Chapter 3 Limits and the Derivative Section 3 Continuity (Part 1)

Chapter 3

Limits and the Derivative

Section 3

Continuity

(Part 1)

2Barnett/Ziegler/Byleen Business Calculus 12e

Learning Objectives for Section 3.3Continuity

The student will understand the concept of continuity

The student will be able to apply the continuity properties

The student will be able to solve word problems.

The student will be able to solve inequalities


Continuity

In this lesson, we’ll take a closer look at graphs that are discontinuous due to:

• Holes

• Gaps

• Asymptotes


Definition of Continuity

A function f is continuous at a point x = c if it meets these three criteria:

1.

2. f (c) undefined

3. )()(lim cfxfcx

lim𝑥→𝑐

𝑓 (𝑥)≠𝐷𝑁𝐸

5

Example 1

Barnett/Ziegler/Byleen Business Calculus 12e

Continuous over the interval: (−∞ ,2 )∪ (2 ,3)∪ (3 ,∞ )

Is f(x) continuous at x = 2 ?

Is f(x) continuous at x = 3 ?

lim𝑥→ 2

𝑓 (𝑥 )= 𝑓 (2)?

𝐷𝑁𝐸 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑f(x) is not continuous at x = 2

lim𝑥→ 3

𝑓 (𝑥 )= 𝑓 (3)?

𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑1f(x) is not continuous at x = 3

6

Example 2


Continuous over the interval: (−∞ ,−2 )∪ (−2 ,∞ )

Is f(x) continuous at x = -2 ?

lim𝑥→−2

𝑓 (𝑥 )= 𝑓 (−2)

3

?

−1f(x) is not continuous at x = -2


Example 3

𝑓 (𝑥 )=2𝑥2+𝑥−1 Is f(x) continuous at x = 3?

lim𝑥→ 3

𝑓 (𝑥 )= 𝑓 (3)

f(x) is continuous at x = 3

2020

?


Example 4

𝑓 (𝑥 )=𝑥2−4𝑥+2 Is f(x) continuous at x = -2?

1. lim𝑥→− 2

𝑥2−4𝑥+2

=¿¿

2 . 𝑓 (−2 )=¿

f(x) is NOT continuous at x = -2

−4

𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

lim𝑥→−2

(𝑥+2)(𝑥−2)𝑥+2

¿ lim𝑥→−2

(𝑥−2)=¿

lim𝑥→−2

𝑥2−4𝑥+2

= 𝑓 (−2)?


Example 5

𝑓 (𝑥 )=|𝑥−5|𝑥−5

Is f(x) continuous at x = 5? Explain.

1. lim𝑥→5

|𝑥−5|𝑥−5

2 . 𝑓 (5 )=¿f(x) is NOT continuous at x = 5

𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

lim𝑥→ 5−

|𝑥−5|𝑥−5

=¿¿

lim𝑥→ 5+¿ |𝑥− 5|

𝑥− 5=¿ ¿¿

¿

lim𝑥→5−

−(𝑥−5)𝑥−5

=¿¿

lim𝑥→ 5+¿ (𝑥−5)

𝑥−5=¿¿ ¿

¿

−1

1lim𝑥→ 5

𝑓 (𝑥 )=𝐷𝑁𝐸

lim𝑥→ 5

|𝑥−5|𝑥−5

= 𝑓 (5)?


Example 6

𝑓 (𝑥 )=|𝑥−5|𝑥−5

Is f(x) continuous at x = 2?

1. lim𝑥→2

|𝑥−5|𝑥−5

=¿

2 . 𝑓 (2 )=¿

f(x) is continuous at x = 2

−1

−1|2−5|2−5

=¿

lim𝑥→ 2

|𝑥−5|𝑥−5

= 𝑓 (2)?

11

Continuity

Where is a graph continuous?• Where there are no asymptotes or holes.• Where the function is defined.


