Question 1

Find
$$\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$$
by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 13 } } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 13 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 13 } } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 13 }$

Question 2

Sketch the graph of the equation $y = 2 | x - 3 | + 2$.

Accepted Answer

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

Question 3

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$, if it exists.

Accepted Answer

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

Question 4

The average salary (in thousands of dollars) at a certain company for the first ten years of its existence can be approximated by the model $S = 0.2 t ^ { 2 } + 2.1 t + 32.5,0 \leq t \leq 10$. How many years did it take for the average salary to reach $\$ 41,000$ ?
Round the answer to the nearest hundredth of a year.

Accepted Answer

A)  $447.13$ 
B)  $447.06$ 
C)  $447.24$ 
D)  $447.37$ 
E)  $447.72$ 
A)  $447.13$ 
B)  $447.06$ 
C)  $447.24$ 
D)  $447.37$ 
E)  $447.72$

Question 5

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$.

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\infty$ 
D)  $\frac { 1 } { 8 }$ 
E)  the slope is undefined at this point 
A)  $\frac { 1 } { 4 }$ 
B)  4 
C)  $\infty$ 
D)  $\frac { 1 } { 8 }$ 
E)  the slope is undefined at this point

Question 6

Use the figure below to approximate the slope of the curve at the point $( x , y )$.

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  The slope is undefined. 
C)  1 
D)  $\frac { 1 } { 2 }$ 
E)  0 
A)  $- \frac { 1 } { 2 }$ 
B)  The slope is undefined. 
C)  1 
D)  $\frac { 1 } { 2 }$ 
E)  0

Question 7

Determine any points on the graph of the following function at which the tangent line is horizontal.
$$f ( x ) = x ^ { 2 } + 6 x - 8$$

Accepted Answer

A)  $( 3,19 )$ 
B)  $( 3,0 )$ 
C)  $( - 3,0 )$ 
D)  $( 1 , - 1 )$ 
E)  $( - 3 , - 17 )$ 
A)  $( 3,19 )$ 
B)  $( 3,0 )$ 
C)  $( - 3,0 )$ 
D)  $( 1 , - 1 )$ 
E)  $( - 3 , - 17 )$

Question 8

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals.
$$f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$$

Accepted Answer

A)  $5.71$ square units 
B)  $2.86$ square units 
C)  $3.69$ square units 
D)  $8.57$ square units 
E)  $17.13$ square units 
A)  $5.71$ square units 
B)  $2.86$ square units 
C)  $3.69$ square units 
D)  $8.57$ square units 
E)  $17.13$ square units

Question 9

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Accepted Answer

A)  6 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $2 \sqrt { 3 }$ 
E)  limit does not exist 
A)  6 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $2 \sqrt { 3 }$ 
E)  limit does not exist

Question 10

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$-axis on the interval $[ 1,8 ]$.

Accepted Answer

A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$ 
A)  $\frac { 1087 } { 2 }$ 
B)  $\frac { 2023 } { 24 }$ 
C)  $\frac { 191 } { 2 }$ 
D)  574 
E)  $\frac { 1267 } { 24 }$

Question 11

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 12

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$, if it exists.

Accepted Answer

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

Question 13

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 14

Find
$$\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$$
by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 7 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 3 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 10 }$ 
A)  $\frac { \sqrt { 7 } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 3 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\sqrt { 10 }$

Question 15

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $\frac { 6 } { 5 }$ 
C)  $\infty$ 
D)  $\frac { 1 } { 30 }$ 
E)  limit does not exist 
A)  0 
B)  $\frac { 6 } { 5 }$ 
C)  $\infty$ 
D)  $\frac { 1 } { 30 }$ 
E)  limit does not exist

Question 16

Approximate the area of the indicated region under the given curve using five rectangles.
$$f ( x ) = 5 - x ^ { 2 }$$

Accepted Answer

A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $\quad 9.25$ 
A)  $9.28$ 
B)  $4.64$ 
C)  $5.48$ 
D)  $4.83$ 
E)  $\quad 9.25$

Question 17

Find all solutions of the equation
$x ^ { 3 } + 3 x ^ { 2 } - 10 x - 30 = 0$

Accepted Answer

A)  
$x = \sqrt { 10 }$
$x = \sqrt { 10 }$
$x = - 3$ 
B) $x = \sqrt { 7 }$
 $x = - \sqrt { 7 }$
$x = - 3$ 
C)  $x = \sqrt { 10 }$
$x = - \sqrt { 7 }$
$$x = 3$$ 
D)  $x = - \sqrt { 10 }$
$x = - \sqrt { 10 }$
$x = 3$ 
E) $x = \sqrt { 7 }$
 $x = - \sqrt { 7 }$
$x = 3$ 
A)  
$x = \sqrt { 10 }$
$x = \sqrt { 10 }$
$x = - 3$ 
B) $x = \sqrt { 7 }$
 $x = - \sqrt { 7 }$
$x = - 3$ 
C)  $x = \sqrt { 10 }$
$x = - \sqrt { 7 }$
$$x = 3$$ 
D)  $x = - \sqrt { 10 }$
$x = - \sqrt { 10 }$
$x = 3$ 
E) $x = \sqrt { 7 }$
 $x = - \sqrt { 7 }$
$x = 3$

Question 18

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 19

Find the coordinates of the point that is located 8 units below the $x$-axis and 5 units to the left of the y-axis.

