 2 }$ 
E)    $f ^ { \prime } ( x ) = 10 x ^ { 2 }$

Question 11

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ .   $f ( x ) = \frac { 3 } { 2 x ^ { 2 } - x }$ ; $a = 0$

Accepted Answer

A)  $f ( 0 ) = - 1$ 
B)  $f ( 0 ) = - 2$ 
C)  $f ( 0 ) = 1$ 
D)  $f ( 0 ) = 0$ 
E)  No value of $f ( 0 )$ will make it continuous at 0. 
A)  $f ( 0 ) = - 1$ 
B)  $f ( 0 ) = - 2$ 
C)  $f ( 0 ) = 1$ 
D)  $f ( 0 ) = 0$ 
E)  No value of $f ( 0 )$ will make it continuous at 0.

Question 12

Estimate $f ^ { \prime } ( 7 )$ of the function $f = 7 \ln x$ .
?
Please round the answer to the nearest whole number.

Accepted Answer

The answer of Estimate \(f ^ { \prime } (...

Question 13

Use the graph to compute $\lim_ { x \rightarrow 0^+ } f ( x )$ and $\lim_ { x \rightarrow 0^- } f ( x )$ .

Accepted Answer

A)  $\lim_ { x \rightarrow 0^+ } f ( x ) = 2$ , $\lim_ { x \rightarrow 0^- } f ( x ) = - 1$ . 
B)  $\lim_ { x \rightarrow 0^+ } f ( x ) = 2$ , $\lim_ { x \rightarrow 0^- } f ( x ) = 2$ . 
C)  $\lim_ { x \rightarrow 0^+ } f ( x ) = - 1$ , $\lim_ { x \rightarrow 0^- } f ( x ) = - 1$ . 
D)  $\lim _ { x \rightarrow 0 ^ { + } } f ( x ) = + \infty$ , $\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = + \infty$ . 
E)  does not exist 
A)  $\lim_ { x \rightarrow 0^+ } f ( x ) = 2$ , $\lim_ { x \rightarrow 0^- } f ( x ) = - 1$ . 
B)  $\lim_ { x \rightarrow 0^+ } f ( x ) = 2$ , $\lim_ { x \rightarrow 0^- } f ( x ) = 2$ . 
C)  $\lim_ { x \rightarrow 0^+ } f ( x ) = - 1$ , $\lim_ { x \rightarrow 0^- } f ( x ) = - 1$ . 
D)  $\lim _ { x \rightarrow 0 ^ { + } } f ( x ) = + \infty$ , $\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = + \infty$ . 
E)  does not exist

Question 14

Calculate the limit algebraically. 
  $\lim _ { x \rightarrow - \infty } \frac { 5 x ^ { 5 } - 4,000 x ^ { 4 } } { 12 x ^ { 5 } + 40,000 }$

Accepted Answer

A) 12 
B)    $\frac { 1 } { 12 }$ 
C)    $- \frac { 5 } { 12 }$ 
D)    $\frac { 11 } { 12 }$ 
E)    $\frac { 5 } { 12 }$ 
A) 12 
B)    $\frac { 1 } { 12 }$ 
C)    $- \frac { 5 } { 12 }$ 
D)    $\frac { 11 } { 12 }$ 
E)    $\frac { 5 } { 12 }$

Question 15

Compute $f ^ { \prime } ( a )$ for $a = 0$ .  
  $f ( x ) = - x ^ { 2 } - 8 x$

Accepted Answer

A) 8 
B)    $- 10$ 
C)   0 
D)    $- 8$ 
E)   10 
A) 8 
B)    $- 10$ 
C)   0 
D)    $- 8$ 
E)   10

Question 16

Draw the graph of a function that is continuous on its domain but not continuous at any integer. ​

Accepted Answer

A) ​   ​ 
B)  ​  
​ 
C)  ​  
​ 
D)  ​  
​ 
E)  ​  
​ 
A) ​   ​ 
B)  ​  
​ 
C)  ​  
​ 
D)  ​  
​ 
E)  ​  
​

Question 17

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
​
​ $f ( x ) = 3 x ^ { 2 } + x$

Accepted Answer

The answer of Compute the derivative function \(f ^ {...

Question 18

Compute $f ^ { \prime } ( a )$ for $a = 3$ .
​ $f ( x ) = \frac { 2 } { x }$ ​
Enter your answer in fraction form.

Accepted Answer

The answer of Compute \(f ^ { \prime } (...

Question 19

Compute $f ^ { \prime } ( a )$ for $a = 3$ .  
  $f ( x ) = x ^ { 2 } - 5$

Accepted Answer

A) -6 
B)  6 
C)  -9 
D)  3 
E)  8 
A) -6 
B)  6 
C)  -9 
D)  3 
E)  8

Question 20

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ .
​ $f ( x ) = 6 x ^ { 2 } + x$

Accepted Answer

The answer of Compute \(f ^ { \prime } (...

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = \frac { 7 } { x }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } - 5 x }$

Estimate the slope of the tangent line to the graph of the following function at the point $x = - 6$ . $f ( x ) = x ^ { 3 }$ Please round the answer to the nearest whole number.

