Question 1

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { \frac { 1 } { x ^ { 2 } } - 1 } { x - 1 }$

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 0 
E)  $- \frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 3 }$ 
G) $- \frac { 1 } { 4 }$ 
H) $- \frac { 1 } { 8 }$ 
A) -3 
B) -2 
C) -1 
D) 0 
E)  $- \frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 3 }$ 
G) $- \frac { 1 } { 4 }$ 
H) $- \frac { 1 } { 8 }$

Question 2

A cellular phone company has a roaming charge of 32 cents for every minute or fraction of a minute when you are out of your zone.(a) Sketch a graph of the "out-of-your-zone" costs of cellular phone usage as a function of the length of the call.(b) Discuss the discontinuities of this function and their significance to the cell phone user.

Accepted Answer

(a) @#IMG-DLM& (b) We consider @#IMG-DLM& These limits show

Question 3

Use the given graph to find the indicated quantities:   (a) $\lim _ { x \rightarrow 3 ^ { + } } f ( x )$ (b) $\lim _ { x \rightarrow 3 } f ( x )$ (c) f (3)
(d) $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ (e) $\varliminf _ { x \rightarrow - 2 } f ( x )$ (f) f (-2)

Accepted Answer

(a) -2
(b) Does not

Question 4

$\text { Average Daily Temperatures in Moorhead, Minnesota }$
$\begin{array} { | l | c | c | c | c | c | c | } 
\hline \text { Month } & 1 ( \mathrm { Jan } ) & 2 ( \mathrm { Feb } ) & 3 ( \mathrm { Mar } ) & 4 ( \mathrm { Apr } ) & 5 ( \mathrm { May } ) & 6 ( \mathrm { Jun } ) \
\hline \text { Temperature } \left( { } ^ { \circ } \mathbf { F } \right) & 6 & 14 & 26 & 43 & 57 & 66 \
\hline \text { Month } & 7 ( \mathrm { Jul } ) & 8 ( \mathrm { Aug } ) & 9 ( \mathrm { Sep } ) & 10 ( \mathrm { Oct } ) & 11 ( \mathrm { Nov } ) & 12 ( \mathrm { Dec } ) \
\hline \text { Temperature } \left( { } ^ { \circ } \mathbf { F } \right) & 71 & 70 & 58 & 46 & 27 & 13 \
\hline
\end{array}$
-During which of the following periods was the rate of change of the average daily temperature the greatest?

Accepted Answer

A) [2, 3] 
B) [3, 4] 
C) [4, 5] 
D) [5, 6] 
E) [6, 7]
F)[2, 4]
G)[3, 5]
H)[3, 6] 
A) [2, 3] 
B) [3, 4] 
C) [4, 5] 
D) [5, 6] 
E) [6, 7]
F)[2, 4]
G)[3, 5]
H)[3, 6]

Question 5

Find the value of the limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { | x - 2 | } { x - 2 }$

Accepted Answer

A) 1 
B) -2 
C)  $\frac { 1 } { 2 }$ 
D) -4 
E) -1
F)4
G)
$- \frac { 1 } { 2 }$ 
H)2 
A) 1 
B) -2 
C)  $\frac { 1 } { 2 }$ 
D) -4 
E) -1
F)4
G)
$- \frac { 1 } { 2 }$ 
H)2

Question 6

Find the limit.(a) $\lim _ { x \rightarrow \infty } \sin x$ (b) $\lim _ { x \rightarrow \infty } e ^ { - x }$ (c) $\lim _ { t \rightarrow \infty } \frac { t ^ { 3 } + 3 t } { 1 - t ^ { 4 } }$

Accepted Answer

(a) Does n

Question 7

At what value(s) of x is the function $f ( x ) = \left\{ \begin{array} { l l } 
x + 2 & \text { if } x \leq - 1 \
x ^ { 2 } & \text { if } - 1 < x < 1 \
3 - x & \text { if } x \geq 1
\end{array} \right.$ discontinuous?

