chapter 3 limits and the derivative section 2 infinite limits and limits at infinity (part 1)
TRANSCRIPT
Chapter 3
Limits and the Derivative
Section 2
Infinite Limits and Limits at Infinity
(Part 1)
2Barnett/Ziegler/Byleen Business Calculus 12e
Objectives for Section 3.2 Infinite Limits and Limits at Infinity
The student will understand the concept of infinite limits. The student will be able to calculate limits at infinity.
3Barnett/Ziegler/Byleen Business Calculus 12e
Example 1
Recall from the first lesson:
lim๐ฅโ 0โ
1๐ฅ
=ยฟ lim๐ฅโ 0+ยฟ 1
๐ฅ=ยฟยฟ
ยฟ lim๐ฅโ 0
1๐ฅ
=ยฟโ โ โ ๐ท๐๐ธ
4Barnett/Ziegler/Byleen Business Calculus 12e
Infinite Limits and Vertical Asymptotes
Definition:
If the graph of y = f (x) has a vertical asymptote of x = a, then as x approaches a from the left or right, then f(x) approaches either or -.
Vertical asymptotes (and holes) are called points of discontinuity.
5Barnett/Ziegler/Byleen Business Calculus 12e
Example 2
Let
Identify all holes and asymptotes and find the left and right hand limits as x approaches the vertical asymptotes.
1
22
2
x
xxxf
๐ (๐ฅ )=(๐ฅ+2)(๐ฅโ1)(๐ฅ+1)(๐ฅโ 1)
ยฟ๐ฅ+2๐ฅ+1
๐ป๐๐๐ :(1 ,1.5)๐๐ด :๐ฅ=โ1
๐ป๐ด : ๐ฆ=1
6Barnett/Ziegler/Byleen Business Calculus 12e
Example 2 (continued)
2
2
2( )
1
x xf x
x
Vertical Asymptote
Hole
lim๐ฅโ โ1โ
๐ฅ+2๐ฅ+1
=ยฟยฟlim
๐ฅโ โ1+ยฟ ๐ฅ+2๐ฅ+1
=ยฟ ยฟยฟ
ยฟlim๐ฅโ โ1
๐ฅ+2๐ฅ+1
=ยฟยฟโ โ โ ๐ท๐๐ธ
Horizontal Asymptote
7Barnett/Ziegler/Byleen Business Calculus 12e
Example 3
Let
Identify all holes and asymptotes and find the left and right hand limits as x approaches the vertical asymptotes.
๐ ( x )= 1
(๐ฅโ 2)2 ๐๐๐ป๐๐๐๐
๐๐ด :๐ฅ=2
๐ (๐ฅ )= 1
(๐ฅโ2)2
๐ป๐ด : ๐ฆ=0
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Example 3 (continued)
Barnett/Ziegler/Byleen Business Calculus 12e
lim๐ฅโ 2โ
1
(๐ฅโ2)2 =ยฟ ยฟโ โlim
๐ฅโ 2+ยฟ 1
(๐ฅโ 2)2 =ยฟยฟ ยฟ
ยฟโlim
๐ฅโ 2
1
(๐ฅโ 2)2 =ยฟยฟ
9Barnett/Ziegler/Byleen Business Calculus 12e
Limits at Infinity
โข We will now study limits as x ยฑ.
โข This is the same concept as the end behavior of a graph.
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End Behavior Review
Barnett/Ziegler/Byleen Business Calculus 12e
Odd degreePositiveleading
coefficient
Odd degreeNegativeleading
coefficient
Even degreePositiveleading
coefficient
Even degreeNegativeleading
coefficient
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Polynomial Functions
Ex 4: Evaluate each limit.
