Question 1

Estimate the limit numerically.   $\lim _ { x \rightarrow - 7 } \frac { x ^ { 2 } + 14 x + 49 } { x + 7 }$

Accepted Answer

A)  $+ \infty$ 
B)  49 
C)  0 
D)  -7 
E)  8 
A)  $+ \infty$ 
B)  49 
C)  0 
D)  -7 
E)  8

Question 2

At which labeled point is the slope of the tangent line greatest  ?   ?

Accepted Answer

A) R 
B)  P 
C)  Q 
A) R 
B)  P 
C)  Q

Question 3

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 12 }$ at the point $x = - 1$ .
​
Please round your answer to the nearest hundredth.

Accepted Answer

The answer of Estimate the derivative of the function \(f...

Question 4

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . $$\begin{array} { | l | l | l | l | } 
\hline t \text { (month) } & 2 & 4 & 6 \
\hline \boldsymbol { R } ( t ) ( \text { ( millions) } & 13.2 & 13.8 & 12.6 \
\hline
\end{array}$$

Accepted Answer

A) -$300,000 
B)  -$0 
C)  $150 
D)  -$150,000 
E)  $150,000 
A) -$300,000 
B)  -$0 
C)  $150 
D)  -$150,000 
E)  $150,000

Question 5

Calculate the limit algebraically.  
  $$\lim _ { x \rightarrow + \infty } \frac { 5 x ^ { 6 } + 8,000 x ^ { 3 } + 5,000,000 } { 8 x ^ { 6 } + 4,000 x ^ { 3 } }$$

Accepted Answer

A)  8 
B)    $\frac { 7 } { 8 }$ 
C)    $\frac { 1 } { 8 }$ 
D)    $\frac { 5 } { 8 }$ 
E)    $- \frac { 5 } { 8 }$ 
A)  8 
B)    $\frac { 7 } { 8 }$ 
C)    $\frac { 1 } { 8 }$ 
D)    $\frac { 5 } { 8 }$ 
E)    $- \frac { 5 } { 8 }$

Question 6

The function given below gives the cost to manufacture $\chi$ items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the cost at the production level $x = 1,000$ .   $C ( x ) = 10,400 + 4 x - \frac { x ^ { 2 } } { 12,000 }$  
Select your answer rounded to the nearest tenth.

Accepted Answer

A) 3.8 
B)  5.0 
C)  4.0 
D)  2.8 
E)  4.8 
A) 3.8 
B)  5.0 
C)  4.0 
D)  2.8 
E)  4.8

Question 7

Estimate the limit numerically.   $$\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 270 x + 5 } { 2 x ^ { 3 } + 15 }$$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $- \infty$ 
C)  0 
D)  $+ \infty$ 
E)  1 
A)  $\frac { 1 } { 5 }$ 
B)  $- \infty$ 
C)  0 
D)  $+ \infty$ 
E)  1

Question 8

Calculate the average rate of change of the given function over the interval $[ - 2,0 ]$ .   $f ( x ) = 6 x + 10$

Accepted Answer

A) -22 
B)  6 
C)  -6 
D)  -12 
E)  12 
A) -22 
B)  6 
C)  -6 
D)  -12 
E)  12

Question 9

Give an example of a function that is not continuous at $x = - 8$ but is not discontinuous there either.

Accepted Answer

Answers may vary. @#IMG-DLM& is such a f

Question 10

Calculate the limit algebraically.   $\lim _ { x \rightarrow - 2 } \frac { 8 x ^ { 2 } + 15 } { x }$

Accepted Answer

A) 8 
B)  3 
C)  -23.5 
D)  -47 
E)  16 
A) 8 
B)  3 
C)  -23.5 
D)  -47 
E)  16

Question 11

Use the graph to compute $\lim_ { x \rightarrow -4 } f ( x )$ and $f ( 0 )$ .

