Question 1

Given the following information about limits, select a graph which could be the graph of y = f (x). $$\begin{array} { c } 
\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow - \infty } f ( x ) = - 1\quad\quad\quad \lim _ { x \rightarrow - 1 ^ { + } } f ( x ) = \lim _ { x \rightarrow - 1 ^ { - } } f ( x ) = \infty \
\lim _ { x \rightarrow - 1 ^ { - } } f ( x ) = \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = - \infty
\end{array}$$

Accepted Answer

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

Question 2

Sketch the graph of a function f on [-5, 5] that satisfies all of the following conditions:
f (-4) = 2, f (-3) = -1, f(-2) = 2, f(1) = -3, f(2) = 1, f(3) = 0, f(4) = -3, $\lim _ { x \rightarrow - 2 } f ( x ) = - 3$ , $\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = - 1$ , and $\lim _ { x \rightarrow 2 ^ { + } } f ( x ) = 1$

Accepted Answer

(Answers w

Question 3

The graph of f is given below. State, with reasons, the number(s) at which
(a) f is not differentiable. (b) f is not continuous.

Accepted Answer

(a) f is not differentiable at x = 6 or

Question 4

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } - 3 x + 5 } { 7 + 2 x - 2 x ^ { 2 } }$

Accepted Answer

A) 
$- \frac { 1 } { 2 }$ 
B) 1 
C) 2 
D)  $\frac { 1 } { 2 }$ 
E) -1 $\frac { 4 } { 7 }$

F) $- \frac { 4 } { 7 }$ 
G)-2
H)$\frac { 4 } { 7 }$ 
A) 
$- \frac { 1 } { 2 }$ 
B) 1 
C) 2 
D)  $\frac { 1 } { 2 }$ 
E) -1 $\frac { 4 } { 7 }$

F) $- \frac { 4 } { 7 }$ 
G)-2
H)$\frac { 4 } { 7 }$

Question 5

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 6

Use the graphs of f and g below to evaluate each limit, if it exists. If it does not exist, explain why.     (a) $\lim _ { x \rightarrow - 2 } [ f ( x ) + g ( x ) ]$ (b) $\lim _ { x \rightarrow 2 } \left[ \frac { g ( x ) } { f ( x ) } \right]$ (c) $\lim _ { x \rightarrow 1 } [ f ( x ) \cdot g ( x ) ]$ (d) $\lim _ { x \rightarrow 0 } \left[ ( x - 3 ) ^ { 2 } \cdot g ( x ) \right]$

Accepted Answer

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

Question 7

Determine whether each function is continuous or discontinuous. Explain your choice.(a) Postage charges to send a letter by first class mail.(b) The altitude of an airplane as a function of the time it has been in the air.(c) The temperature of an oven as it is run through its self cleaning cycle.(d) The number of people waiting in a queue for a bank teller.

Accepted Answer

(a) The cost of sending a letter changes

Question 8

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 9

Determine $\lim _ { x \rightarrow 1 } \frac { x ^ { 1000 } - 1 } { x - 1 }$ by producing an appropriate table.

Accepted Answer

x
f(x)
x @#IMG-DLM& 1.001
1716.924
0.99
99.995683

Question 10

A projectile is launched vertically upward from the surface of Mars. The table below gives the height of the object at the indicated time following launch.$$\begin{array}{l}
\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Time (seconds): } & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2.0 & 2.4 & 2.8 & 3.2 & 3.6 \
\hline \text { Height (feet): } & 0 & 18.2 & 34.4 & 48.4 & 60.4 & 70.4 & 78.3 & 84.2 & 88.1 & 90.0 \
\hline
\end{array}\\
\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Time (seconds): } & 4.0 & 4.4 & 4.8 & 5.2 & 5.6 & 6.0 & 6.4 & 6.8 & 7.2 & 7.6 \
\hline \text { Height (feet): } & 89.6 & 87.3 & 82.9 & 76.5 & 68.1 & 57.6 & 45.1 & 30.5 & 13.8 & 0 \
\hline
\end{array}
\end{array}$$
(a) Graph the data.
(b) Using the data, compute the average velocity of the projectile on the following time intervals:
(i) [0; 4.0]
(ii) [0.8; 3.2]
(iii) [1.2; 2.8]
(iv) [1.6; 2.4]
(v) [2.0; 2.4]
(vi) [1.6; 2.0]
(c) Estimate the velocity of the projectile when t = 2.0. Justify your results.(d) Using your graph and the table of values, determine when the projectile reaches its maximum height. Justify your answer.(e) Using the graph and table of values, estimate the velocity of the projectile throughout the interval [0; 7.6] and sketch a graph of this velocity.