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ . $f ( x ) = \frac { x ^ { 2 } - 5 x + 6 } { x - 3 }$ ; $a = 3$

The graph of a function f is given. Determine whether f is continuous on its domain.

Compute $f ^ { \prime } ( 5 )$ . $f ( x ) = \frac { 2.1 } { x }$

Use a graph to determine whether the function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. $h ( x ) = \left\{ \begin{array} { l l } \frac { | x | } { x } & \text { if } x \neq 0 \\0 & \text { if } x = 0\end{array} \right.$

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . $f ( x ) = x ^ { 2 } - 9$

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = 8 - 5 x ^ { 3 }$

Determine what, if any, value to assign to $f ( a )$ to make f continuous at $x = a$ . $f ( x ) = \frac { 3 } { 2 x ^ { 2 } - x }$ ; $a = 0$

Estimate $f ^ { \prime } ( 7 )$ of the function $f = 7 \ln x$ . ? Please round the answer to the nearest whole number.

Use the graph to compute $\lim_ { x \rightarrow 0^+ } f ( x )$ and $\lim_ { x \rightarrow 0^- } f ( x )$ . $Use the graph to compute \lim_ { x \rightarrow 0^+ } f ( x ) and \lim_ { x \rightarrow 0^- } f ( x ) .$

Calculate the limit algebraically. $\lim _ { x \rightarrow - \infty } \frac { 5 x ^ { 5 } - 4,000 x ^ { 4 } } { 12 x ^ { 5 } + 40,000 }$

Compute $f ^ { \prime } ( a )$ for $a = 0$ . $f ( x ) = - x ^ { 2 } - 8 x$

Draw the graph of a function that is continuous on its domain but not continuous at any integer.

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = 3 x ^ { 2 } + x$

Compute $f ^ { \prime } ( a )$ for $a = 3$ . $f ( x ) = \frac { 2 } { x }$ Enter your answer in fraction form.

Compute $f ^ { \prime } ( a )$ for $a = 3$ . $f ( x ) = x ^ { 2 } - 5$

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ . $f ( x ) = 6 x ^ { 2 } + x$

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Exam 10: Introduction to the Derivative

Compute the derivative function f′(x)f ^ { \prime } ( x )f′(x) algebraically. ​ f(x)=7xf ( x ) = \frac { 7 } { x }f(x)=x7​

Calculate the limit algebraically. lim⁡x→5x2−12x+35x2−5x\lim _ { x \rightarrow 5 } \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } - 5 x }limx→5​x2−5xx2−12x+35​

Estimate the slope of the tangent line to the graph of the following function at the point x=−6x = - 6x=−6 . ​ f(x)=x3f ( x ) = x ^ { 3 }f(x)=x3 ​ Please round the answer to the nearest whole number.

Determine what, if any, value to assign to f(a)f ( a )f(a) to make f continuous at x=ax = ax=a . ​ f(x)=x2−5x+6x−3f ( x ) = \frac { x ^ { 2 } - 5 x + 6 } { x - 3 }f(x)=x−3x2−5x+6​ ; a=3a = 3a=3

Use the graph to compute the following quantities. ​ lim⁡t→−2f(x)\lim_ { t \rightarrow -2 } f ( x )limt→−2​f(x) and lim⁡x→2f(x)\lim _ { x \rightarrow 2 } f ( x )limx→2​f(x) ​ ​ Please enter your answer as two numbers, separated by a comma.

The graph of a function f is given. Determine whether f is continuous on its domain. ​ ​

Compute f′(5)f ^ { \prime } ( 5 )f′(5) . f(x)=2.1xf ( x ) = \frac { 2.1 } { x }f(x)=x2.1​

Calculate the average rate of change of the given function over the interval [2,6][ 2,6 ][2,6] . f(x)=x2−9f ( x ) = x ^ { 2 } - 9f(x)=x2−9

Compute the derivative function f′(x)f ^ { \prime } ( x )f′(x) algebraically. f(x)=8−5x3f ( x ) = 8 - 5 x ^ { 3 }f(x)=8−5x3

Determine what, if any, value to assign to f(a)f ( a )f(a) to make f continuous at x=ax = ax=a . f(x)=32x2−xf ( x ) = \frac { 3 } { 2 x ^ { 2 } - x }f(x)=2x2−x3​ ; a=0a = 0a=0

Estimate f′(7)f ^ { \prime } ( 7 )f′(7) of the function f=7ln⁡xf = 7 \ln xf=7lnx . ? Please round the answer to the nearest whole number.

Use the graph to compute lim⁡x→0+f(x)\lim_ { x \rightarrow 0^+ } f ( x )limx→0+​f(x) and lim⁡x→0−f(x)\lim_ { x \rightarrow 0^- } f ( x )limx→0−​f(x) .

Calculate the limit algebraically. lim⁡x→−∞5x5−4,000x412x5+40,000\lim _ { x \rightarrow - \infty } \frac { 5 x ^ { 5 } - 4,000 x ^ { 4 } } { 12 x ^ { 5 } + 40,000 }limx→−∞​12x5+40,0005x5−4,000x4​

Compute f′(a)f ^ { \prime } ( a )f′(a) for a=0a = 0a=0 . f(x)=−x2−8xf ( x ) = - x ^ { 2 } - 8 xf(x)=−x2−8x