Question 1

Determine any points on the graph of the following function at which the tangent line is horizontal.
$$f ( x ) = x ^ { 2 } + 8 x - 8$$

Accepted Answer

A)  $( 4,40 )$ 
B)  $( 4,0 )$ 
C)  $( - 4,0 )$ 
D)  $( 0 , - 8 )$ 
E)  $( - 4 , - 24 )$ 
A)  $( 4,40 )$ 
B)  $( 4,0 )$ 
C)  $( - 4,0 )$ 
D)  $( 0 , - 8 )$ 
E)  $( - 4 , - 24 )$

Question 2

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$.

Accepted Answer

A)  $\frac { 1 } { 7 \sqrt { x - 7 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 7 } }$ 
C)  $\frac { \sqrt { x - 7 } } { 2 }$ 
D)  $\sqrt { x - 7 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 7 \sqrt { x - 7 } }$ 
B)  $\frac { 1 } { 2 \sqrt { x - 7 } }$ 
C)  $\frac { \sqrt { x - 7 } } { 2 }$ 
D)  $\sqrt { x - 7 }$ 
E)  limit does not exist

Question 3

Find the following limit, if it exists.
$$\lim _ { x \rightarrow - 2 } x ^ { 8 }$$

Accepted Answer

A)  $- 2$ 
B)  $- 16$ 
C)  2 
D)  256 
E)  The limit does not exist. 
A)  $- 2$ 
B)  $- 16$ 
C)  2 
D)  256 
E)  The limit does not exist.

Question 4

Find
$$\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$$
by direct substitution.

Accepted Answer

A)  $\frac { 1 } { 160 }$ 
B)  $- \frac { 1 } { 160 }$ 
C)  0 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 1 } { 160 }$ 
B)  $- \frac { 1 } { 160 }$ 
C)  0 
D)  $\infty$ 
E)  limit does not exist

Question 5

Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 2 } } { 15 } - 108$ 
B)  $\frac { \sqrt { 10 } } { 5 } - 108$ 
C)  $\frac { 36 \sqrt { 2 } } { 5 }$ 
D)  $\frac { \sqrt { 10 } } { 5 } + 108$ 
E)  limit does not exist

Question 6

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$, if it exists.

Accepted Answer

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

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 9

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Accepted Answer

A)  $- \frac { 5 } { 49 }$ 
B)  6 
C)  $- 6$ 
D)  $\infty$ 
E)  limit does not exist 
A)  $- \frac { 5 } { 49 }$ 
B)  6 
C)  $- 6$ 
D)  $\infty$ 
E)  limit does not exist

Question 10

Evaluate $\sum _ { j = 1 } ^ { 15 } \left( j ^ { 3 } - 5 j ^ { 2 } \right)$ using the summation formulas and properties.

Accepted Answer

A)  $\quad 14,400$ 
B)  13,275 
C)  14,395 
D)  14,325 
E)  8,200 
A)  $\quad 14,400$ 
B)  13,275 
C)  14,395 
D)  14,325 
E)  8,200

Question 11

Find
$$\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$$
for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$.

Accepted Answer

A)  $\frac { \sqrt { 11 } } { 54 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 6 } - 81$ 
C)  $\frac { 3 \sqrt { 11 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 6 } + 81$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 11 } } { 54 } - 81$ 
B)  $\frac { \sqrt { 3 } } { 6 } - 81$ 
C)  $\frac { 3 \sqrt { 11 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 6 } + 81$ 
E)  limit does not exist

Question 12

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 13

Complete the table and use the result to estimate $$\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & -3.1 & -3.01 & -3.001 & -3 & -2.999 & -2.99 & -2.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 7 }$ 
B)  7 
C)  $\infty$ 
D)  $- 7$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 7 }$ 
B)  7 
C)  $\infty$ 
D)  $- 7$ 
E)  limit does not exist

Question 14

Find an equation of the tangent line to the graph of the following function at the point $( - 1 , - 5 )$.
$- 3 x ^ { 2 } - 2$

Accepted Answer

A)  $y = 6 x - 11$ 
B)  $y = 6 x + 11$ 
C)  $y = 6 x + 1$ 
D)  $y = 30 x - 25$ 
E)  $y = 30 x + 25$ 
A)  $y = 6 x - 11$ 
B)  $y = 6 x + 11$ 
C)  $y = 6 x + 1$ 
D)  $y = 30 x - 25$ 
E)  $y = 30 x + 25$

Question 15

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)  $\$ 484.62$

Question 16

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Accepted Answer

A)  $\frac { 9 } { 25 }$ 
B)  $- 8$ 
C)  8 
D)  $\infty$ 
E)  limit does not exist 
A)  $\frac { 9 } { 25 }$ 
B)  $- 8$ 
C)  8 
D)  $\infty$ 
E)  limit does not exist

Question 17

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line. 
A)  $( 1,20 )$ 
B)  $( 1,20 )$ and $( - 1 , - 20 )$ 
C)  $( 5,700 )$ and $( - 5,700 )$ 
D)  $( 0,0 )$ 
E)  $f ( x )$ has no points with a horizontal tangent line.

Question 18

Find the following limit, if it exists.
$\lim _ { x \rightarrow - 3 } x ^ { 8 }$

Accepted Answer

A)  $- 3$ 
B)  $- 24$ 
C)  3 
D)  6561 
E)  The limit does not exist. 
A)  $- 3$ 
B)  $- 24$ 
C)  3 
D)  6561 
E)  The limit does not exist.

Question 19

Use the graph to find $\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$.

