Deck 2: Limits and Derivatives

Question 1

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

A)
B)
C)

D)

E)

F)
G)
H)

Answer 1

Question 2

In order to determine an appropriate delivery schedule to a group of rural homes in North Dakota, a fuel oil distributor monitors fuel oil consumption and the daily outdoor temperature (in degrees Fahrenheit). A table was constructed for a function F (T) of fuel oil consumption (in gallons per day) as a function of the temperature T.(a) Sketch a graph which you believe would approximate the graph of y = F (T).(b) What is the meaning of $F ^ { \prime } ( T ) ?$ What are its units?
(c) Write a sentence that would explain to an intelligent layperson the meaning of $F ^ { \prime } ( 0 ) = - 0.4$

Answer 2

(a)
(There are many possible graphs, though presumably they are all decreasing!)
(b) $F ^ { \prime } ( T )$ represents the rate of change of fuel oil consumption with respect to temperature. Its units are (gallons/day) / ( ^$\circ$ F).
(c) $F ^ { \prime } ( 0 ) = - 0.4$ means that as the temperature increases past 0 ^$\circ$ F, the fuel consumption is decreasing by 0.4 (gallons/day) / (F).

Question 3

Given the graph of y = f (x) below, select a graph which best represents the graph of a. e. b. f. c. g. d. h.

Answer 3

D

Question 4

Given the graph of , answer the questions that follow. (a) Find all values of x at which
(i) f is increasing.(iv) (vii) (ii) f is decreasing.(v) f has an inflection point.(viii) f has a local maximum.(iii) (b) Sketch a graph which could represent y = f (x).

Question 5

Given the graph of y = f (x), sketch the graph of y = (a) (b) (c) (d)

Question 6

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

A)

B)

C)

D)

E)

F)
G)
H)

Question 7

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

A)
B)

C)

D)

E)
F)

G)
H)

Question 8

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

A)

B)

C)

D)

E)

F)
G)

H)

Question 9

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

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

Question 10

Given the graph of , answer the questions that follow. (a) Find all values of x at which
(i) f is increasing.(iv) (vii) (ii) f is decreasing.(v) f has an inflection point.(viii) f has a local maximum.(iii) (vi) (ix) f has a local minimum.(b) Sketch a graph which could represent y = f (x).

Question 11

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

A)
B)
C)

D)
E)

F)
G)
H)

Question 12

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

A)
B)
C)

D)
E)
F)
G)
H)

Question 13

Given the graph of y = f (x), answer the following questions. Find all values of x at which
(a) (b) (c) (d) f is increasing.(e) f is decreasing.(f) (g) (h) f has an inflection point.(i) (j)

Question 14

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

A)

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

Question 15

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

A)
B)
C)
D)
E)

F)

G)
H)

Question 16

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

A)

B)

C) 1eaa8e2_0f37_9ed6_96ab_237635657d14_TB2033_11
D)
E)

F)

G)

H)

Question 17

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

A)
B)

C)
D)
E)
F)

G)
H)

Question 18

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

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

Question 19

Given the graph of , answer the questions that follow. (a) Find all values of x at which
(i) f is increasing.(iv) (vii) (ii) f is decreasing.(v) (viii) f has a local maximum.(iii) (vi) (ix) f has a local minimum.(b) Sketch a graph which could represent y = f (x).

Question 20

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

A)

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

Question 21

Given the graph of y = f (x) below, sketch the graph of .

Question 22

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

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 23

Find an equation of the line tangent to f (x) = x² - 4x at the point (3, - 3).

A)2x - y = 9
E)x - y = 9
B)x - 2y = 9
F)y - x = 9
C)y - 2x = 9
G)2x - y = 4
D)2y - x = 9
H)x - 2y = 4

Question 24

Given the graph of y = f (x) below, sketch the graph of .

Question 25

If f (-1) = 3 and $f ^ { \prime } ( - 1 ) = 2$ , find an equation of the tangent line at x = -1.

A)y = -5 + 2x
B)y = 1 + 2x
C)y = -2x + 1
D)y = 5 - 2x
E)y = -1 + 2x
F)y = 5 + 2x
G)Does not exist
H)None of the above

Question 26

For the function f whose graph is given, arrange the following values in increasing order and explain your reasoning.

Question 27

Consider the function (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.

Question 28

Below are the graphs of a function and its first and second derivatives. Identify which of the following graphs (a, b, and c) is f (x), which is , and which is . Justify your choices.

Question 29

Determine a function f and a number a where the given limit represents the derivative of f at a.(a) (b) (c) (d)

Question 30

Find the derivative of f (x) = 3x - 5 using the definition of the derivative.

