Question 1

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 6 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,8 )$ 
B)  $( 1,8 )$ and $( - 1 , - 8 )$ 
C)  $f ( x )$ has no points with a horizontal tangent line. 
D)  $( 2,28 )$ and $( - 2,28 )$ 
E)  $( 0,0 )$ 
A)  $( 1,8 )$ 
B)  $( 1,8 )$ and $( - 1 , - 8 )$ 
C)  $f ( x )$ has no points with a horizontal tangent line. 
D)  $( 2,28 )$ and $( - 2,28 )$ 
E)  $( 0,0 )$

Question 2

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { 1 - x } - 3 } { x + 8 }$

Accepted Answer

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

Question 3

Find the limit by direct substitution.   $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$

Accepted Answer

A)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 7 
B)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 9 
C)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 84 
D)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = $\infty$ 
E)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 81 
A)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 7 
B)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 9 
C)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 84 
D)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = $\infty$ 
E)  $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$ = 81

Question 4

Determine what statement is true.
The sum of the first n positive integers is

Accepted Answer

A)  $\frac { n ( n + 1 ) } { 2 }$ 
B)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 }$ 
A)  $\frac { n ( n + 1 ) } { 2 }$ 
B)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 }$

Question 5

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ .   $f ( x ) = x ^ { 2 } - 4$

Accepted Answer

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

Question 6

Find the derivative of the function.   $f ( x ) = \frac { 1 } { 14 x ^ { 3 } }$

Accepted Answer

A)  $\frac { 1 } { 14 }$ 
B)  $2 x + 14$ 
C)  $2 x - 14$ 
D)  $\frac { 42 } { x ^ { 4 } }$ 
E)  $- \frac { 3 } { 14 x ^ { 4 } }$ 
A)  $\frac { 1 } { 14 }$ 
B)  $2 x + 14$ 
C)  $2 x - 14$ 
D)  $\frac { 42 } { x ^ { 4 } }$ 
E)  $- \frac { 3 } { 14 x ^ { 4 } }$

Question 7

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not.   $f ( x ) = \frac { \sqrt { x + 4 } - 5 } { x - 6 }$ , $\lim _ { x \rightarrow 6 } f ( x )$

Accepted Answer

A)     $\lim _ { x \rightarrow 6 } f ( x )$ =4 
B)     $\lim _ { x \rightarrow 6 } f ( x )$ does not exist 
C)     $\lim _ { x \rightarrow 6 } f ( x )$ does not exist 
D)    $\lim _ { x \rightarrow 6 } f ( x )$ =0 
E)     $\lim _ { x \rightarrow 6 } f ( x )$ =6 
A)     $\lim _ { x \rightarrow 6 } f ( x )$ =4 
B)     $\lim _ { x \rightarrow 6 } f ( x )$ does not exist 
C)     $\lim _ { x \rightarrow 6 } f ( x )$ does not exist 
D)    $\lim _ { x \rightarrow 6 } f ( x )$ =0 
E)     $\lim _ { x \rightarrow 6 } f ( x )$ =6

Question 8

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { x + 8 } - \sqrt { 8 } } { x }$

Accepted Answer

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

Question 9

Find the derivative of the function.   $f ( x ) = \frac { 1 } { 3 x ^ { 2 } }$

Accepted Answer

A)  $2 x - 3$ 
B)  $- \frac { 2 } { 3 x ^ { 3 } }$ 
C)  $\frac { 6 } { x ^ { 3 } }$ 
D)  $2 x + 3$ 
E)  $\frac { 1 } { 3 }$ 
A)  $2 x - 3$ 
B)  $- \frac { 2 } { 3 x ^ { 3 } }$ 
C)  $\frac { 6 } { x ^ { 3 } }$ 
D)  $2 x + 3$ 
E)  $\frac { 1 } { 3 }$

Question 10

Use the given information to evaluate the limit.   $\lim _ { x \rightarrow c } f ( x ) = 9$ , $\lim _ { x \rightarrow 6 } g ( x ) = - 4$ $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$

Accepted Answer

A)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 4 
B)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = -13 
C)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 169 
D)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 5 
E)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = -216 
A)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 4 
B)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = -13 
C)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 169 
D)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = 5 
E)  $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$ = -216

