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Chapter 2

Limits and Derivatives

Educators

AD
+ 6 more educators

Problem 1

Write an equation that expresses the fact that a function $ f $ is continuous at the number 4.

Ma. Theresa A.
Numerade Educator

Problem 2

If $ f $ is continuous on $ (-\infty, \infty) $, what can you say about its graph?

Leon D.
Numerade Educator

Problem 3

(a) From the graph of $ f $, state the numbers at which $ f $ is discontinuous and explain why.
(b) For each of the numbers stated in part (a), determine whether $ f $ is continuous from the right, or from the left, nor neither.

DM
David M.
Numerade Educator

Problem 4

From the graph of $ g $, state the intervals on which $ g $ is continuous.

DM
David M.
Numerade Educator

Problem 5

Sketch the graph of a function $ f $ that is continuous except for the stated discontinuity.

Discontinuous, but continuous from the right, at 2.

Leon D.
Numerade Educator

Problem 6

Sketch the graph of a function $ f $ that is continuous except for the stated discontinuity.

Discontinuities at -1 and 4, but continuous from the left at -1 and from the right of 4

Leon D.
Numerade Educator

Problem 7

Sketch the graph of a function $ f $ that is continuous except for the stated discontinuity.

Removable discontinuity at 3, jump discontinuity at 5.

Leon D.
Numerade Educator

Problem 8

Sketch the graph of a function $ f $ that is continuous except for the stated discontinuity.

Neither left nor right continuous at -2, continuous only from the left at 2.

Babita K.
Numerade Educator

Problem 9

The toll $ T $ charged for driving on a certain stretch of a toll road is 5 dollars except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is 7 dollars.
(a) Sketch a graph of $ T $ as a function of the time $ t $, measured in hours past midnight.
(b) Discuss the discontinuities of this function and their significance to someone who uses the road.

KS
Keyan S.
Numerade Educator

Problem 10

Explain why each function is continuous or discontinuous.
(a) The temperature at a specific location as a function of time
(b) The temperature at a specific time as a function of the distance due west from New York City
(c) The altitude above sea level as a function of the distance due west from New York City
(d) The cost of a taxi ride as a function of the distance traveled
(e) The current in the circuit for the lights in a room as a function of time

Michael C.
Numerade Educator

Problem 11

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ f(x) = (x + 2x^3)^4, \hspace{5mm} a = -1 $

Stark L.
Numerade Educator

Problem 12

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ g(t) = \frac{t^2 + 5t}{2t + 1}, \hspace{5mm} a = 2 $

Ma. Theresa A.
Numerade Educator

Problem 13

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ p(v) = 2 \sqrt{3v^2 + 1}, \hspace{5mm} a = 1 $

Ma. Theresa A.
Numerade Educator

Problem 14

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ f(x) = 3x^4 - 5x + \sqrt[3]{x^2 + 4}, \hspace{5mm} a = 2 $

Daniel J.
Numerade Educator

Problem 15

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

$ f(x) = x + \sqrt{x - 4}, \hspace{5mm} [4, \infty) $

Leon D.
Numerade Educator

Problem 16

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

$ g(x) = \frac{x - 1}{3x + 6}, \hspace{5mm} (-\infty, -2) $

Leon D.
Numerade Educator

Problem 17

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \frac{1}{x + 2} \hspace{55mm} a = -2 $

AK
Anjali K.
Numerade Educator

Problem 18

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\dfrac{1}{x + 2} & \mbox{if $ x \neq -2 $} \hspace{40mm} a = -2\\
1 & \mbox{if $ x = -2 $}
\end{array} \right.$

Suman Saurav T.
Numerade Educator

Problem 19

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
x + 3 & \mbox{if $ x \le -1 $} \hspace{40mm} a = -1\\
2^x & \mbox{if $ x > -1 $}
\end{array} \right.$