Question 31

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.

Question 32

Given the graph of y = f (x), find all values of x for which
(a) (b) (c) (d)

Question 33

If f (x) = 5, which of the following represents a. e. b.3
F) c. g.Does not exist
D)5
H)None of the above

Question 34

Find the derivative of using the definition of the derivative.

Question 35

Water is flowing into a large cylindrical 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

Question 36

Given the graph of y = f (x) below, sketch the graph of .

Question 37

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

Question 38

Given the graph of y = f (x) below, sketch the graph of .

Question 39

Find an equation of the line tangent to the curve y = x + (1/x) at the point $\left( 5 , \frac { 26 } { 25 } \right)$

A)24x - 25y = 94
B)24x - 25y = -94
C)25x - 24y = 94
D)25x - 24y = -94
E)24x - 24y = 94
F)24x - 24y = -94
G)25x - 25y = 94
H)25x - 25y = -94

Question 40

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.

Question 41

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

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 42

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

-During which of the following periods was the rate of change of the average daily temperature the greatest?

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 43

Given the following information about limits, select a graph which could be the graph of y = f (x). $$\begin{array} { l } \lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow - \infty } f ( x ) = 0 \quad\quad\quad \lim _ { x \rightarrow - 1 ^ { + } } f ( x ) = \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = \infty \\\lim _ { x \rightarrow - 1 ^ { - } } f ( x ) = \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - \infty \\\end{array}$$

A)

B)

C)

D)

E)

F)

G)
H)

Question 44

(a) Find the slope of the tangent line to the curve (i)
using Definition 1 from Section 2.6.(ii)
using Definition 2 from Section 2.6.(b) Using your results from part (a), find an equation of the tangent line at (1, 0).

Question 45

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

A)-2
B)1
C)2
D)4
E)-1
F)-3
G)0
H)3

Question 46

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

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

Question 47

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

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

Question 48

Suppose that . f is not defined for .(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.

Question 49

Below are the sunrise and sunset times (in standard time) for Moorhead, Minnesota on the 21st of each month in 2007.Sunrise and Sunset in Moorhead, Minnesota on the 21st of each month in 2007
Date
Day of Year
Sunrise
Sunset
Day Length (h)
Jan 21
21
08:03
17:14
9.18
Feb 21
52
07:21
18:01
10.67
Mar 21
80
06:28
18:41
12.22
Apr 21
111
05:29
19:24
13.92
May 21
141
04:45
20:02
15.28
Jun 21
172
04:32
20:25
15.88
Jul 21
202
04:54
20:12
15.30
Aug 21
233
05:32
19:27
13.92
Sep 21
264
06:12
18:27
12.25
Oct 21
294
06:53
17:29
10.60
Nov 21
325
07:39
16:47
9.13
Dec 21
355
08:09
16:41
8.53
(a) Find the rate of change in day length with respect to time (in hours per day)
(i)
from March 21 through July 21.(ii)
from March 21 through May 21.(iii)
from March 21 through April 21.(b) Estimate the instantaneous rate of change in day length per day (in hours per day) for March 21.(c) During which one-month period did the day length change the greatest amount?
(d) During which one month period did the average change in day length change the greatest amount? Is this the same time period as in part (c)? Explain.

Question 50

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

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

Question 51

The following table shows the relationship between pressure (in atmospheres) and volume (in liters) of hydrogen gas at 0 ^$\circ$ C.
$\begin{array} { | l | c | c | c | c | c | c | } \hline \text { Pressure (atm) } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Volume (L) } & 22.4 & 11.2 & 7.5 & 5.6 & 4.5 & 3.7 \\\hline\end{array}$
(a) Find the average rate of change of volume with respect to pressure for the following pressure intervals:
(i) [3, 5]
(ii) [3, 4]
(iii) [2, 3]
(b) Estimate the rate of change of volume with respect to pressure if the pressure is 2 atmospheres.

Question 52

Using the graph below, determine the following: (a) (b) (c) (d) (e) (f) (g) Find the horizontal asymptote(s) of the graph of y = f (x).(h) Find the vertical asymptote(s) of the graph of y = f (x).

Question 53

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

-During which of the following periods was the rate of change of the average daily temperature the smallest?

A)[2, 6]
B)[4, 8]
C)[4, 9]
D)[6, 10]
E)[8, 10]
F)[9, 10]
G)[10, 11]
H)[11, 12]

Question 54

Find the value of x at which the curve $y = \frac { x ^ { 2 } - 16 } { x ^ { 2 } - 5 x + 4 }$ has a vertical asymptote.

