Question 1

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0 
A)  $\frac { 1 } { 2 }$ 
B)  $- 1$ 
C)  3 
D)  The limit does not exist. 
E)  0 D

Question 2

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$, where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$. Round your answer to the nearest cent.

Accepted Answer

A)  $\$ 49,650.00$ 
B)  $\$ 484.50$ 
C)  $\$ 12.00$ 
D)  $\$ 496.50$ 
E)  $\$ 484.62$ 
A)  $\$ 49,650.00$ 
B)  $\$ 484.50$ 
C)  $\$ 12.00$ 
D)  $\$ 496.50$ 
E)  $\$ 484.62$ D

Question 3

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 3 }$ 
B)  3 
C)  $\infty$ 
D)  $\frac { 1 } { 6 }$ 
E)  the slope is undefined at this point D

Question 4

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ a which the tangent line is horizontal.

Accepted Answer

A)  $( 1,12 )$ 
B)  $( 1,12 )$ and $( - 1 , - 12 )$ 
C)  $( 3,108 )$ and $( - 3,108 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line. 
A)  $( 1,12 )$ 
B)  $( 1,12 )$ and $( - 1 , - 12 )$ 
C)  $( 3,108 )$ and $( - 3,108 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line.

Question 5

Using the summation formulas and properties, evaluate the following expression.
$$\sum _ { i = 1 } ^ { 30 } 8$$

Accepted Answer

A)  3720 
B)  465 
C)  232 
D)  240 
E)  38 
A)  3720 
B)  465 
C)  232 
D)  240 
E)  38

Question 6

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $- \frac { 3 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $- \infty$ 
E)  limit does not exist 
A)  0 
B)  $- \frac { 3 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $- \infty$ 
E)  limit does not exist

Question 7

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$-axis on the interval $[ 1,4 ]$.

Accepted Answer

A)  $\frac { 143 } { 2 }$ 
B)  $\frac { 81 } { 4 }$ 
C)  $\frac { 47 } { 2 }$ 
D)  78 
E)  $\frac { 51 } { 4 }$ 
A)  $\frac { 143 } { 2 }$ 
B)  $\frac { 81 } { 4 }$ 
C)  $\frac { 47 } { 2 }$ 
D)  78 
E)  $\frac { 51 } { 4 }$

Question 8

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$, if it exists.

Accepted Answer

A)  0 
B)  2 
C)  4 
D)  $- 1$ 
E)  The limit does not exist. 
A)  0 
B)  2 
C)  4 
D)  $- 1$ 
E)  The limit does not exist.

Question 9

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$, if it exists.

Accepted Answer

A)  1 
B)  $- 10$ 
C)  10 
D)  $- 1$ 
E)  The limit does not exist. 
A)  1 
B)  $- 10$ 
C)  10 
D)  $- 1$ 
E)  The limit does not exist.

Question 10

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$.

Accepted Answer

A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist 
A)  24 
B)  $\infty$ 
C)  12 
D)  0 
E)  limit does not exist

Question 11

Find the following limit, if it exists. $$\lim _ { x \rightarrow - 3 } x ^ { 5 }$$

Accepted Answer

A)  -3 
B)  -15 
C)  3 
D)  -243 
E)  The limit does not exist. 
A)  -3 
B)  -15 
C)  3 
D)  -243 
E)  The limit does not exist.

Question 12

Find the slope of the graph of the following function at the point (-1,1) . $$3 x ^ { 2 } - 2$$

Accepted Answer

A)  4 
B)  -6 
C)  6 
D)  1 
E)  -8 
A)  4 
B)  -6 
C)  6 
D)  1 
E)  -8

Question 13

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

Accepted Answer

A)  14 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { \sqrt { 7 } } { 2 }$ 
D)  $2 \sqrt { 7 }$ 
E)  limit does not exist 
A)  14 
B)  $\frac { 7 } { 2 }$ 
C)  $\frac { \sqrt { 7 } } { 2 }$ 
D)  $2 \sqrt { 7 }$ 
E)  limit does not exist

Question 14

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$, where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$. Round your answer to the nearest cent.