Leon D.
Numerade Educator

Problem 20

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\dfrac{x^2 - x}{x^2 - 1} & \mbox{if $ x \neq 1 $} \hspace{40mm} a = 1\\
1 & \mbox{if $ x = 1 $}
\end{array} \right.$

Leon D.
Numerade Educator

Problem 21

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\cos x & \mbox{if $ x < 0 $}\\
0 & \mbox{if $ x = 0 $} \hspace{42mm} a = 0\\
1 - x^2 & \mbox{if $ x > 0 $}
\end{array} \right.$

AK
Anjali K.
Numerade Educator

Problem 22

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\dfrac{2x^2 - 5x - 3}{x - 3} & \mbox{if $ x \neq 3 $} \hspace{30mm} a = 3\\
6 & \mbox{if $ x = 3 $}
\end{array} \right.$

Ma. Theresa A.
Numerade Educator

Problem 23

How would you "remove the discontinuity" of $ f $? In other words, how would you define $ f(2) $ in order to make $ f $ continuous at 2?

$ f(x) = \dfrac {x^2 - x - 2}{x - 2} $

Leon D.
Numerade Educator

Problem 24

How would you "remove the discontinuity" of $ f $? In other words, how would you define $ f(2) $ in order to make $ f $ continuous at 2?

$ f(x) = \dfrac {x^3 - 8}{x^2 - 4} $

Leon D.
Numerade Educator

Problem 25

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ F(x) = \dfrac{2x^2 - x - 1}{x^2 + 1} $

DM
David M.
Numerade Educator

Problem 26

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ G(x) = \dfrac{x^2 + 1}{2x^2 - x - 1} $

Linda H.
Numerade Educator

Problem 27

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ Q(x) = \dfrac{\sqrt[3]{x - 2}}{x^3 - 2} $

Linda H.
Numerade Educator

Problem 28

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ R(t) = \dfrac{e^{\sin t}}{2 + \cos \pi t} $

Daniel J.
Numerade Educator

Problem 29

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ A(t) = \arcsin(1 + 2t) $

DM
David M.
Numerade Educator

Problem 30

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ B(x) = \dfrac{\tan x}{\sqrt{4 - x^2}} $

DM
David M.
Numerade Educator

Problem 31

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ \displaystyle M(x) = \sqrt{1 + \frac{1}{x}} $

Linda H.
Numerade Educator

Problem 32

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ N(r) = \tan^{-1}(1 + e^{-r^2}) $

Daniel J.
Numerade Educator

Problem 33

Locate the discontinuities of the function and illustrate by graphing.

$ y = \dfrac{1}{1 + e^{1/x}} $

Daniel J.
Numerade Educator

Problem 34

Locate the discontinuities of the function and illustrate by graphing.

$ y = \ln (\tan^2 x) $

Leon D.
Numerade Educator

Problem 35

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to 2} x \sqrt{20 - x^2} $

Daniel J.
Numerade Educator

Problem 36

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to \pi} \sin(x + \sin x) $

Leon D.
Numerade Educator

Problem 37

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to 1} \ln\biggl( \dfrac{5 - x^2}{1 + x} \biggr) $

Leon D.
Numerade Educator

Problem 38

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to 4} 3^{\sqrt{x^2 - 2x - 4}} $

Daniel J.
Numerade Educator

Problem 39

Show that $ f $ is continuous on $ (-\infty, \infty ) $.

$ f(x) = \left\{
\begin{array}{ll}
1 - x^2 & \mbox{if $ x \le 1 $}\\
\ln x & \mbox{if $ x > 1 $}
\end{array} \right.$

Ma. Theresa A.
Numerade Educator

Problem 40

Show that $ f $ is continuous on $ (-\infty, \infty ) $.