Accepted Answer

A)  $\lim_ { x \rightarrow -4 } f ( x ) = - 2$ , $f ( 0 ) = - \infty$ 
B)  $\lim _ { x \rightarrow - 4 } f ( x ) = - 3$ , $f ( 0 ) = 2$ 
C)  $\lim _ { \rightarrow - 4 } f ( x ) = 7$ , $f ( 0 ) = + \infty$ 
D)  $\lim _ { x \rightarrow - 4 } f ( x ) = - 3$ , $f ( 0 ) = + \infty$ 
E)  does not exist 
A)  $\lim_ { x \rightarrow -4 } f ( x ) = - 2$ , $f ( 0 ) = - \infty$ 
B)  $\lim _ { x \rightarrow - 4 } f ( x ) = - 3$ , $f ( 0 ) = 2$ 
C)  $\lim _ { \rightarrow - 4 } f ( x ) = 7$ , $f ( 0 ) = + \infty$ 
D)  $\lim _ { x \rightarrow - 4 } f ( x ) = - 3$ , $f ( 0 ) = + \infty$ 
E)  does not exist

Question 12

Calculate the limit algebraically.  
  $\lim _ { x \rightarrow 2 } \frac { x - 11 } { x + 3 }$

Accepted Answer

A) -0.6 
B)  3 
C)  -1.8 
D)  1.8 
E)  None of these 
A) -0.6 
B)  3 
C)  -1.8 
D)  1.8 
E)  None of these

Question 13

Calculate the average rate of change of the given function over the interval $[ - 3,5 ]$ .   $f ( x ) = 2 x ^ { 2 } + 8$

Accepted Answer

A) -42 
B)  42 
C)  4 
D)  10.5 
E)  16 
A) -42 
B)  42 
C)  4 
D)  10.5 
E)  16

Question 14

Estimate the limit numerically.   $$\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 4 } + 40 x ^ { 3 } } { 1,500 x ^ { 3 } + 3 }$$

Accepted Answer

A) 0 
B)  $- \infty$ 
C)  1 
D)  $+ \infty$ 
E)  4 
A) 0 
B)  $- \infty$ 
C)  1 
D)  $+ \infty$ 
E)  4

Question 15

Does this limit exist
    $\lim _ { x \rightarrow - 1 } \frac { x ^ { 2 } + 6 } { x + 1 }$  
Select the correct answer.

Accepted Answer

A) does not exist 
B)  exists 
A) does not exist 
B)  exists

Question 16

The function given below gives the cost to manufacture x items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the total cost at the production level $x = 120$ .   $C ( x ) = 14,600 + 100 x + \frac { 1,300 } { x }$  
Select your answer rounded to the nearest tenth.

Accepted Answer

A) 100.1 
B)  99.9 
C)  100.9 
D)  98.9 
E)  101.1 
A) 100.1 
B)  99.9 
C)  100.9 
D)  98.9 
E)  101.1

Question 17

Compute the derivative function $f ^ { \prime } ( x )$ algebraically.
​
​ $f ( x ) = - x ^ { 2 } - 7 x$

Accepted Answer

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

Question 18

Compute $f ^ { \prime } ( a )$ for $a = 1$ .  
  $f ( x ) = 2 - 6 x ^ { 3 }$

Accepted Answer

A)   $12$ 
B)    $- 18 x ^ { 2 }$ 
C)    $- 18$ 
D)    $18$ 
E)    $36$ 
A)   $12$ 
B)    $- 18 x ^ { 2 }$ 
C)    $- 18$ 
D)    $18$ 
E)    $36$

Question 19

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

Estimate the limit numerically. $\lim _ { x \rightarrow - 7 } \frac { x ^ { 2 } + 14 x + 49 } { x + 7 }$

At which labeled point is the slope of the tangent line greatest ? ?

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 12 }$ at the point $x = - 1$ . Please round your answer to the nearest hundredth.