Question 11

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 66 - x ^ { 3 }&[1,4]\
\end{array}$

Accepted Answer

A) A = 85.5 square units 
B) A = 36.75 square units 
C) A = 231.75 square units 
D) A = 183 square units 
E) A = 134.25 square units 
A) A = 85.5 square units 
B) A = 36.75 square units 
C) A = 231.75 square units 
D) A = 183 square units 
E) A = 134.25 square units

Question 12

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point.   $f ( x ) = \sqrt { x - 4 } , \quad ( 8,2 )$

Accepted Answer

A)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 4 } } \
A t ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
B)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 4 } } \
\text { At } ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 4 } } \
A t ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
D)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 4 } } \
A t ( 8,2 ) , m = - \frac { 1 } { 4 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 4 } } \
\text { At } ( 8,2 ) , m = - \frac { 1 } { 4 }
\end{array}$ 
A)  $\begin{array} { l } 
m = \frac { 1 } { \sqrt { x - 4 } } \
A t ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
B)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 4 } } \
\text { At } ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
C)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 4 } } \
A t ( 8,2 ) , m = \frac { 1 } { 4 }
\end{array}$ 
D)  $\begin{array} { l } 
m = \frac { 1 } { 2 \sqrt { x - 4 } } \
A t ( 8,2 ) , m = - \frac { 1 } { 4 }
\end{array}$ 
E)  $\begin{array} { l } 
m = - \frac { 1 } { 2 \sqrt { x - 4 } } \
\text { At } ( 8,2 ) , m = - \frac { 1 } { 4 }
\end{array}$

Question 13

Use the derivative of $f ( x ) = 3 x ^ { 4 } + 4 x ^ { 3 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Accepted Answer

A)  $( 1,7 )$ and $( - 1 , - 1 )$ 
B)  $( - 1 , - 12 )$ 
C)  $f ( x )$ has no points with a horizontal tangent line. 
D)  $( 0,0 )$ and $( - 1 , - 1 )$ 
E)  $( 3,0 )$ 
A)  $( 1,7 )$ and $( - 1 , - 1 )$ 
B)  $( - 1 , - 12 )$ 
C)  $f ( x )$ has no points with a horizontal tangent line. 
D)  $( 0,0 )$ and $( - 1 , - 1 )$ 
E)  $( 3,0 )$

Question 14

Find  $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)$

Accepted Answer

A)  $\lim_{n\rightarrow \infty } S ( n ) = 3$ 
B)  $\lim _{n\rightarrow \infty }S ( n ) = 7$ 
C)  $\lim_{n\rightarrow \infty } S ( n ) = 4$ 
D)   $\lim_{n\rightarrow\infty } S ( n ) = 5$ 
E)   $\lim_{n\rightarrow \infty } S ( n ) = 6$ 
A)  $\lim_{n\rightarrow \infty } S ( n ) = 3$ 
B)  $\lim _{n\rightarrow \infty }S ( n ) = 7$ 
C)  $\lim_{n\rightarrow \infty } S ( n ) = 4$ 
D)   $\lim_{n\rightarrow\infty } S ( n ) = 5$ 
E)   $\lim_{n\rightarrow \infty } S ( n ) = 6$

Question 15

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \ln ( 11 - x ) , \lim _ { x \rightarrow - 1 } f ( x )$

Accepted Answer

A)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
B)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 11$ 
C)    $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
D)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
E)     $\lim _ { x \rightarrow - 1 } f ( x )$ does not exist. 
A)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
B)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 11$ 
C)    $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
D)     $\lim _ { x \rightarrow - 1 } f ( x ) = \ln 12$ 
E)     $\lim _ { x \rightarrow - 1 } f ( x )$ does not exist.

Question 16

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 4 - x ^ { 2 }$

Accepted Answer

A)  $7.28$ 
B)  $3.64$ 
C)  $4.48$ 
D)  $3.83$ 
E)  $7.25$ 
A)  $7.28$ 
B)  $3.64$ 
C)  $4.48$ 
D)  $3.83$ 
E)  $7.25$

Question 17

Consider the function $f ( x ) = x ^ { 9 }$ ,select the graph of $f$ and $f ^ { \prime }$ on the same set of coordinate axes.