12

Example 7

Where is continuous?

is continuous over the interval:


13

Example 8




14

Example 9




15

Homework

#3-3A

Pg 161

(7-14, 23, 27,

29, 31, 49-59 odd)


Chapter 3

Limits and the Derivative

Section 3

Continuity

(Part 2)


Learning Objectives for Section 3.3Continuity

The student will understand the concept of continuity

The student will be able to apply the continuity properties

The student will be able to solve word problems.

The student will be able to solve inequalities

18

Application: Media

A music website called MyTunes charges $0.99 per song if you download less than 100 songs per month and $0.89 per song if you download 100 or more songs per month.

Write a piecewise function f(x) for the cost of downloading x songs per month.

Graph the function. Is f(x) continuous at x = 100? Explain.


19

Solution


𝑓 (𝑥 )={0.99𝑥 0≤𝑥<1000.89𝑥 𝑥 ≥100

$99$89$79

100 110 f(x) is NOT continuous at x=100

lim𝑥→ 100

𝑓 (𝑥 )= 𝑓 (100)

89𝐷𝑁𝐸

?

20

Application: Natural Gas Rates

The table shows the monthly rates for natural gas charged by MyGas Company. The charge is based on the number of therms used per month. (1 therm = 100,000 Btu)

Write a piecewise function f(x) of the monthly charge for x therms.

Graph f(x). Is f(x) continuous at x = 40? Give a mathematical reason.


Amount Cost

Base Charge $8.00

First 40 therms $0.60 per therm

Over 40 therms $0.35 per therm

21

Solution


𝑓 (𝑥 )={ 8+0.60 𝑥0≤ 𝑥≤ 408+.60 ( 40 )+.35 (𝑥−40)𝑥>40

¿ {8+0.60 𝑥0≤ 𝑥≤ 4 0.35𝑥+18 𝑥>40

Amount Cost

Base Charge $8.00

First 40 therms $0.60 per therm

Over 40 therms $0.35 per therm

22

Solution


$90$60$30

40 80 f(x) IS continuous at x=40

𝑓 (𝑥)={8+0.60 𝑥0≤ 𝑥≤4 0.35𝑥+18 𝑥>40

lim𝑥→ 40

𝑓 (𝑥 )= 𝑓 (40)

3232

?

23

Solving Inequalities

Up until now, we have solved inequalities using a graphical approach.• Where is the graph below the x-axis?• Where is the graph above the x-axis?

Now we will learn an algebraic approach that is based on continuity properties.



Constructing Sign Charts

1. Find all numbers which are:

a. Holes or vertical asymptotes. Plot these as open circles on the number line.

b. x-interceptsPlot these according to the inequality symbol.

2. Select a test number in each interval and determine if f (x) is positive (+) or negative (–) in the interval.

3. Determine your answer using the signs and the inequality symbol and write it using interval notation.

25

Polynomial Inequalities

Ex 3: Solve and write your answer in interval notation.


𝑥2−4 𝑥−12>0

(x −6)( x+2)>0

6−2+ - +

𝐴𝑛𝑠𝑤𝑒𝑟 :(−∞,−2)∪(6 ,∞ )

Factor f(x) and solve for zeros.

Graph the zeros on a number line. Use open circles for < or >, use closed circles for or .

Test numbers on all sides of the zeros by plugging them into the inequality.

Since f(x) > 0 we want the positive intervals.

26

Rational Inequalities

Ex 4: Solve and write your answer in interval notation.


𝑥2−4𝑥+4

≤0

Graph the x-intercepts. Use open circles for < or >, use closed circles for or .

(𝑥+2)(𝑥−2)𝑥+4

≤0

Graph the holes and vertical asymptotes as open circles.

2−2−4

𝐴𝑛𝑠𝑤𝑒𝑟 : (−∞ ,−4 )∪[−2 ,2]

Test numbers on all sides of the points by plugging them into the reduced inequality.

Since f(x) 0, we want the negative intervals.

− +¿ − +¿

Factor the top and bottom.

27

Homework


#3-3BPg. 162

(36-43, 45, 81, 85, 86)

Documents

Chapter 3 Limits and the Derivative Section 3 Continuity (Part 1)