Question 1

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 2

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity.   $g ( x ) = \frac { 3 } { x ^ { 2 } - 16 }$

Accepted Answer

A) The function is continuous on its domain. 
B)  The points of discontinuity: 16, 0. 
C)  The points of discontinuity: $\frac { 1 } { 3 }$ , 0. 
D)  The point of discontinuity: 0. 
E)  The point of discontinuity: 16. 
A) The function is continuous on its domain. 
B)  The points of discontinuity: 16, 0. 
C)  The points of discontinuity: $\frac { 1 } { 3 }$ , 0. 
D)  The point of discontinuity: 0. 
E)  The point of discontinuity: 16.

Question 3

Calculate the limit algebraically.   $\lim _ { x \rightarrow 0 } ( x + 2 )$

Accepted Answer

A) 4 
B)  2 
C)  -2 
D)  1 
E)  -4 
A) 4 
B)  2 
C)  -2 
D)  1 
E)  -4

Question 4

The number of visitors, in millions of visitors per year, to Hawaii in the years 1985 to 1993 can be approximated by  
 $n ( t ) = - 0.068 t ^ { 2 } + 0.572 t + 4.39$  
Where $t = 0$ represents June 30,1985. During the same period, visitor spending can be approximated by
  $r ( t ) = - 0.453 t ^ { 2 } + 5.6 t + 6.65$  
Where $t = 0$ represents June 30,1985. Assuming the trends in the models above continue indefinitely, numerically estimate $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) }$ .
 
Please round the answers to the nearest hundredth.

Accepted Answer

A)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 5.21$ 
B)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 6.66$ 
C)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 8.66$ 
D)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 6.11$ 
E)  does not exist 
A)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 5.21$ 
B)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 6.66$ 
C)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 8.66$ 
D)  $\lim _ { t \rightarrow + \infty } \frac { r ( t ) } { n ( t ) } = 6.11$ 
E)  does not exist

Question 5

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

Accepted Answer

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

Question 6

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.   $g ( x ) = \left\{ \begin{array} { l l } 
7 x + 2 & \text { if } x < 0 \
6 x + 2 & \text { if } x \geq 0
\end{array} \right.$

Accepted Answer

A) The points of discontinuity: 2, 0. 
B)  The point of discontinuity: 0. 
C)  The point of discontinuity: 2. 
D)  The function is continuous on its domain. 
E)  The points of discontinuity: -2, 0. 
A) The points of discontinuity: 2, 0. 
B)  The point of discontinuity: 0. 
C)  The point of discontinuity: 2. 
D)  The function is continuous on its domain. 
E)  The points of discontinuity: -2, 0.

Question 7

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 6 }$ at the point $x = - 2$ .  
Please select the correct answer rounded to the nearest hundredth.

Accepted Answer

A) -2.5 
B)  -2.55 
C)  -3 
D)  -0.5 
E)  -2.45 
A) -2.5 
B)  -2.55 
C)  -3 
D)  -0.5 
E)  -2.45

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.
​ $$g ( x ) = \left\{ \begin{array} { l l } 
5 x + 2 & \text { if } x < 0 \
2 x + 2 & \text { if } x \geq 0
\end{array} \right.$$

Accepted Answer

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

Question 9

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

Accepted Answer

A)  $f ^ { \prime } ( x ) = 4$ 
B)    $f ^ { \prime } ( x ) = 4 x + 1$ 
C)    $f ^ { \prime } ( x ) = 4 + x$ 
D)    $f ^ { \prime } ( x ) = 4 x$ 
E)    $f ^ { \prime } ( x ) = 2 x + 1$ 
A)  $f ^ { \prime } ( x ) = 4$ 
B)    $f ^ { \prime } ( x ) = 4 x + 1$ 
C)    $f ^ { \prime } ( x ) = 4 + x$ 
D)    $f ^ { \prime } ( x ) = 4 x$ 
E)    $f ^ { \prime } ( x ) = 2 x + 1$

Question 10

Calculate the limit algebraically.
​ $\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 5 x - 2 } { 4 x ^ { 2 } - 5 }$ ​

Accepted Answer

The answer of Calculate the limit algebraically.
​ \(\lim _ {...