$ f(x) = \left\{
\begin{array}{ll}
\sin x & \mbox{if $ x < \pi/4 $}\\
\cos x & \mbox{if $ x \ge \pi/4 $}
\end{array} \right.$

Daniel J.
Numerade Educator

Problem 41

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{
\begin{array}{ll}
x^2 & \mbox{if $ x < -1 $}\\
x & \mbox{if $ -1 \le x < 1 $} \\
1/x & \mbox{if $ x \ge 1 $}
\end{array} \right.$

Cindy R.
Numerade Educator

Problem 42

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{
\begin{array}{ll}
2^x & \mbox{if $ x \le 1 $}\\
3 - x & \mbox{if $ 1 < x \le 4 $} \\
\sqrt{x} & \mbox{if $ x > 4 $}
\end{array} \right.$

Leon D.
Numerade Educator

Problem 43

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{
\begin{array}{ll}
x + 2 & \mbox{if $ x < 0 $}\\
e^x & \mbox{if $ 0 \le x \le 1 $} \\
2 - x & \mbox{if $ x > 1 $}
\end{array} \right.$

AK
Anjali K.
Numerade Educator

Problem 44

The gravitational force exerted by the planet Earth on a unit mass at a distance $ r $ from the center of the planet is

$ F(r) = \left\{
\begin{array}{ll}
\frac{GMr}{R^3} & \mbox{if $ r < R $}\\
\frac{GM}{r^2} & \mbox{if $ r \ge R $}
\end{array} \right.$

where $ M $ is the mass of Earth, $ R $ is its radius, and $ G $ is the gravitational constant. Is $ F $ a continuous function of $ r $?

Linda H.
Numerade Educator

Problem 45

For what value of the constant $ c $ is the function $ f $ continuous on $ (-\infty, \infty) $?

$ f(x) = \left\{
\begin{array}{ll}
cx^2 + 2x & \mbox{if $ x < 2 $}\\
x^3 - cx & \mbox{if $ x \ge 2 $}
\end{array} \right.$

Ma. Theresa A.
Numerade Educator

Problem 46

Find the values of $ a $ and $ b $ that make $ f $ continuous everywhere.

$ f(x) = \left\{
\begin{array}{ll}
\dfrac{x^2 - 4}{x - 2} & \mbox{if $ x < 2 $}\\
ax^2 - bx + 3 & \mbox{if $ 2 \le x < 3 $} \\
2x - a + b & \mbox{if $ x \ge 3 $}
\end{array} \right.$

Ma. Theresa A.
Numerade Educator

Problem 47

Suppose $ f $ and $ g $ are continuous functions such that $ g(2) = 6 $ and $ \displaystyle \lim_{x \to 2} [3f(x) + f(x)g(x)] = 36 $. Find $ f(2) $.

Ma. Theresa A.
Numerade Educator

Problem 48

Let $ f(x) = 1/x $ and $ g(x) = 1/x^2 $.

(a) Find $ (f \circ g)(x) $.
(b) Is $ f \circ g $ continuous everywhere? Explain.

Daniel J.
Numerade Educator

Problem 49

Which of the following functions $ f $ has a removable discontinuity at $ a $? If the discontinuity is removable, find a function $ g $ that agrees with $ f $ for $ x \neq a $ and is continuous at $ a $.

(a) $ f(x) = \dfrac{x^4 -1}{x - 1}$, $ a = 1 $
(b) $ f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2} $, $ a = 2 $
(c) $ f(x) = [ \sin x ] $, $ a = \pi $

Carolyn B.
Numerade Educator

Problem 50

Suppose that a function $ f $ is continuous on $ [0, 1] $ except at 0.25 and that $ f(0) = 1 $ and $ f(1) = 3 $. Let $ N = 2 $. Sketch two possible graphs of $ f $, one showing that $ f $ might not satisfy the conclusion of the Intermediate Value Theorem and one showing that $ f $ might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn't satisfy the hypothesis.)

Daniel J.
Numerade Educator

Problem 51

If $ f(x) = x^2 + 10 \sin x $, show that there is a number $ c $ such that $ f(c) = 1000 $.

