Question 1

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$.

Accepted Answer

A)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 9 } { 4 }$ 
B)  $\quad S ( n ) = \frac { 9 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 9$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 36 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$, the limit does not exist 
A)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 9 } { 4 }$ 
B)  $\quad S ( n ) = \frac { 9 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 9$ 
C)  $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 36 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
D)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$ 
E)  $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$, the limit does not exist D

Question 2

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. B

Question 3

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

Accepted Answer

A)  $- \frac { 3 } { 2 }$ 
B)  0 
C)  1 
D)  $\frac { 3 } { 2 }$ 
E)  $- 1$ 
A)  $- \frac { 3 } { 2 }$ 
B)  0 
C)  1 
D)  $\frac { 3 } { 2 }$ 
E)  $- 1$ E

Question 4

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$-axis on the interval $[ 0,6 ]$.

Accepted Answer

A)  246 
B)  72 
C)  221 
D)  102 
E)  77 
A)  246 
B)  72 
C)  221 
D)  102 
E)  77

Question 5

Find all solutions of the equation
$$1 - \frac { 4 } { x } - \frac { 21 } { x ^ { 2 } } = 0$$

Accepted Answer

A) 
$x = - 7$
$x = - 3$ 
B) 
$x = 7$
$x = - 3$ 
C) 
$$\begin{array} { l } 
x = - 7 \
x = 3
\end{array}$$ 
D) 
$x = 7$

$x = 3$ 
E) 
None of the above 
A) 
$x = - 7$
$x = - 3$ 
B) 
$x = 7$
$x = - 3$ 
C) 
$$\begin{array} { l } 
x = - 7 \
x = 3
\end{array}$$ 
D) 
$x = 7$

$x = 3$ 
E) 
None of the above

Question 6

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 7

Find the values of $x$ that solve the equation
$$( 13 x - 2 ) ^ { 2 } = 25$$

Accepted Answer

A)  $x = \frac { 7 } { 13 } , \quad x = \frac { - 7 } { 13 }$ 
B)  $x = \frac { 7 } { 13 } , x = \frac { - 3 } { 13 }$ 
C)  $x = 7 , \quad x = - 7$ 
D)  $x = \frac { 3 } { 13 } , \quad x = \frac { - 7 } { 13 }$ 
E)  $x = \frac { 3 } { 13 } , \quad x = \frac { - 3 } { 13 }$ 
A)  $x = \frac { 7 } { 13 } , \quad x = \frac { - 7 } { 13 }$ 
B)  $x = \frac { 7 } { 13 } , x = \frac { - 3 } { 13 }$ 
C)  $x = 7 , \quad x = - 7$ 
D)  $x = \frac { 3 } { 13 } , \quad x = \frac { - 7 } { 13 }$ 
E)  $x = \frac { 3 } { 13 } , \quad x = \frac { - 3 } { 13 }$

Question 8

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$.

Accepted Answer

A)  - 7 
B)  - 11 
C)  7 
D)  - 4 
E)  limit does not exist 
A)  - 7 
B)  - 11 
C)  7 
D)  - 4 
E)  limit does not exist

Question 9

If $f ( x ) = - 3 x ^ { 2 } - 4 x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$

Accepted Answer

A)  $- 6 x - 4$ 
B)  $- 3 x - 4$ 
C)  $2 x - 4$ 
D)  $- 6 x$ 
E)  The limit does not exist. 
A)  $- 6 x - 4$ 
B)  $- 3 x - 4$ 
C)  $2 x - 4$ 
D)  $- 6 x$ 
E)  The limit does not exist.

Question 10

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

Accepted Answer

A)  2 
B)  $\frac { 3 } { 2 }$ 
C)  $- 2$ 
D)  $- 1$ 
E)  1 
A)  2 
B)  $\frac { 3 } { 2 }$ 
C)  $- 2$ 
D)  $- 1$ 
E)  1

Question 11

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

Accepted Answer

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

Question 12

$$\text { Graph the circle } ( x - 4 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 9$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)    
E) 
$\text { None of the above. }$ 
A) 
  
B) 
  
C) 
  
D)    
E) 
$\text { None of the above. }$

Question 13

Let $M$ denote the midpoint of the line segment joining $( 4,3 )$ and $( 9,6 )$. Find the distance from $M$ to the point $( - 3 , - 6 )$. Round the answer to the nearest tenth.

Accepted Answer

A)  $14.4$ 
B)  $13.8$ 
C)  $14.2$ 
D)  $13.6$ 
E)  $13.9$ 
A)  $14.4$ 
B)  $13.8$ 
C)  $14.2$ 
D)  $13.6$ 
E)  $13.9$

Question 14

Complete the table and use the result to estimate
$$\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$$
numerically.
$\begin{array}{|c|c|c|c|c|c|c|r|}
\hline x & -8.1 & -8.01 & -8.001 & -8 & -7.999 & -7.99 & -7.9 \
\hline f(x) & & & & ? & & & \
\hline
\end{array}$

Accepted Answer

A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist 
A)  $- \frac { 1 } { 12 }$ 
B)  12 
C)  $\infty$ 
D)  $- 12$ 
E)  limit does not exist

Question 15

Find the distance between the points $( 3 , - 2 )$ and $( 8 , - 2 )$.

Accepted Answer

A)  7 
B)  $\sqrt { 7 }$ 
C)  2 
D)  $\sqrt { 5 }$ 
E)  5 
A)  7 
B)  $\sqrt { 7 }$ 
C)  2 
D)  $\sqrt { 5 }$ 
E)  5

Question 16

Estimate the following limit numerically, if it exists.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist. 
A)  $\frac { 1 } { 6 }$ 
B)  0 
C)  $- \frac { 1 } { 4 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  The limit does not exist.