Accepted Answer

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

Question 18

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln ( 5 x - 1 ) } { x - 1 }$

Accepted Answer

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

Question 19

Find the limit (if it exists).  $\lim _ { x \rightarrow \infty } \left( \frac { 4 + x } { 7 + x } \right)$

Accepted Answer

A) 4 
B) 1 
C) -1 
D) -4 
E) does not exist 
A) 4 
B) 1 
C) -1 
D) -4 
E) does not exist

Question 20

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval.
 $\begin{array} {l } 
 \text {Function }& \text {Interval }\
f ( x ) = 9 - x ^ { 3 }&[1,2]\
\end{array}$

Accepted Answer

A) A = 5.375 square units 
B) A = 5.5 square units 
C) A = 5.25 square units 
D) A = 5.625 square units 
E) A = 5.125 square units 
A) A = 5.375 square units 
B) A = 5.5 square units 
C) A = 5.25 square units 
D) A = 5.625 square units 
E) A = 5.125 square units

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 6 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { 1 - x } - 3 } { x + 8 }$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$

Determine what statement is true. The sum of the first n positive integers is

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ . $f ( x ) = x ^ { 2 } - 4$

Find the derivative of the function. $f ( x ) = \frac { 1 } { 14 x ^ { 3 } }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { \sqrt { x + 4 } - 5 } { x - 6 }$ , $\lim _ { x \rightarrow 6 } f ( x )$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { x + 8 } - \sqrt { 8 } } { x }$

Find the derivative of the function. $f ( x ) = \frac { 1 } { 3 x ^ { 2 } }$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 9$ , $\lim _ { x \rightarrow 6 } g ( x ) = - 4$ $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=66- [1,4]

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 4 } , \quad ( 8,2 )$

Use the derivative of $f ( x ) = 3 x ^ { 4 } + 4 x ^ { 3 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \ln ( 11 - x ) , \lim _ { x \rightarrow - 1 } f ( x )$

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 4 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 4 - x ^ { 2 }$

Consider the function $f ( x ) = x ^ { 9 }$ ,select the graph of $f$ and $f ^ { \prime }$ on the same set of coordinate axes.

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln ( 5 x - 1 ) } { x - 1 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 + x } { 7 + x } \right)$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=9- [1,2]

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

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Exam 12: Limits and An Introduction To Calculus

Use the derivative of f(x)=2x3+6xf ( x ) = 2 x ^ { 3 } + 6 xf(x)=2x3+6x to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

Select the correct graph for the following function using a graphing utility. f(x)=1−x−3x+8f ( x ) = \frac { \sqrt { 1 - x } - 3 } { x + 8 }f(x)=x+81−x​−3​

Find the limit by direct substitution. lim⁡x→−3x+84\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }limx→−3​x+84​

Determine what statement is true. The sum of the first n positive integers is

Select a graph of the function and the tangent line at the point (1,f(1))( 1 , f ( 1 ) )(1,f(1)) . f(x)=x2−4f ( x ) = x ^ { 2 } - 4f(x)=x2−4

Find the derivative of the function. f(x)=114x3f ( x ) = \frac { 1 } { 14 x ^ { 3 } }f(x)=14x31​

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. f(x)=x+4−5x−6f ( x ) = \frac { \sqrt { x + 4 } - 5 } { x - 6 }f(x)=x−6x+4​−5​ , lim⁡x→6f(x)\lim _ { x \rightarrow 6 } f ( x )limx→6​f(x)

Select the correct graph for the following function using a graphing utility. f(x)=x+8−8xf ( x ) = \frac { \sqrt { x + 8 } - \sqrt { 8 } } { x }f(x)=xx+8​−8​​

Find the derivative of the function. f(x)=13x2f ( x ) = \frac { 1 } { 3 x ^ { 2 } }f(x)=3x21​

Use the given information to evaluate the limit. lim⁡x→cf(x)=9\lim _ { x \rightarrow c } f ( x ) = 9limx→c​f(x)=9 , lim⁡x→6g(x)=−4\lim _ { x \rightarrow 6 } g ( x ) = - 4limx→6​g(x)=−4 lim⁡x→c[6f(x)g(x)]\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]limx→c​[6f(x)g(x)]