Question 11

Compute $f ^ { \prime } ( 3 )$ .  
  $f ( x ) = 2.9 x ^ { 2 } - 5.7 x$

Accepted Answer

A)   $f ^ { \prime } ( 3 ) = 17.4$ 
B)    $f ^ { \prime } ( 3 ) = 0.1$ 
C)    $f ^ { \prime } ( 3 ) = 11.7$ 
D)    $f ^ { \prime } ( 3 ) = 3$ 
E)    $f ^ { \prime } ( 3 ) = 11.5$ 
A)   $f ^ { \prime } ( 3 ) = 17.4$ 
B)    $f ^ { \prime } ( 3 ) = 0.1$ 
C)    $f ^ { \prime } ( 3 ) = 11.7$ 
D)    $f ^ { \prime } ( 3 ) = 3$ 
E)    $f ^ { \prime } ( 3 ) = 11.5$

Question 12

Calculate the limit algebraically.  
  $\lim _ { x \rightarrow 125 } ( x - \sqrt [ 3 ] { x } )$

Accepted Answer

A) 0 
B)  120 
C)  125 
D)  -5 
E)  130 
A) 0 
B)  120 
C)  125 
D)  -5 
E)  130

Question 13

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

Accepted Answer

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

Question 14

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ .  
  $f ( x ) = 7 x ^ { 2 } + x$

Accepted Answer

A) 129 
B)  66 
C)  126 
D)  45 
E)  43 
A) 129 
B)  66 
C)  126 
D)  45 
E)  43

Question 15

Calculate the limit algebraically.  
  $\lim _ { h \rightarrow 5 } \left( h ^ { 2 } + 10 h + 2 \right)$

Accepted Answer

A) 75 
B)  77 
C)  38.5 
D)  2 
E)  -77 
A) 75 
B)  77 
C)  38.5 
D)  2 
E)  -77

Question 16

Estimate the limit numerically.   $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x - 3 }$

Accepted Answer

A) -3 
B)  -6 
C)  diverges to $+ \infty$ 
D)  3 
E)  6 
A) -3 
B)  -6 
C)  diverges to $+ \infty$ 
D)  3 
E)  6

Question 17

Calculate the average rate of change of the given function over the interval $[ 7,13 ]$ .   $f ( x ) = 6 x ^ { 2 } - \frac { x } { 2 }$

Accepted Answer

A) 64.90 
B)  216.33 
C)  717.00 
D)  35.85 
E)  119.50 
A) 64.90 
B)  216.33 
C)  717.00 
D)  35.85 
E)  119.50

Question 18

Calculate the limit algebraically.    $\lim _ { x \rightarrow 2 } \frac { 13 + x } { x }$

Accepted Answer

A) 7.5 
B)  -2 
C)  15 
D)  13 
E)  6.5 
A) 7.5 
B)  -2 
C)  15 
D)  13 
E)  6.5

Question 19

Use any method to find the slope of the tangent line to the graph of the function $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point that has x-coordinate $x = 5$ .

Accepted Answer

A) -125 
B)  25 
C)  $\frac { 1 } { 25 }$ 
D)  $- \frac { 2 } { 125 }$ 
E)  $\frac { 1 } { 125 }$ 
A) -125 
B)  25 
C)  $\frac { 1 } { 25 }$ 
D)  $- \frac { 2 } { 125 }$ 
E)  $\frac { 1 } { 125 }$

Question 20

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

Accepted Answer

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

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

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. $g ( x ) = \frac { 3 } { x ^ { 2 } - 16 }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 0 } ( x + 2 )$

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

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. $g ( x ) = \left\{ \begin{array} { l l } 7 x + 2 & \text { if } x < 0 \\6 x + 2 & \text { if } x \geq 0\end{array} \right.$

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 6 }$ at the point $x = - 2$ . Please select the correct answer rounded to the nearest hundredth.

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. $g ( x ) = \left\{ \begin{array} { l l } 5 x + 2 & \text { if } x < 0 \\2 x + 2 & \text { if } x \geq 0\end{array} \right.$

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

Calculate the limit algebraically. $\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 5 x - 2 } { 4 x ^ { 2 } - 5 }$

Compute $f ^ { \prime } ( 3 )$ . $f ( x ) = 2.9 x ^ { 2 } - 5.7 x$

Calculate the limit algebraically. $\lim _ { x \rightarrow 125 } ( x - \sqrt [ 3 ] { x } )$

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

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ . $f ( x ) = 7 x ^ { 2 } + x$

Calculate the limit algebraically. $\lim _ { h \rightarrow 5 } \left( h ^ { 2 } + 10 h + 2 \right)$

Estimate the limit numerically. $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x - 3 }$

Calculate the average rate of change of the given function over the interval $[ 7,13 ]$ . $f ( x ) = 6 x ^ { 2 } - \frac { x } { 2 }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 2 } \frac { 13 + x } { x }$

Use any method to find the slope of the tangent line to the graph of the function $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point that has x-coordinate $x = 5$ .