Leon D.
Numerade Educator

Problem 52

Suppose $ f $ is continuous on $ [1, 5] $ and the only solutions of the equation $ f(x) = 6 $ are $ x = 1 $ and $ x = 4 $. If $ f(2) = 8$, explain why $ f(3) > 6 $.

Stark L.
Numerade Educator

Problem 53

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

$ x^4 + x - 3 = 0 $, $ (1, 2) $

Stark L.
Numerade Educator

Problem 54

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

$ \ln x = x - \sqrt{x} $, $ (2, 3) $

Stark L.
Numerade Educator

Problem 55

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

$ e^x = 3 - 2x $, $ (0, 1) $

Leon D.
Numerade Educator

Problem 56

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

$ \sin x = x^2 - x $, $ (1, 2) $

Daniel J.
Numerade Educator

Problem 57

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

$ \cos x = x^3 $

Oswaldo J.
Numerade Educator

Problem 58

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

$ \ln x = 3 - 2x $

Daniel J.
Numerade Educator

Problem 59

(a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three decimal places.

$ 100e^{-x/100} = 0.01x^2 $

Leon D.
Numerade Educator

Problem 60

(a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three decimal places.

$ \arctan x = 1 - x $

Daniel J.
Numerade Educator

Problem 61

Prove, without graphing, that the graph of the function has at least two $ x $-intercepts in the specified interval.

$ y = \sin x^3 $, $ (1, 2) $

AD
Anupa D.
Numerade Educator

Problem 62

Prove, without graphing, that the graph of the function has at least two $ x $-intercepts in the specified interval.

$ y = x^2 - 3 + 1/x $, $ (0, 2) $

Daniel J.
Numerade Educator

Problem 63

Prove that $ f $ is continuous at $ a $ if and only if
$$ \lim_{h \to 0}f(a + h) = f(a) $$

Daniel J.
Numerade Educator

Problem 64

To prove that sine is continuous, we need to show that $ \lim_{x \to a} \sin x = \sin a $ for every real number $ a $. By Exercise 63 an equivalent statement is that $$ \lim_{h \to 0} \sin (a + h) = \sin a $$.

Use (6) to show that this is true.

Daniel J.
Numerade Educator

Problem 65

Prove that cosine is a continuous function.

Daniel J.
Numerade Educator

Problem 66

(a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5.

Daniel J.
Numerade Educator

Problem 67

For what values of $ x $ is $ f $ continuous?
$$ f(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x $ is rational}\\
1 & \mbox{if $ x $ is irrational}
\end{array} \right.$$

Daniel J.
Numerade Educator

Problem 68

For what values of $ x $ is $ g $ continuous?
$$ g(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x $ is rational}\\
x & \mbox{if $ x $ is irrational}
\end{array} \right.$$

Daniel J.
Numerade Educator

Problem 69

Is there a number that is exactly 1 more than its cube?

Daniel J.
Numerade Educator

Problem 70

If $ a $ and $ b $ are positive numbers, prove that the equation

$$ \dfrac{a}{x^3 + 2x^2 - 1} + \dfrac{b}{x^3 + x - 2} = 0 $$

has at least one solution in the interval $ (-1, 1) $.

Daniel J.
Numerade Educator

Problem 71

Show that the function
$ f(x) = \left\{
\begin{array}{ll}
x^4 \sin (1/x) & \mbox{if $ x \neq 0 $}\\
0 & \mbox{if $ x = 0 $}
\end{array} \right.$

is continuous on $ (-\infty, \infty) $.

Daniel J.
Numerade Educator

Problem 72

(a) Show that the absolute value function $ F(x) = | x | $ is continuous everywhere.
(b) Prove that if $ f $ is a continuous function on an interval, then so is $ | f | $.
(c) Is the converse of the statement in part (b) also true? In other words, if $ | f | $ is continuous, does it follow that $ f $ is continuous? If so, prove it. If not, find a counterexample.

Jimmy Y.
Numerade Educator

Problem 73

A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of the day on both days.

Daniel J.
Numerade Educator