A)-4
B)-3
C)-2
D)-1
E)0
F)1
G)2
H)3

Question 55

Given the following information about limits, select a graph which could be the graph of y = f (x). $$\begin{array} { l } \lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow - \infty } f ( x ) = 0 \quad\quad\quad \lim _ { x \rightarrow - 1 ^ { - } } f ( x ) = \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = \infty \\\quad\quad\quad\quad\quad\lim _ { x \rightarrow - 1 ^ { + } } f ( x ) = \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - \infty \\\end{array}$$

A)

B)

C)

D)

E)

F)

G)

H)

Question 56

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

A)
$\frac { 3 } { 4 }$
B)3
C)7
D) $- \frac { 3 } { 4 }$
E)-3

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

Question 57

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

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 58

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

A)

B)

C)


D)

E)

F)

G)

H)

Question 59

The following table shows the concentration (in mol/l) of a certain chemical in terms of reaction time (in hours) during a decomposition reaction.Time (hours)
0
5
10
20
30
50
Concentration (mol/L)
2.32
1.86
1.49
0.98
0.62
0.25
(a) Find the average rate of change of concentration with respect to time for the following time intervals:
(i) [0; 5] (ii) [10; 20] (iii) [30; 50]
(b) Plot the points from the table and fit an appropriate exponential model to the data.(c) From your model in part (b), determine the instantaneous rate of change of concentration with respect to time.(d) Is the rate of change of concentration increasing or decreasing with respect to time? Justify your answer.

Question 60

The following table contains data from the historic flood of the Cedar River at Cedar Rapids, Iowa during June 2008. Readings were taken at 6:00 A.M. each day.Day
Stage (ft)
08
10.46
09
13.79
10
17.54
11
19.62
12
25.88
13
31.03
14
29.23
15
24.93
16
21.95
17
20.13
18
18.85
19
16.25
20
13.63
21
12.12
22
10.94
Source: US Army Corps of Engineers - Rock Island District - Water Control Center
(a) Find the average daily rate of change of the river level over each of the following periods:
(i) [08, 13]
(ii) [13, 22]
(iii) [12, 13]
(iv) [13, 14]
(b) Estimate the daily rate of change of the stage level on June 14th.

Question 61

At what value(s) of x is the function

Question 62

Given the following information about limits, sketch a graph which could be the graph of y = f (x). Label all horizontal and vertical asymptotes.

Question 63

At what value(s) of x is the function discontinuous?

Question 64

Find the distance between the two values of x at which the function $\frac { 1 } { x ^ { 2 } - 3 x + 2 }$ is discontinuous.

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

Question 65

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.

Question 66

Use the Intermediate Value Theorem to show that x³ - 5x - 7 = 0 for some value of x in (2, 3).

Question 67

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?

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

Question 68

Given the following information about limits, sketch a graph which could be the graph of y = f (x). Label all horizontal and vertical asymptotes.

Question 69

Suppose that f (x) is defined on [1, 3] and that f (1) = 3 and f (3) = 5. Sketch a possible graph of f that does not satisfy the conclusion of the Intermediate Value Theorem.

Question 70

Find the limit.(a) (b) (c)

Question 71

Given the following information about limits, sketch a graph which could be the graph of y = f (x). Label all horizontal and vertical asymptotes.

Question 72

At what value(s) of x does the function $\frac { ( x + 1 ) ^ { 2 } } { x ^ { 2 } - 1 }$ have a removable discontinuity?

A)-3
B)3
C)2
D)-1
E)1
F)-4
G)-2
H)4

Question 73

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.

Question 74

Find the limit.(a) (b) (c)

Question 75

The definition of continuity of f (x) at a point requires three things. List these three conditions, and in each case give an example (a graph or a formula) which illustrates how this condition can fail at x = a.

Question 76

Find the constant(s) c that make(s) the function $f ( x ) = \left\{ \begin{array} { l l } c ^ { 2 } - x ^ { 2 } & \text { if } x < 2 \\2 ( c - x ) & \text { if } x \geq 2\end{array} \right.$ continuous on $( - \infty , \infty )$ .

A)-4, -2
B)2, 0
C)2
D)4
E)-2, 4
F)-2
G)0
H)Does not exist

Question 77

At what value(s) of x does the function $\frac { ( x + 2 ) ^ { 2 } } { x ^ { 2 } - 4 }$ have an infinite discontinuity?

A)3
B)-4
C)-2
D)-3
E)-1
F)1
G)2
H)4

Question 78

Find the vertical asymptote(s) of the curve

Question 79

Given the graph of f below, state the intervals on which f is continuous.

Question 80

Find the limit.(a) (b) (c)