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=66- [1,4]

Find a formula for the slope of the graph of fff at the point (x,f(x))( x , f ( x ) )(x,f(x)) .Then use it to find the slope at the given point. f(x)=x−4,(8,2)f ( x ) = \sqrt { x - 4 } , \quad ( 8,2 )f(x)=x−4​,(8,2)

Use the derivative of f(x)=3x4+4x3f ( x ) = 3 x ^ { 4 } + 4 x ^ { 3 }f(x)=3x4+4x3 to determine any points on the graph of f(x)f ( x )f(x) at which the tangent line is horizontal.

Find lim⁡n→∞S(n)∑i=1n[4−2(in)](1n)\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)limn→∞​S(n)∑i=1n​[4−2(ni​)](n1​)

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. f(x)=ln⁡(11−x),lim⁡x→−1f(x)f ( x ) = \ln ( 11 - x ) , \lim _ { x \rightarrow - 1 } f ( x )f(x)=ln(11−x),limx→−1​f(x)

Approximate the area of the indicated region under the given curve using five rectangles. f(x)=4−x2f ( x ) = 4 - x ^ { 2 }f(x)=4−x2

Consider the function f(x)=x9f ( x ) = x ^ { 9 }f(x)=x9 ,select the graph of fff and f′f ^ { \prime }f′ on the same set of coordinate axes.

Select the correct graph for the following function using a graphing utility. f(x)=ln⁡(5x−1)x−1f ( x ) = \frac { \ln ( 5 x - 1 ) } { x - 1 }f(x)=x−1ln(5x−1)​

Find the limit (if it exists). lim⁡x→∞(4+x7+x)\lim _ { x \rightarrow \infty } \left( \frac { 4 + x } { 7 + x } \right)limx→∞​(7+x4+x​)

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=9- [1,2]

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Filters

Use the derivative of $f ( x ) = 2 x ^ { 3 } + 6 x$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { 1 - x } - 3 } { x + 8 }$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 3 } \sqrt { x + 84 }$

Select a graph of the function and the tangent line at the point $( 1 , f ( 1 ) )$ . $f ( x ) = x ^ { 2 } - 4$

Find the derivative of the function. $f ( x ) = \frac { 1 } { 14 x ^ { 3 } }$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \frac { \sqrt { x + 4 } - 5 } { x - 6 }$ , $\lim _ { x \rightarrow 6 } f ( x )$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \sqrt { x + 8 } - \sqrt { 8 } } { x }$

Find the derivative of the function. $f ( x ) = \frac { 1 } { 3 x ^ { 2 } }$

Use the given information to evaluate the limit. $\lim _ { x \rightarrow c } f ( x ) = 9$ , $\lim _ { x \rightarrow 6 } g ( x ) = - 4$ $\lim _ { x \rightarrow c } [ 6 f ( x ) g ( x ) ]$

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \sqrt { x - 4 } , \quad ( 8,2 )$

Use the derivative of $f ( x ) = 3 x ^ { 4 } + 4 x ^ { 3 }$ to determine any points on the graph of $f ( x )$ at which the tangent line is horizontal.

Find $\lim _{n \rightarrow \infty} S(n) \sum_{i=1}^{n}\left[4-2\left(\frac{i}{n}\right)\right]\left(\frac{1}{n}\right)$

Select the correct graph for the following function using a graphing utility.Determine whether the limit exists or not. $f ( x ) = \ln ( 11 - x ) , \lim _ { x \rightarrow - 1 } f ( x )$

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 4 - x ^ { 2 }$ $Approximate the area of the indicated region under the given curve using five rectangles. f ( x ) = 4 - x ^ { 2 }$

Consider the function $f ( x ) = x ^ { 9 }$ ,select the graph of $f$ and $f ^ { \prime }$ on the same set of coordinate axes.

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { \ln ( 5 x - 1 ) } { x - 1 }$

Find the limit (if it exists). $\lim _ { x \rightarrow \infty } \left( \frac { 4 + x } { 7 + x } \right)$