Accepted Answer

A)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
B)  $\infty$ 
C)  0 
D)  3 
E)  limit does not exist 
A)  $\frac { 3 \sqrt { 3 } } { 2 }$ 
B)  $\infty$ 
C)  0 
D)  3 
E)  limit does not exist

Question 20

Find the derivative of $f ( x ) = 7 x ^ { 2 } - 9 x + 4$.

Accepted Answer

A)  $14 x - 9$ 
B)  $7 x - 9$ 
C)  $14 x - 5$ 
D)  $7 x - 5$ 
E)  $14 x$ 
A)  $14 x - 9$ 
B)  $7 x - 9$ 
C)  $14 x - 5$ 
D)  $7 x - 5$ 
E)  $14 x$

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 8 x - 8$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow - 2 } x ^ { 8 }$

Find $\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$ by direct substitution.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

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 $[ 1,4 ]$ .

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

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Evaluate $\sum _ { j = 1 } ^ { 15 } \left( j ^ { 3 } - 5 j ^ { 2 } \right)$ using the summation formulas and properties.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$ .

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

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Find an equation of the tangent line to the graph of the following function at the point $( - 1 , - 5 )$ . $- 3 x ^ { 2 } - 2$

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.

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 8 }$

Use the graph to find $\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$ . $Use the graph to find \lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right) .$

Find the derivative of $f ( x ) = 7 x ^ { 2 } - 9 x + 4$ .

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

Filters

Exam 11: Limits and an Introduction to Calculus

Determine any points on the graph of the following function at which the tangent line is horizontal. f(x)=x2+8x−8f ( x ) = x ^ { 2 } + 8 x - 8f(x)=x2+8x−8

Find lim⁡h→0f(x+h)−f(x)h\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }limh→0​hf(x+h)−f(x)​ for f(x)=x−7f ( x ) = \sqrt { x - 7 }f(x)=x−7​ .

Find the following limit, if it exists. lim⁡x→−2x8\lim _ { x \rightarrow - 2 } x ^ { 8 }limx→−2​x8

Find lim⁡x→7x−8x2+12x+27\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }limx→7​x2+12x+27x−8​ by direct substitution.

Find lim⁡x→3[g(x)−f(x)]\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]limx→3​[g(x)−f(x)] for f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 and g(x)=x2+95x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }g(x)=5x2x2+9​​ .

Consider the following graph of the function and approximate lim⁡x→0e−4x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }limx→0​xe−4x−1​ , 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] .

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

Find lim⁡x→−∞[x(7+x)2−6]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]limx→−∞​[(7+x)2x​−6] (if it exists).

Evaluate ∑j=115(j3−5j2)\sum _ { j = 1 } ^ { 15 } \left( j ^ { 3 } - 5 j ^ { 2 } \right)∑j=115​(j3−5j2) using the summation formulas and properties.

Find lim⁡x→3[g(x)−f(x)]\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]limx→3​[g(x)−f(x)] for f(x)=3x3f ( x ) = 3 x ^ { 3 }f(x)=3x3 and g(x)=x2+26x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }g(x)=6x2x2+2​​ .

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​

Complete the table and use the result to estimate lim⁡x→−3x+3x2−x−12\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }limx→−3​x2−x−12x+3​ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Find an equation of the tangent line to the graph of the following function at the point (−1,−5)( - 1 , - 5 )(−1,−5) . −3x2−2- 3 x ^ { 2 } - 2−3x2−2

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.

Find lim⁡x→−∞[x(5+x)2+8]\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]limx→−∞​[(5+x)2x​+8] (if it exists).

Use the derivative of f(x)=5x3+15xf ( x ) = 5 x ^ { 3 } + 15 xf(x)=5x3+15x to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

Find the following limit, if it exists. lim⁡x→−3x8\lim _ { x \rightarrow - 3 } x ^ { 8 }limx→−3​x8

Use the graph to find lim⁡x→03sin⁡(π3x)\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)limx→0​3sin(3xπ​) .

Find the derivative of f(x)=7x2−9x+4f ( x ) = 7 x ^ { 2 } - 9 x + 4f(x)=7x2−9x+4 .

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

Filters

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 8 x - 8$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = \sqrt { x - 7 }$ .

Find the following limit, if it exists. $\lim _ { x \rightarrow - 2 } x ^ { 8 }$

Find $\lim _ { x \rightarrow 7 } \frac { x - 8 } { x ^ { 2 } + 12 x + 27 }$ by direct substitution.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } }$ .

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 4 x } - 1 } { x } , if it exists.$

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 $[ 1,4 ]$ .

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

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 7 + x ) ^ { 2 } } - 6 \right]$ (if it exists).

Evaluate $\sum _ { j = 1 } ^ { 15 } \left( j ^ { 3 } - 5 j ^ { 2 } \right)$ using the summation formulas and properties.

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 6 x ^ { 2 } }$ .

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

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 3 } \frac { x + 3 } { x ^ { 2 } - x - 12 }$ numerically. x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 f(x) ?

Find an equation of the tangent line to the graph of the following function at the point $( - 1 , - 5 )$ . $- 3 x ^ { 2 } - 2$

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.

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Use the derivative of $f ( x ) = 5 x ^ { 3 } + 15 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 8 }$

Use the graph to find $\lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right)$ . $Use the graph to find \lim _ { x \rightarrow 0 } 3 \sin \left( \frac { \pi } { 3 x } \right) .$

Find the derivative of $f ( x ) = 7 x ^ { 2 } - 9 x + 4$ .