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . t (month) 2 4 6 (t)( ( millions) 13.2 13.8 12.6

Calculate the limit algebraically. $\lim _ { x \rightarrow + \infty } \frac { 5 x ^ { 6 } + 8,000 x ^ { 3 } + 5,000,000 } { 8 x ^ { 6 } + 4,000 x ^ { 3 } }$

The function given below gives the cost to manufacture $\chi$ items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the cost at the production level $x = 1,000$ . $C ( x ) = 10,400 + 4 x - \frac { x ^ { 2 } } { 12,000 }$ Select your answer rounded to the nearest tenth.

Estimate the limit numerically. $\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 270 x + 5 } { 2 x ^ { 3 } + 15 }$

Calculate the average rate of change of the given function over the interval $[ - 2,0 ]$ . $f ( x ) = 6 x + 10$

Give an example of a function that is not continuous at $x = - 8$ but is not discontinuous there either.

Calculate the limit algebraically. $\lim _ { x \rightarrow - 2 } \frac { 8 x ^ { 2 } + 15 } { x }$

Use the graph to compute $\lim_ { x \rightarrow -4 } f ( x )$ and $f ( 0 )$ . $Use the graph to compute \lim_ { x \rightarrow -4 } f ( x ) and f ( 0 ) .$

Calculate the limit algebraically. $\lim _ { x \rightarrow 2 } \frac { x - 11 } { x + 3 }$

Calculate the average rate of change of the given function over the interval $[ - 3,5 ]$ . $f ( x ) = 2 x ^ { 2 } + 8$

Estimate the limit numerically. $\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 4 } + 40 x ^ { 3 } } { 1,500 x ^ { 3 } + 3 }$

Does this limit exist $\lim _ { x \rightarrow - 1 } \frac { x ^ { 2 } + 6 } { x + 1 }$ Select the correct answer.

The function given below gives the cost to manufacture x items. Estimate (using $h = 0.0001$ ) the instantaneous rate of change of the total cost at the production level $x = 120$ . $C ( x ) = 14,600 + 100 x + \frac { 1,300 } { x }$ Select your answer rounded to the nearest tenth.

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = - x ^ { 2 } - 7 x$

Compute $f ^ { \prime } ( a )$ for $a = 1$ . $f ( x ) = 2 - 6 x ^ { 3 }$

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

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Exam 10: Introduction to the Derivative

Estimate the limit numerically. lim⁡x→−7x2+14x+49x+7\lim _ { x \rightarrow - 7 } \frac { x ^ { 2 } + 14 x + 49 } { x + 7 }limx→−7​x+7x2+14x+49​

At which labeled point is the slope of the tangent line greatest ? ?

Estimate the derivative of the function f(x)=x22−x312f ( x ) = \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 12 }f(x)=2x2​−12x3​ at the point x=−1x = - 1x=−1 . ​ Please round your answer to the nearest hundredth.

Calculate the average rate of change of the given function over the interval [2,6][ 2,6 ][2,6] . t (month) 2 4 6 (t)( ( millions) 13.2 13.8 12.6

Calculate the limit algebraically. lim⁡x→+∞5x6+8,000x3+5,000,0008x6+4,000x3\lim _ { x \rightarrow + \infty } \frac { 5 x ^ { 6 } + 8,000 x ^ { 3 } + 5,000,000 } { 8 x ^ { 6 } + 4,000 x ^ { 3 } }limx→+∞​8x6+4,000x35x6+8,000x3+5,000,000​

Estimate the limit numerically. lim⁡x→+∞3x2+270x+52x3+15\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 270 x + 5 } { 2 x ^ { 3 } + 15 }limx→+∞​2x3+153x2+270x+5​

Calculate the average rate of change of the given function over the interval [−2,0][ - 2,0 ][−2,0] . f(x)=6x+10f ( x ) = 6 x + 10f(x)=6x+10

Give an example of a function that is not continuous at x=−8x = - 8x=−8 but is not discontinuous there either.