Accepted Answer

A)  $( - 8 , - 5 )$ 
B)  $( 8 , - 5 )$ 
C)  $( - 5,8 )$ 
D)  $( 5 , - 8 )$ 
E)  $( - 5 , - 8 )$ 
A)  $( - 8 , - 5 )$ 
B)  $( 8 , - 5 )$ 
C)  $( - 5,8 )$ 
D)  $( 5 , - 8 )$ 
E)  $( - 5 , - 8 )$

Question 20

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)  $( - 4 , - 24 )$

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

Sketch the graph of the equation $y = 2 | x - 3 | + 2$ .

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 } , if it exists.$

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals. $f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

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

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x } , if it exists.$

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

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$ by direct substitution.

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Find all solutions of the equation $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 30 = 0$

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

Find the coordinates of the point that is located 8 units below the $x$ -axis and 5 units to the left of the y-axis.

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

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

Limits and an Introduction to Calculus

Filters

Exam 12: Review of Graphs, Equations, and Inequalities

Find lim⁡x→6x+7x−2\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }limx→6​x−2x+7​​ by direct substitution.

Sketch the graph of the equation y=2∣x−3∣+2y = 2 | x - 3 | + 2y=2∣x−3∣+2 .

Use the graph below to find lim⁡x→−4∣x+4∣x+4\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }limx→−4​x+4∣x+4∣​ , if it exists.

Use the limit process to find the slope of the graph of x+12\sqrt { x + 12 }x+12​ at (4,4)( 4,4 )(4,4) .

Use the figure below to approximate the slope of the curve at the point (x,y)( x , y )(x,y) .

Determine any points on the graph of the following function at which the tangent line is horizontal. f(x)=x2+6x−8f ( x ) = x ^ { 2 } + 6 x - 8f(x)=x2+6x−8

Approximate the area of the region under the function below on the interval [0,2][ 0,2 ][0,2] using 8 rectangles. Round your answer to two decimals. f(x)=13x4f ( x ) = \frac { 1 } { 3 } x ^ { 4 }f(x)=31​x4

Find lim⁡x→0−xx+3−3\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }limx→0−​x+3​−3​x​

Use the limit process to find the area of the region between f(x)=18(x2+8x)f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)f(x)=81​(x2+8x) and the xxx -axis on the interval [1,8][ 1,8 ][1,8] .

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​ .

Consider the following graph of the function and approximate lim⁡x→0e−2x−1x\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }limx→0​xe−2x−1​ , if it exists.

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

Find lim⁡x→7x+3x−3\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }limx→7​x−3x+3​​ by direct substitution.

Find lim⁡x→∞65x\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }limx→∞​5x6​ (if it exists).

Approximate the area of the indicated region under the given curve using five rectangles. f(x)=5−x2f ( x ) = 5 - x ^ { 2 }f(x)=5−x2

Find all solutions of the equation x3+3x2−10x−30=0x ^ { 3 } + 3 x ^ { 2 } - 10 x - 30 = 0x3+3x2−10x−30=0

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​​ .

Find the coordinates of the point that is located 8 units below the xxx -axis and 5 units to the left of the y-axis.

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

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

Limits and an Introduction to Calculus

Filters

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 7 } } { x - 2 }$ by direct substitution.

Sketch the graph of the equation $y = 2 | x - 3 | + 2$ .

Use the graph below to find $\lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 4 } \frac { | x + 4 | } { x + 4 } , if it exists.$

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$ .

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Determine any points on the graph of the following function at which the tangent line is horizontal. $f ( x ) = x ^ { 2 } + 6 x - 8$

Approximate the area of the region under the function below on the interval $[ 0,2 ]$ using 8 rectangles. Round your answer to two decimals. $f ( x ) = \frac { 1 } { 3 } x ^ { 4 }$

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 3 } - \sqrt { 3 } }$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the $x$ -axis on the interval $[ 1,8 ]$ .

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

Consider the following graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x }$ , if it exists. $Consider the following graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x } , if it exists.$

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

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$ by direct substitution.

Find $\lim _ { x \rightarrow \infty } \frac { 6 } { 5 x }$ (if it exists).

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 5 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 5 - x ^ { 2 }$

Find all solutions of the equation $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 30 = 0$

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

Find the coordinates of the point that is located 8 units below the $x$ -axis and 5 units to the left of the y-axis.

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