Accepted Answer

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

Question 11

Given the graph of y = f (x), answer the following questions.   Find all values of x at which
(a) $f ^ { \prime } ( x ) > 0$ (b) $f ^ { \prime } ( x )  0$ (g) $f ^ { \prime \prime } ( x ) < 0$ (h) f has an inflection point.(i) $f ^ { \prime } \text { is increasing. }$ (j) $f ^ { \prime } \text { is decreasing. }$

Accepted Answer

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

Question 12

A tank is in the shape of a large, inverted (point-down) cone. Water is owing into the tank
at a constant rate. Let H (t) represent the height of the water level at time t.(a) Sketch a possible graph of H (t).(b) Describe how the rate of change of H with respect to t varies as t increases.(c) Sketch a graph of $H ^ { \prime } ( t )$

Accepted Answer

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

Question 13

Suppose that $f ( x ) = \frac { x - 3 } { x ^ { 2 } - 9 }$ . f is not defined for $x = \pm 3$ .(a) For which of these two values does f have an infinite discontinuity? Explain.(b) For which of these two values does f have a removable discontinuity? Explain.

Accepted Answer

(a) Since @#IMG-DLM& f has an infinite d

Question 14

Find the value of the limit $\lim _ { x \rightarrow 4 ^ { + } } \frac { | x - 4 | } { x - 4 }$

Accepted Answer

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

Question 15

If f (1) = 5 and $f ^ { \prime } ( 1 ) = - 3$ , find an equation of the tangent line at x = 1.

Accepted Answer

A) y = -3x + 5 
B) y = 5x - 3 
C) y = -3x + 8 
D) y = -3x - 8 
E) y = 3x + 8
F)y = 3x - 8
G)Does not exist
H)None of the above 
A) y = -3x + 5 
B) y = 5x - 3 
C) y = -3x + 8 
D) y = -3x - 8 
E) y = 3x + 8
F)y = 3x - 8
G)Does not exist
H)None of the above

Question 16

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - x }$

Accepted Answer

A) 5 
B) 4 
C) 3 
D) 2 
E)  $\frac { 7 } { 2 }$
F) $\frac { 3 } { 2 }$ 
G) $\frac { 1 } { 2 }$ 
H)0 
A) 5 
B) 4 
C) 3 
D) 2 
E)  $\frac { 7 } { 2 }$
F) $\frac { 3 } { 2 }$ 
G) $\frac { 1 } { 2 }$ 
H)0

Question 17

-For the function whose graph is given above, determine $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist 
A) -3 
B) -2 
C) -1 
D) 0 
E) 1
F)2
G)3
H)Does not exist

Question 18

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 19

(a) Explain in your own words what is meant by $\lim _ { x \rightarrow - 2 } f ( x ) = 3$ (b) Is it possible for this statement to be true yet for f -(2) = 5? Explain.

Accepted Answer

The answer of (a) Explain in your own words what...

Question 20

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, C, of cellular phone usage as a function of the length of the call, t, for $0 \leq t \leq 5$ .(b) Evaluate:
(i) $\lim _ { t \rightarrow 2 ^ { - } } C ( t )$ (ii) $\lim _ { t \rightarrow 2 ^ { + } } C ( t )$ (c) Explain the significance of the left limit (i) and the right limit (ii) to the cell phone user.(d) For what values of t does C (t) not have a limit? Justify your answer.

Accepted Answer

(a) @#IMG-DLM& (b) (i) 64 cents
(ii) 96

Given the following information about limits, select a graph which could be the graph of y = f (x). f(x)=f(x)=-1f(x)=f(x)=\infty f(x)=f(x)=-\infty

The graph of f is given below. State, with reasons, the number(s) at which (a) f is not differentiable. (b) f is not continuous.

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } - 3 x + 5 } { 7 + 2 x - 2 x ^ { 2 } }$

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

Determine $\lim _ { x \rightarrow 1 } \frac { x ^ { 1000 } - 1 } { x - 1 }$ by producing an appropriate table.

