Question 1

Using the summation formulas and properties, evaluate the following expression.
$$\sum _ { i = 1 } ^ { 60 } 4$$

Accepted Answer

A)  7320 
B)  1830 
C)  236 
D)  240 
E)  64 
A)  7320 
B)  1830 
C)  236 
D)  240 
E)  64

Question 2

Solve the inequality and graph the solution on the real number line.
$$\frac { x + 12 } { x + 2 } - 2 < 0$$

Accepted Answer

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

Question 3

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)  $14 x$

Question 4

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$.

Accepted Answer

A)  $\frac { \sqrt { 15 } } { 30 }$ 
B)  $\frac { \sqrt { 15 } } { 15 }$ 
C)  0 
D)  $\frac { \sqrt { 15 } } { 2 }$ 
E)  limit does not exist 
A)  $\frac { \sqrt { 15 } } { 30 }$ 
B)  $\frac { \sqrt { 15 } } { 15 }$ 
C)  0 
D)  $\frac { \sqrt { 15 } } { 2 }$ 
E)  limit does not exist

Question 5

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$.

Accepted Answer

A)  $\frac { 1 } { 26 }$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 2 } { 13 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 26 }$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 2 } { 13 }$ 
E)  limit does not exist

Question 6

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$.

Accepted Answer

A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist 
A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 15 }$ 
C)  $\frac { 15 } { 2 }$ 
D)  $\frac { 2 } { 15 }$ 
E)  limit does not exist

Question 7

Sketch the solution to the following inequality on the real number line:
$$3 | 7 - 3 x | \leq 9$$

Accepted Answer

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

Question 8

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 9

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$$

Accepted Answer

A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist. 
A)  $- \frac { 7 } { 3 }$ 
B)  $- 4$ 
C)  4 
D)  7 
E)  The limit does not exist.

Question 10

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$, if it exists.

Accepted Answer

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

Question 11

Find the $y$-intercept of the graph of the equation $5 y = 6 x + 30$.

Accepted Answer

A)  $( 0 , - 6 )$ 
B)  $( - 5,0 )$ 
C)  $( 0,6 )$ 
D)  $( 0 , - 5 )$ 
E)  $( 0,30 )$ 
A)  $( 0 , - 6 )$ 
B)  $( - 5,0 )$ 
C)  $( 0,6 )$ 
D)  $( 0 , - 5 )$ 
E)  $( 0,30 )$

Question 12

Find the following limit of the sequence as $n$ approaches infinity, if it exists.
$$a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$$

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 11 } { 6 }$ 
E)  The limit does not exist. 
A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 11 } { 6 }$ 
E)  The limit does not exist.

Question 13

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$.

Accepted Answer

A)  -12 
B)  52 
C)  28 
D)  48 
E)  - 44 
A)  -12 
B)  52 
C)  28 
D)  48 
E)  - 44

Question 14

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$, if it exists.

Accepted Answer

A)  1 
B)  4 
C)  8 
D)  $- 1$ 
E)  The limit does not exist. 
A)  1 
B)  4 
C)  8 
D)  $- 1$ 
E)  The limit does not exist.

Question 15

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 16

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Accepted Answer

A)  0 
B)  $- \frac { 3 } { 7 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $- \infty$ 
E)  limit does not exist 
A)  0 
B)  $- \frac { 3 } { 7 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $- \infty$ 
E)  limit does not exist

Question 17

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- 6$ 
C)  6 
D)  $- 4$ 
E)  The limit does not exist. 
A)  $- \frac { 1 } { 2 }$ 
B)  $- 6$ 
C)  6 
D)  $- 4$ 
E)  The limit does not exist.

Question 18

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$.

Accepted Answer

A)  12 
B)  $\infty$ 
C)  6 
D)  0 
E)  limit does not exist 
A)  12 
B)  $\infty$ 
C)  6 
D)  0 
E)  limit does not exist

Question 19

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

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 60 } 4$

Solve the inequality and graph the solution on the real number line. $\frac { x + 12 } { x + 2 } - 2 < 0$

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

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$ .