Accepted Answer

A) -1 
B) 0 
C) 1 
D) -1, 0 
E) 0, 1
F)-1, 1
G)-1, 0, 1
H)None 
A) -1 
B) 0 
C) 1 
D) -1, 0 
E) 0, 1
F)-1, 1
G)-1, 0, 1
H)None

Question 8

Find the value of the limit $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)$

Accepted Answer

A) 
$- \frac { 1 } { 2 }$ 
B)  $\sqrt { 2 }$ 
C) 
$- \sqrt { 2 }$ 
D)  $\frac { 1 } { 4 }$ 
E) 
$\infty$ 
F) $- \frac { 1 } { 4 }$
G)0
H)$\frac { 1 } { 2 }$ 
A) 
$- \frac { 1 } { 2 }$ 
B)  $\sqrt { 2 }$ 
C) 
$- \sqrt { 2 }$ 
D)  $\frac { 1 } { 4 }$ 
E) 
$\infty$ 
F) $- \frac { 1 } { 4 }$
G)0
H)$\frac { 1 } { 2 }$

Question 9

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car’s velocity is given in the table below:
$\begin{array} { | c | c | c | c | c | c | c | c | c | c | } 
\hline t ( s ) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \
\hline v ( t ) ( f t / s ) & 0 & 18 & 47 & 77 & 104 & 132 & 163 & 184 & 208 \
\hline
\end{array}$
-On what time interval does the car have the lowest average acceleration?

Accepted Answer

A) [0, 1] 
B) [1, 2] 
C) [2, 3] 
D) [3, 4] 
E) [4, 5]
F)[5, 6]
G)[6, 7]
H)[7, 8] 
A) [0, 1] 
B) [1, 2] 
C) [2, 3] 
D) [3, 4] 
E) [4, 5]
F)[5, 6]
G)[6, 7]
H)[7, 8]

Question 10

Given the graph of $$y = f ^ { \prime } ( x )$$ below, select a graph which could be that of y = f (x).

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
 
F)   
G)  
H)
  
A) 
  
B) 
  
C) 
  
D) 
  
E) 
 
F)   
G)  
H)

Question 11

If f (x) = 2x, how close to 3 does x have to be to ensure that f (x) is within 0.1 of 8?

Accepted Answer

Answers will vary. One reasona

Question 12

Given the graph of y = f (x) below, sketch the graph of $y = f ^ { \prime } ( x )$ .

Accepted Answer

The answer of Given the graph of y = f...

Question 13

Evaluate the limit, if it exists. If it does not exist, explain why.(a) $\lim _ { x \rightarrow 2 } \left( 5 x ^ { 3 } - 3 x ^ { 2 } + 4 \right)$ (b) $\lim _ { x \rightarrow 9 } \frac { x - 9 } { \sqrt { x } - 3 }$ (c) $\lim _ { x \rightarrow - 1 } \frac { x ^ { 4 } - 1 } { x ^ { 2 } - 1 }$ (d) $\lim _ { x \rightarrow 3 } \frac { \frac { 2 } { x } - \frac { 2 } { 3 } } { x - 3 }$ (e) $\lim _ { x \rightarrow - 3 } \frac { | x + 3 | } { 2 x + 6 }$ (f) $\lim _ { x \rightarrow - 3 ^ { - } } \frac { | x + 3 | } { 2 x + 6 }$

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) @#IMG-DLM& (d) @#IMG-DLM&

Question 14

Consider the function $f ( x ) = | x - 2 |$ (a) What is the domain of f?
(b) At what number(s), if any, is f discontinuous? Explain your answer.(c) At what number(s), if any, is f not differentiable? Explain your answer.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) f is n

Question 15

Find the value of x at which the curve $y = \frac { 2 x + 1 } { 2 x ^ { 2 } + 7 x + 3 }$ has a vertical asymptote.

Accepted Answer

A) -4 
B) -3 
C) -2 
D) -1 
E)  $- \frac { 1 } { 2 }$ 
F)1
G)2
H)3 
A) -4 
B) -3 
C) -2 
D) -1 
E)  $- \frac { 1 } { 2 }$ 
F)1
G)2
H)3

Question 16

Find the limit.(a) $\lim _ { x \rightarrow 2 ^ { + } } \frac { 1 } { x - 2 }$ (b) $\lim _ { x \rightarrow \infty } \frac { 1 - 2 x ^ { 3 } } { x ^ { 2 } + x ^ { 3 } }$ (c) $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)$

Accepted Answer

The answer of Find the limit.(a) \(\lim _ { x...