Question 17

Find the following limit, if it exists.
$$\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$$

Accepted Answer

A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist. 
A)  $\frac { 5 } { 8 }$ 
B)  $- 5$ 
C)  $- 1$ 
D)  0 
E)  The limit does not exist.

Question 18

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

Accepted Answer

A)  3 
B)  $\frac { 1 } { 2 }$ 
C)  $- 3$ 
D)  0 
E)  The slope is undefined. 
A)  3 
B)  $\frac { 1 } { 2 }$ 
C)  $- 3$ 
D)  0 
E)  The slope is undefined.

Question 19

If $f ( x ) = - 2 x ^ { 2 } + x$, find the following limit, if it exists.
$$\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$$

Accepted Answer

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

Question 20

Find $\lim _ { t \rightarrow 3 } \frac { t ^ { 3 } - 27 } { t - 3 }$.

Accepted Answer

A)  9 
B)  18 
C)  27 
D)  3 
E)  limit does not exist 
A)  9 
B)  18 
C)  27 
D)  3 
E)  limit does not exist

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

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

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$ -axis on the interval $[ 0,6 ]$ .

Find all solutions of the equation $1 - \frac { 4 } { x } - \frac { 21 } { x ^ { 2 } } = 0$

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 values of $x$ that solve the equation $( 13 x - 2 ) ^ { 2 } = 25$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$ .

If $f ( x ) = - 3 x ^ { 2 } - 4 x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

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

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

$\text { Graph the circle } ( x - 4 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 9$

Let $M$ denote the midpoint of the line segment joining $( 4,3 )$ and $( 9,6 )$ . Find the distance from $M$ to the point $( - 3 , - 6 )$ . Round the answer to the nearest tenth.

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find the distance between the points $( 3 , - 2 )$ and $( 8 , - 2 )$ .

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

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

If $f ( x ) = - 2 x ^ { 2 } + x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

Find $\lim _ { t \rightarrow 3 } \frac { t ^ { 3 } - 27 } { t - 3 }$ .

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

Rewrite ∑i=1n9i3n5\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }∑i=1n​n59i3​ as a rational function S(n)S ( n )S(n) and find lim⁡n→∞S(n)\lim _ { n \rightarrow \infty } S ( n )limn→∞​S(n) .

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

Use the limit process to find the area of the region between f(x)=x2+5f ( x ) = x ^ { 2 } + 5f(x)=x2+5 and the xxx -axis on the interval [0,6][ 0,6 ][0,6] .

Find all solutions of the equation 1−4x−21x2=01 - \frac { 4 } { x } - \frac { 21 } { x ^ { 2 } } = 01−x4​−x221​=0

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 values of xxx that solve the equation (13x−2)2=25( 13 x - 2 ) ^ { 2 } = 25(13x−2)2=25

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)=−4+7xf ( x ) = - 4 + 7 xf(x)=−4+7x .

If f(x)=−3x2−4xf ( x ) = - 3 x ^ { 2 } - 4 xf(x)=−3x2−4x , find the following limit, if it exists. 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)​

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

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

Graph the circle (x−4)2+(y−6)2=9\text { Graph the circle } ( x - 4 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 9 Graph the circle (x−4)2+(y−6)2=9

Let MMM denote the midpoint of the line segment joining (4,3)( 4,3 )(4,3) and (9,6)( 9,6 )(9,6) . Find the distance from MMM to the point (−3,−6)( - 3 , - 6 )(−3,−6) . Round the answer to the nearest tenth.

Complete the table and use the result to estimate lim⁡x→−8x+8x2+4x−32\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }limx→−8​x2+4x−32x+8​ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find the distance between the points (3,−2)( 3 , - 2 )(3,−2) and (8,−2)( 8 , - 2 )(8,−2) .

Estimate the following limit numerically, if it exists. lim⁡x→1x−1x2+4x−5\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }limx→1​x2+4x−5x−1​

Find the following limit, if it exists. lim⁡x→∞5x8\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }limx→∞​x85​

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

If f(x)=−2x2+xf ( x ) = - 2 x ^ { 2 } + xf(x)=−2x2+x , find the following limit, if it exists. 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)​

Find lim⁡t→3t3−27t−3\lim _ { t \rightarrow 3 } \frac { t ^ { 3 } - 27 } { t - 3 }limt→3​t−3t3−27​ .

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

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

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

Use the limit process to find the area of the region between $f ( x ) = x ^ { 2 } + 5$ and the $x$ -axis on the interval $[ 0,6 ]$ .

Find all solutions of the equation $1 - \frac { 4 } { x } - \frac { 21 } { x ^ { 2 } } = 0$

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 values of $x$ that solve the equation $( 13 x - 2 ) ^ { 2 } = 25$

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = - 4 + 7 x$ .

If $f ( x ) = - 3 x ^ { 2 } - 4 x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

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

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

$\text { Graph the circle } ( x - 4 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 9$

Let $M$ denote the midpoint of the line segment joining $( 4,3 )$ and $( 9,6 )$ . Find the distance from $M$ to the point $( - 3 , - 6 )$ . Round the answer to the nearest tenth.

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } + 4 x - 32 }$ numerically. x -8.1 -8.01 -8.001 -8 -7.999 -7.99 -7.9 f(x) ?

Find the distance between the points $( 3 , - 2 )$ and $( 8 , - 2 )$ .

Estimate the following limit numerically, if it exists. $\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 4 x - 5 }$

Find the following limit, if it exists. $\lim _ { x \rightarrow \infty } \frac { 5 } { x ^ { 8 } }$

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

If $f ( x ) = - 2 x ^ { 2 } + x$ , find the following limit, if it exists. $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

Find $\lim _ { t \rightarrow 3 } \frac { t ^ { 3 } - 27 } { t - 3 }$ .