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

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

Exam 10: Introduction to the Derivative

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

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. g(x)=3x2−16g ( x ) = \frac { 3 } { x ^ { 2 } - 16 }g(x)=x2−163​

Calculate the limit algebraically. lim⁡x→0(x+2)\lim _ { x \rightarrow 0 } ( x + 2 )limx→0​(x+2)

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

Estimate the derivative of the function f(x)=x28−x36f ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 6 }f(x)=8x2​−6x3​ at the point x=−2x = - 2x=−2 . Please select the correct answer rounded to the nearest hundredth.

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

Calculate the limit algebraically. ​ lim⁡x→+∞3x2+5x−24x2−5\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 5 x - 2 } { 4 x ^ { 2 } - 5 }limx→+∞​4x2−53x2+5x−2​ ​

Compute f′(3)f ^ { \prime } ( 3 )f′(3) . f(x)=2.9x2−5.7xf ( x ) = 2.9 x ^ { 2 } - 5.7 xf(x)=2.9x2−5.7x

Calculate the limit algebraically. lim⁡x→125(x−x3)\lim _ { x \rightarrow 125 } ( x - \sqrt [ 3 ] { x } )limx→125​(x−3x​)

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

Compute f′(a)f ^ { \prime } ( a )f′(a) algebraically for a=3a = 3a=3 . f(x)=7x2+xf ( x ) = 7 x ^ { 2 } + xf(x)=7x2+x

Calculate the limit algebraically. lim⁡h→5(h2+10h+2)\lim _ { h \rightarrow 5 } \left( h ^ { 2 } + 10 h + 2 \right)limh→5​(h2+10h+2)

Estimate the limit numerically. lim⁡x→3x2−9x−3\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x - 3 }limx→3​x−3x2−9​

Calculate the average rate of change of the given function over the interval [7,13][ 7,13 ][7,13] . f(x)=6x2−x2f ( x ) = 6 x ^ { 2 } - \frac { x } { 2 }f(x)=6x2−2x​

Calculate the limit algebraically. lim⁡x→213+xx\lim _ { x \rightarrow 2 } \frac { 13 + x } { x }limx→2​x13+x​

Use any method to find the slope of the tangent line to the graph of the function f(x)=1x2f ( x ) = \frac { 1 } { x ^ { 2 } }f(x)=x21​ at the point that has x-coordinate x=5x = 5x=5 .

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

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

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

Use a graph to determine whether the given function is continuous on its domain. If it is not continuous on its domain, list the points of discontinuity. $g ( x ) = \frac { 3 } { x ^ { 2 } - 16 }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 0 } ( x + 2 )$

Estimate the derivative of the function $f ( x ) = \frac { x ^ { 2 } } { 8 } - \frac { x ^ { 3 } } { 6 }$ at the point $x = - 2$ . Please select the correct answer rounded to the nearest hundredth.

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

Calculate the limit algebraically. $\lim _ { x \rightarrow + \infty } \frac { 3 x ^ { 2 } + 5 x - 2 } { 4 x ^ { 2 } - 5 }$

Compute $f ^ { \prime } ( 3 )$ . $f ( x ) = 2.9 x ^ { 2 } - 5.7 x$

Calculate the limit algebraically. $\lim _ { x \rightarrow 125 } ( x - \sqrt [ 3 ] { x } )$

Compute $f ^ { \prime } ( a )$ algebraically for $a = 3$ . $f ( x ) = 7 x ^ { 2 } + x$

Calculate the limit algebraically. $\lim _ { h \rightarrow 5 } \left( h ^ { 2 } + 10 h + 2 \right)$

Estimate the limit numerically. $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x - 3 }$

Calculate the average rate of change of the given function over the interval $[ 7,13 ]$ . $f ( x ) = 6 x ^ { 2 } - \frac { x } { 2 }$

Calculate the limit algebraically. $\lim _ { x \rightarrow 2 } \frac { 13 + x } { x }$

Use any method to find the slope of the tangent line to the graph of the function $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point that has x-coordinate $x = 5$ .

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