Question 17

Given the graph of $y = f ^ { \prime } ( x )$ , answer the questions that follow.   (a) Find all values of x at which
(i) f is increasing.(iv) $f ^ { \prime \prime } ( x )  0$ (vi) $f ^ { \prime } \text { is increasing. }$ (ix) f has a local minimum.(b) Sketch a graph which could represent y = f (x).

Accepted Answer

(a) (i) f is increasing on [x₀, x₁] and [x

Question 18

Find the value of the limit $\lim _ { x \rightarrow \infty } \sqrt { \frac { x + 8 x ^ { 2 } } { 2 x ^ { 2 } - 1 } }$

Accepted Answer

A) -2 
B) 1 
C)  $- \frac { 1 } { 2 }$ 
D) 4 
E)  $\frac { 1 } { 2 }$
F)2
G)0
H)-1 
A) -2 
B) 1 
C)  $- \frac { 1 } { 2 }$ 
D) 4 
E)  $\frac { 1 } { 2 }$
F)2
G)0
H)-1

Question 19

Given the graph of y = f (x) below, select a graph which best represents the graph of $$y = f ^ { \prime } ( x )$$

Accepted Answer

A)   
B)   
C)    
D)    
E)   
F)  
G)  
H)   
A)   
B)   
C)    
D)    
E)   
F)  
G)  
H)

Question 20

Given the graph of y = f (x) below, select a graph which best represents the graph of $$y = f ^ { \prime } ( x )$$

Accepted Answer

A) 
  
B)    
C)    
D)    
E)   
F)   
G)   
H)   
A) 
  
B)    
C)    
D)    
E)   
F)   
G)   
H)

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { \frac { 1 } { x ^ { 2 } } - 1 } { x - 1 }$

$\text { Average Daily Temperatures in Moorhead, Minnesota }$ Month 1() 2() 3() 4() 5() 6() Temperature 6 14 26 43 57 66 Month 7() 8() 9() 10() 11() 12() Temperature 71 70 58 46 27 13 -During which of the following periods was the rate of change of the average daily temperature the greatest?

Find the value of the limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { | x - 2 | } { x - 2 }$

Find the limit.(a) $\lim _ { x \rightarrow \infty } \sin x$ (b) $\lim _ { x \rightarrow \infty } e ^ { - x }$ (c) $\lim _ { t \rightarrow \infty } \frac { t ^ { 3 } + 3 t } { 1 - t ^ { 4 } }$

At what value(s) of x is the function $f ( x ) = \left\{ \begin{array} { l l } x + 2 & \text { if } x \leq - 1 \\x ^ { 2 } & \text { if } - 1 < x < 1 \\3 - x & \text { if } x \geq 1\end{array} \right.$ discontinuous?

Find the value of the limit $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)$

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car’s velocity is given in the table below: t(s) 0 1 2 3 4 5 6 7 8 v(t)(ft/s) 0 18 47 77 104 132 163 184 208 -On what time interval does the car have the lowest average acceleration?

Given the graph of $y = f ^ { \prime } ( x )$ below, select a graph which could be that of y = f (x). $Given the graph of y = f ^ { \prime } ( x ) below, select a graph which could be that of y = f (x).$

If f (x) = 2^x, how close to 3 does x have to be to ensure that f (x) is within 0.1 of 8?

Given the graph of y = f (x) below, sketch the graph of $y = f ^ { \prime } ( x )$ . $Given the graph of y = f (x) below, sketch the graph of y = f ^ { \prime } ( x ) .$

Consider the function $f ( x ) = | x - 2 |$ (a) What is the domain of f? (b) At what number(s), if any, is f discontinuous? Explain your answer.(c) At what number(s), if any, is f not differentiable? Explain your answer.

Find the value of x at which the curve $y = \frac { 2 x + 1 } { 2 x ^ { 2 } + 7 x + 3 }$ has a vertical asymptote.

