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

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)

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)  $14 x$

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

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

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

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)

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.

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.

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.

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)  $( 0,30 )$

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.

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

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.

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.

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

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.

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

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

Exam 12: Review of Graphs, Equations, and Inequalities

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: $3 | 7 - 3 x | \leq 9$

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. $Use the graph below to find \lim _ { x \rightarrow - 2 } \frac { | x + 2 | } { x + 2 } , if it exists. A) - 1 B) - 2 C) 2 D) 1 E) The limit does not exist.$

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. $Consider the graph of the function and approximate \lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x } , if it exists. A) 1 B) 4 C) 8 D) - 1 E) The limit does not exist.$

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 $\lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 }$ . $Use the graph to find \lim _ { x \rightarrow 2 } \frac { 3 x ^ { 2 } - 12 } { x - 2 } . A) 12 B) \infty C) 6 D) 0 E) limit does not exist$

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

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.