Suppose that $f ( x ) = \frac { x - 3 } { x ^ { 2 } - 9 }$ . f is not defined for $x = \pm 3$ .(a) For which of these two values does f have an infinite discontinuity? Explain.(b) For which of these two values does f have a removable discontinuity? Explain.

Find the value of the limit $\lim _ { x \rightarrow 4 ^ { + } } \frac { | x - 4 | } { x - 4 }$

If f (1) = 5 and $f ^ { \prime } ( 1 ) = - 3$ , find an equation of the tangent line at x = 1.

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - x }$

$-For the function whose graph is given above, determine \lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ -For the function whose graph is given above, determine $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$

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

(a) Explain in your own words what is meant by $\lim _ { x \rightarrow - 2 } f ( x ) = 3$ (b) Is it possible for this statement to be true yet for f -(2) = 5? Explain.

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

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Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

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Multiple Integrals

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Exam 2: Limits and Derivatives

Given the following information about limits, select a graph which could be the graph of y = f (x). f(x)=f(x)=-1f(x)=f(x)=\infty f(x)=f(x)=-\infty

The graph of f is given below. State, with reasons, the number(s) at which (a) f is not differentiable. (b) f is not continuous.

Find the value of the limit lim⁡x→∞4x2−3x+57+2x−2x2\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } - 3 x + 5 } { 7 + 2 x - 2 x ^ { 2 } }limx→∞​7+2x−2x24x2−3x+5​

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

Determine lim⁡x→1x1000−1x−1\lim _ { x \rightarrow 1 } \frac { x ^ { 1000 } - 1 } { x - 1 }limx→1​x−1x1000−1​ by producing an appropriate table.

Suppose that f(x)=x−3x2−9f ( x ) = \frac { x - 3 } { x ^ { 2 } - 9 }f(x)=x2−9x−3​ . f is not defined for x=±3x = \pm 3x=±3 .(a) For which of these two values does f have an infinite discontinuity? Explain.(b) For which of these two values does f have a removable discontinuity? Explain.

Find the value of the limit lim⁡x→4+∣x−4∣x−4\lim _ { x \rightarrow 4 ^ { + } } \frac { | x - 4 | } { x - 4 }limx→4+​x−4∣x−4∣​

If f (1) = 5 and f′(1)=−3f ^ { \prime } ( 1 ) = - 3f′(1)=−3 , find an equation of the tangent line at x = 1.

Find the value of the limit lim⁡x→12x2+x−3x2−x\lim _ { x \rightarrow 1 } \frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - x }limx→1​x2−x2x2+x−3​

-For the function whose graph is given above, determine lim⁡x→−2−f(x)\lim _ { x \rightarrow - 2 ^ { - } } f ( x )limx→−2−​f(x)

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).

(a) Explain in your own words what is meant by lim⁡x→−2f(x)=3\lim _ { x \rightarrow - 2 } f ( x ) = 3limx→−2​f(x)=3 (b) Is it possible for this statement to be true yet for f -(2) = 5? Explain.

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 $\lim _ { x \rightarrow \infty } \frac { 4 x ^ { 2 } - 3 x + 5 } { 7 + 2 x - 2 x ^ { 2 } }$

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

Determine $\lim _ { x \rightarrow 1 } \frac { x ^ { 1000 } - 1 } { x - 1 }$ by producing an appropriate table.

Suppose that $f ( x ) = \frac { x - 3 } { x ^ { 2 } - 9 }$ . f is not defined for $x = \pm 3$ .(a) For which of these two values does f have an infinite discontinuity? Explain.(b) For which of these two values does f have a removable discontinuity? Explain.

Find the value of the limit $\lim _ { x \rightarrow 4 ^ { + } } \frac { | x - 4 | } { x - 4 }$

If f (1) = 5 and $f ^ { \prime } ( 1 ) = - 3$ , find an equation of the tangent line at x = 1.

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - x }$

$-For the function whose graph is given above, determine \lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ -For the function whose graph is given above, determine $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$

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

(a) Explain in your own words what is meant by $\lim _ { x \rightarrow - 2 } f ( x ) = 3$ (b) Is it possible for this statement to be true yet for f -(2) = 5? Explain.