Find the limit.(a) $\lim _ { x \rightarrow 2 ^ { + } } \frac { 1 } { x - 2 }$ (b) $\lim _ { x \rightarrow \infty } \frac { 1 - 2 x ^ { 3 } } { x ^ { 2 } + x ^ { 3 } }$ (c) $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)$

Find the value of the limit $\lim _ { x \rightarrow \infty } \sqrt { \frac { x + 8 x ^ { 2 } } { 2 x ^ { 2 } - 1 } }$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Exam 2: Limits and Derivatives

Find the value of the limit lim⁡x→11x2−1x−1\lim _ { x \rightarrow 1 } \frac { \frac { 1 } { x ^ { 2 } } - 1 } { x - 1 }limx→1​x−1x21​−1​

Find the value of the limit lim⁡x→2−∣x−2∣x−2\lim _ { x \rightarrow 2 ^ { - } } \frac { | x - 2 | } { x - 2 }limx→2−​x−2∣x−2∣​

Find the value of the limit lim⁡x→∞(x2+x−x)\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)limx→∞​(x2+x​−x)

A car on a test track accelerates from 0 ft/s to 208 ft/s in 8 seconds. The car’s velocity is given in the table below: t(s) 0 1 2 3 4 5 6 7 8 v(t)(ft/s) 0 18 47 77 104 132 163 184 208 -On what time interval does the car have the lowest average acceleration?

Given the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x) below, select a graph which could be that of y = f (x).

If f (x) = 2x, how close to 3 does x have to be to ensure that f (x) is within 0.1 of 8?

Given the graph of y = f (x) below, sketch the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x) .

Consider the function f(x)=∣x−2∣f ( x ) = | x - 2 |f(x)=∣x−2∣ (a) What is the domain of f? (b) At what number(s), if any, is f discontinuous? Explain your answer.(c) At what number(s), if any, is f not differentiable? Explain your answer.

Find the value of x at which the curve y=2x+12x2+7x+3y = \frac { 2 x + 1 } { 2 x ^ { 2 } + 7 x + 3 }y=2x2+7x+32x+1​ has a vertical asymptote.

Find the value of the limit lim⁡x→∞x+8x22x2−1\lim _ { x \rightarrow \infty } \sqrt { \frac { x + 8 x ^ { 2 } } { 2 x ^ { 2 } - 1 } }limx→∞​2x2−1x+8x2​​

Given the graph of y = f (x) below, select a graph which best represents the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x)

Given the graph of y = f (x) below, select a graph which best represents the graph of y=f′(x)y = f ^ { \prime } ( x )y=f′(x)

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { \frac { 1 } { x ^ { 2 } } - 1 } { x - 1 }$

Find the value of the limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { | x - 2 | } { x - 2 }$

Find the value of the limit $\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + x } - x \right)$

Given the graph of $y = f ^ { \prime } ( x )$ below, select a graph which could be that of y = f (x). $Given the graph of y = f ^ { \prime } ( x ) below, select a graph which could be that of y = f (x).$

If f (x) = 2^x, how close to 3 does x have to be to ensure that f (x) is within 0.1 of 8?

Given the graph of y = f (x) below, sketch the graph of $y = f ^ { \prime } ( x )$ . $Given the graph of y = f (x) below, sketch the graph of y = f ^ { \prime } ( x ) .$

Consider the function $f ( x ) = | x - 2 |$ (a) What is the domain of f? (b) At what number(s), if any, is f discontinuous? Explain your answer.(c) At what number(s), if any, is f not differentiable? Explain your answer.

Find the value of x at which the curve $y = \frac { 2 x + 1 } { 2 x ^ { 2 } + 7 x + 3 }$ has a vertical asymptote.

Find the value of the limit $\lim _ { x \rightarrow \infty } \sqrt { \frac { x + 8 x ^ { 2 } } { 2 x ^ { 2 } - 1 } }$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$

Given the graph of y = f (x) below, select a graph which best represents the graph of $y = f ^ { \prime } ( x )$ $Given the graph of y = f (x) below, select a graph which best represents the graph of y = f ^ { \prime } ( x )$