Accepted Answer

A)  $\$ 51,650.00$ 
B)  $\$ 244.75$ 
C)  $\$ 13.50$ 
D)  $\$ 258.25$ 
E)  $\$ 244.82$ 
A)  $\$ 51,650.00$ 
B)  $\$ 244.75$ 
C)  $\$ 13.50$ 
D)  $\$ 258.25$ 
E)  $\$ 244.82$

Question 15

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .

Accepted Answer

A)  185 
B)  110 
C)  65 
D)  75 
E)  82 
A)  185 
B)  110 
C)  65 
D)  75 
E)  82

Question 16

A union contract guarantees a $15 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,000$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is given by
$f ( t ) = \left\{ \begin{array} { l l } 
31.00 , & 0 < t \leq 1 \
35.65 , & 1 < t \leq 2 \
41.00 , & 2 < t \leq 3
\end{array} \right.$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.

Accepted Answer

A)  $31.00$ 
B)  $35.65$ 
C)  $1.00$ 
D)  $41.00$ 
E)  The limit does not exist. 
A)  $31.00$ 
B)  $35.65$ 
C)  $1.00$ 
D)  $41.00$ 
E)  The limit does not exist.

Question 17

Use the function below and its derivative to determine any points on the graph of f at which the tangent line is horizontal. $f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$

Accepted Answer

A)  $( - 1,3 ) , ( 0,0 ) , ( 1,3 )$ 
B)  $( - 1 , - 3 ) , ( 1 , - 12 )$ 
C)  $( - 1,3 ) , ( 1,3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1,3 )$ 
A)  $( - 1,3 ) , ( 0,0 ) , ( 1,3 )$ 
B)  $( - 1 , - 3 ) , ( 1 , - 12 )$ 
C)  $( - 1,3 ) , ( 1,3 )$ 
D)  $( 0,0 ) , ( 1,0 )$ 
E)  $( 0,0 ) , ( 1,3 )$

Question 18

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$.
$- 3 x ^ { 2 } - 3$

Accepted Answer

A)  $y = - 18 x - 84$ 
B)  $y = - 18 x + 84$ 
C)  $y = - 18 x + 24$ 
D)  $y = 180 x + 570$ 
E)  $y = 180 x - 570$ 
A)  $y = - 18 x - 84$ 
B)  $y = - 18 x + 84$ 
C)  $y = - 18 x + 24$ 
D)  $y = 180 x + 570$ 
E)  $y = 180 x - 570$

Question 19

Find
$$\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$$
by direct substitution.

Accepted Answer

A)  $\frac { 6 } { \sqrt { 10 } }$ 
B)  $4 \sqrt { 6 }$ 
C)  $24 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 24 } { \sqrt { 10 } }$ 
A)  $\frac { 6 } { \sqrt { 10 } }$ 
B)  $4 \sqrt { 6 }$ 
C)  $24 \sqrt { 10 }$ 
D)  1 
E)  $\frac { 24 } { \sqrt { 10 } }$

Question 20

A union contract guarantees a $16 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,500$, the salary $f ( t )$ (in thousands of dollars) for the next 3 years is givet by
$$f ( t ) = \left\{ \begin{array} { l l } 
31.50 , & 0 < t \leq 1 \
36.54 , & 1 < t \leq 2 \
42.39 , & 2 < t \leq 3
\end{array} \right.$$
where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$, if it exists.

Accepted Answer

A)  31.50 
B)  36.54 
C)  1.00 
D)  42.39 
E)  The limit does not exist. 
A)  31.50 
B)  36.54 
C)  1.00 
D)  42.39 
E)  The limit does not exist.

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$ . Round your answer to the nearest cent.

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ a which the tangent line is horizontal.

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 30 } 8$

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$ -axis on the interval $[ 1,4 ]$ .

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists. $Use the graph to find \lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 } , if it exists.$

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 } , if it exists.$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the slope of the graph of the following function at the point (-1,1) . $3 x ^ { 2 } - 2$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .

Use the function below and its derivative to determine any points on the graph of f at which the tangent line is horizontal. $f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$ . $- 3 x ^ { 2 } - 3$

Find $\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$ by direct substitution.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Review of Graphs, Equations, and Inequalities

Filters

Exam 11: Limits and an Introduction to Calculus

Find the following limit, if it exists. lim⁡x→∞3x−6\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }limx→∞​x−63​

The cost function for a certain model of a digital camera given by C=12.00x+48,450C = 12.00 x + 48,450C=12.00x+48,450 , where CCC is the cost (in dollars) and xxx is the number of cameras produced. Find the average cost per unit when x=100x = 100x=100 . Round your answer to the nearest cent.

Use the limit process to find the slope of the graph of x+4\sqrt { x + 4 }x+4​ at (5,3)( 5,3 )(5,3) .

Use the derivative of f(x)=3x3+9xf ( x ) = 3 x ^ { 3 } + 9 xf(x)=3x3+9x to determine any points on the graph of f(x)f ( x )f(x) a which the tangent line is horizontal.

Using the summation formulas and properties, evaluate the following expression. ∑i=1308\sum _ { i = 1 } ^ { 30 } 8∑i=130​8

Find lim⁡x→∞1−6x1+8x\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }limx→∞​1+8x1−6x​ (if it exists).

Use the limit process to find the area of the region between f(x)=14(x2+4x)f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)f(x)=41​(x2+4x) and the xxx -axis on the interval [1,4][ 1,4 ][1,4] .

Use the graph to find lim⁡x→−11x+1\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }limx→−1​x+11​ , if it exists.

Use the graph below to find lim⁡x→−10∣x+10∣x+10\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }limx→−10​x+10∣x+10∣​ , if it exists.

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​ .

Find the following limit, if it exists. lim⁡x→−3x5\lim _ { x \rightarrow - 3 } x ^ { 5 }limx→−3​x5

Find the slope of the graph of the following function at the point (-1,1) . 3x2−23 x ^ { 2 } - 23x2−2

Find lim⁡x→0−xx+7−7\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }limx→0−​x+7​−7​x​

The cost function for a certain model of a digital camera given by C=13.50x+48,950C = 13.50 x + 48,950C=13.50x+48,950 , where CCC is the cost (in dollars) and xxx is the number of cameras produced. Find the average cost per unit when x=200x = 200x=200 . Round your answer to the nearest cent.

Use the limit process to find the area of the region between f(x ) =6x+7 and the x-axis on the interval [0,5] .

Use the function below and its derivative to determine any points on the graph of f at which the tangent line is horizontal. f(x)=−3x4+6x2,f′(x)=−12x3+12xf ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 xf(x)=−3x4+6x2,f′(x)=−12x3+12x

Find an equation of the tangent line to the graph of the following function at the point (3,−30)( 3 , - 30 )(3,−30) . −3x2−3- 3 x ^ { 2 } - 3−3x2−3

Find lim⁡x→64xx+4\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }limx→6​x+4​4x​ by direct substitution.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Review of Graphs, Equations, and Inequalities

Filters

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 3 } { x ^ { - 6 } }$

The cost function for a certain model of a digital camera given by $C = 12.00 x + 48,450$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 100$ . Round your answer to the nearest cent.

Use the limit process to find the slope of the graph of $\sqrt { x + 4 }$ at $( 5,3 )$ .

Use the derivative of $f ( x ) = 3 x ^ { 3 } + 9 x$ to determine any points on the graph of $f ( x )$ a which the tangent line is horizontal.

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 30 } 8$

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 4 } \left( x ^ { 2 } + 4 x \right)$ and the $x$ -axis on the interval $[ 1,4 ]$ .

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists. $Use the graph to find \lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 } , if it exists.$

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 } , if it exists.$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ . $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 } .$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 5 }$

Find the slope of the graph of the following function at the point (-1,1) . $3 x ^ { 2 } - 2$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 7 } - \sqrt { 7 } }$

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

Use the function below and its derivative to determine any points on the graph of f at which the tangent line is horizontal. $f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$ . $- 3 x ^ { 2 } - 3$

Find $\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$ by direct substitution.