Calculate the limit algebraically. lim⁡x→−28x2+15x\lim _ { x \rightarrow - 2 } \frac { 8 x ^ { 2 } + 15 } { x }limx→−2​x8x2+15​

Use the graph to compute lim⁡x→−4f(x)\lim_ { x \rightarrow -4 } f ( x )limx→−4​f(x) and f(0)f ( 0 )f(0) .

Calculate the limit algebraically. lim⁡x→2x−11x+3\lim _ { x \rightarrow 2 } \frac { x - 11 } { x + 3 }limx→2​x+3x−11​

Calculate the average rate of change of the given function over the interval [−3,5][ - 3,5 ][−3,5] . f(x)=2x2+8f ( x ) = 2 x ^ { 2 } + 8f(x)=2x2+8

Estimate the limit numerically. lim⁡x→+∞4x4+40x31,500x3+3\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 4 } + 40 x ^ { 3 } } { 1,500 x ^ { 3 } + 3 }limx→+∞​1,500x3+34x4+40x3​

Does this limit exist lim⁡x→−1x2+6x+1\lim _ { x \rightarrow - 1 } \frac { x ^ { 2 } + 6 } { x + 1 }limx→−1​x+1x2+6​ Select the correct answer.

Compute the derivative function f′(x)f ^ { \prime } ( x )f′(x) algebraically. ​ ​ f(x)=−x2−7xf ( x ) = - x ^ { 2 } - 7 xf(x)=−x2−7x

Compute f′(a)f ^ { \prime } ( a )f′(a) for a=1a = 1a=1 . f(x)=2−6x3f ( x ) = 2 - 6 x ^ { 3 }f(x)=2−6x3

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

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Functions of Several Variables

Trigonometric Models

Filters

Estimate the limit numerically. $\lim _ { x \rightarrow - 7 } \frac { x ^ { 2 } + 14 x + 49 } { x + 7 }$

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 3 } } { 12 }$ at the point $x = - 1$ . Please round your answer to the nearest hundredth.

Calculate the average rate of change of the given function over the interval $[ 2,6 ]$ . t (month) 2 4 6 (t)( ( millions) 13.2 13.8 12.6

Calculate the limit algebraically. $\lim _ { x \rightarrow + \infty } \frac { 5 x ^ { 6 } + 8,000 x ^ { 3 } + 5,000,000 } { 8 x ^ { 6 } + 4,000 x ^ { 3 } }$

Estimate the limit numerically. $\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 270 x + 5 } { 2 x ^ { 3 } + 15 }$

Calculate the average rate of change of the given function over the interval $[ - 2,0 ]$ . $f ( x ) = 6 x + 10$

Give an example of a function that is not continuous at $x = - 8$ but is not discontinuous there either.

Calculate the limit algebraically. $\lim _ { x \rightarrow - 2 } \frac { 8 x ^ { 2 } + 15 } { x }$

Use the graph to compute $\lim_ { x \rightarrow -4 } f ( x )$ and $f ( 0 )$ . $Use the graph to compute \lim_ { x \rightarrow -4 } f ( x ) and f ( 0 ) .$

Calculate the limit algebraically. $\lim _ { x \rightarrow 2 } \frac { x - 11 } { x + 3 }$

Calculate the average rate of change of the given function over the interval $[ - 3,5 ]$ . $f ( x ) = 2 x ^ { 2 } + 8$

Estimate the limit numerically. $\lim _ { x \rightarrow + \infty } \frac { 4 x ^ { 4 } + 40 x ^ { 3 } } { 1,500 x ^ { 3 } + 3 }$

Does this limit exist $\lim _ { x \rightarrow - 1 } \frac { x ^ { 2 } + 6 } { x + 1 }$ Select the correct answer.

Compute the derivative function $f ^ { \prime } ( x )$ algebraically. $f ( x ) = - x ^ { 2 } - 7 x$

Compute $f ^ { \prime } ( a )$ for $a = 1$ . $f ( x ) = 2 - 6 x ^ { 3 }$

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