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$ .

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

Sketch the solution to the following inequality on the real number line: $3 | 7 - 3 x | \leq 9$

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

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$

Find the $y$ -intercept of the graph of the equation $5 y = 6 x + 30$ .

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$ .

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$ , if it exists. $Consider the graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x } , if it exists.$

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

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$ . $Use the graph to find \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 } .$

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

Limits and an Introduction to Calculus

Filters

Exam 12: Review of Graphs, Equations, and Inequalities

Using the summation formulas and properties, evaluate the following expression. ∑i=1604\sum _ { i = 1 } ^ { 60 } 4∑i=160​4

Solve the inequality and graph the solution on the real number line. x+12x+2−2<0\frac { x + 12 } { x + 2 } - 2 < 0x+2x+12​−2<0

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

Find lim⁡y→015+y−15y\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }limy→0​y15+y​−15​​ .

Find lim⁡x→13−x−13x2−169\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }limx→13−​x2−169x−13​ .

Find lim⁡x→15−x−15x2−225\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }limx→15−​x2−225x−15​ .

Sketch the solution to the following inequality on the real number line: 3∣7−3x∣≤93 | 7 - 3 x | \leq 93∣7−3x∣≤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.

Find the following limit, if it exists. lim⁡x→∞4x2+3−2−x2\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }limx→∞​−2−x24x2+3​

Use the graph below to find lim⁡x→−2∣x+2∣x+2\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }limx→−2​x+2∣x+2∣​ , if it exists.

Find the yyy -intercept of the graph of the equation 5y=6x+305 y = 6 x + 305y=6x+30 .

Find the following limit of the sequence as nnn approaches infinity, if it exists. an=1n(n+4n[n(n+1)6])a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)an​=n1​(n+n4​[6n(n+1)​])

Use the limit process to find the slope of the graph of 4x−8x24 x - 8 x ^ { 2 }4x−8x2 at (3,−60)( 3 , - 60 )(3,−60) .

Consider the graph of the function and approximate lim⁡x→0sin⁡(8x)x\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }limx→0​xsin(8x)​ , if it exists.

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

Find lim⁡x→∞1−3x1+7x\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }limx→∞​1+7x1−3x​ (if it exists).

Find the following limit, if it exists. lim⁡x→∞6x2−4−3−x2\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }limx→∞​−3−x26x2−4​

Use the graph to find lim⁡x→23x2−12x−2\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }limx→2​x−23x2−12​ .

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.

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

Using the summation formulas and properties, evaluate the following expression. $\sum _ { i = 1 } ^ { 60 } 4$

Solve the inequality and graph the solution on the real number line. $\frac { x + 12 } { x + 2 } - 2 < 0$

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

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 15 + y } - \sqrt { 15 } } { y }$ .

Find $\lim _ { x \rightarrow 13 ^ { - } } \frac { x - 13 } { x ^ { 2 } - 169 }$ .

Find $\lim _ { x \rightarrow 15 ^ { - } } \frac { x - 15 } { x ^ { 2 } - 225 }$ .

Sketch the solution to the following inequality on the real number line: $3 | 7 - 3 x | \leq 9$

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

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } + 3 } { - 2 - x ^ { 2 } }$

Use the graph below to find $\lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 }$ , if it exists. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists.$

Find the $y$ -intercept of the graph of the equation $5 y = 6 x + 30$ .

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$

Use the limit process to find the slope of the graph of $4 x - 8 x ^ { 2 }$ at $( 3 , - 60 )$ .

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$ , if it exists. $Consider the graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x } , if it exists.$

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

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 6 x ^ { 2 } - 4 } { - 3 - x ^ { 2 } }$

Use the graph to find $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$ . $Use the graph to find \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 } .$

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