Barnett/Ziegler/Byleen Business Calculus 12e
lim๐ฅโ โโ
๐ฅ2=ยฟยฟ
lim๐ฅโ+โ
๐ฅ2=ยฟยฟ
lim๐ฅโ โโ
โ3 ๐ฅ4=ยฟยฟ
lim๐ฅโ+โ
โ 3๐ฅ4=ยฟยฟ
lim๐ฅโ โโ
6 ๐ฅ3=ยฟ
lim๐ฅโ+โ
6 ๐ฅ3=ยฟ
lim๐ฅโ โโ
โ5 ๐ฅ3=ยฟ
lim๐ฅโ+โ
โ5 ๐ฅ3=ยฟ
โ
โ
โ โ
โ โ
โ โ
โ
โ
โ โ
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Rational Functions
If a rational function has a horizontal asymptote, then it determines the end behavior of the graph.
If f(x) is a rational function, then
Barnett/Ziegler/Byleen Business Calculus 12e
lim๐ฅโ ยฑ โ
๐ (๐ฅ )=h๐๐๐๐ง๐๐๐ก๐๐๐๐ ๐ฆ๐๐๐ก๐๐ก๐๐ฃ๐๐๐ข๐
13
Rational Functions
Ex 5: Evaluate
Barnett/Ziegler/Byleen Business Calculus 12e
๐ (๐ฅ )= 5๐ฅ+2 ๐ป๐ด : ๐ฆ=0
Because the degree of the numerator < degree of the
denominator.
lim๐ฅโ โ
๐ (๐ฅ )=ยฟยฟ
lim๐ฅโ โโ
๐ (๐ฅ )=ยฟยฟ
0
0
lim๐ฅโ ยฑ โ
๐ (๐ฅ )
14
Rational Functions
Ex 6: Evaluate
Barnett/Ziegler/Byleen Business Calculus 12e
๐ป๐ด : ๐ฆ=32
Because the degree of the numerator = degree of the
denominator.
lim๐ฅโ โ
๐ (๐ฅ )=ยฟยฟ
lim๐ฅโ โโ
๐ (๐ฅ )=ยฟยฟ
32
32
2
2
3 5 9
2 7
x xy
x
lim๐ฅโ ยฑ โ
๐ (๐ฅ )
15
Rational Functions
If a rational function doesnโt have a horizontal asymptote, then to determine its end behavior, take the limit of the ratio of the leading terms of the top and bottom.
Barnett/Ziegler/Byleen Business Calculus 12e
16
Rational Functions
Ex 7: Evaluate
Barnett/Ziegler/Byleen Business Calculus 12e
๐ป๐ด :๐๐๐๐Because the degree of the numerator > degree of the
denominator.
lim๐ฅโ โ
2 ๐ฅ5
6 ๐ฅ3 =ยฟยฟ
๐ (๐ฅ )= 2 ๐ฅ5 โ๐ฅ3โ 16 ๐ฅ3+2 ๐ฅ2โ7
lim๐ฅโ โ
๐ฅ2
3=ยฟยฟโ
lim๐ฅโ โโ
2๐ฅ5
6 ๐ฅ3 =ยฟ ยฟlim๐ฅโ โโ
๐ฅ2
3=ยฟยฟโ
lim๐ฅโ ยฑ โ
๐ (๐ฅ )
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Rational Functions
Ex 8: Evalaute
Barnett/Ziegler/Byleen Business Calculus 12e
๐ป๐ด :๐๐๐๐
lim๐ฅโ โ
5 ๐ฅ6
2 ๐ฅ5 =ยฟยฟ
๐ (๐ฅ )= 5๐ฅ6+3 ๐ฅ2 ๐ฅ5 โ๐ฅโ 5
lim๐ฅโ โ
5 ๐ฅ2
=ยฟยฟโ
lim๐ฅโ โโ
5 ๐ฅ6
2๐ฅ5 =ยฟยฟlim๐ฅโ โโ
5 ๐ฅ2
=ยฟยฟโ โ
lim๐ฅโ ยฑ โ
๐ (๐ฅ )
18
Homework
Barnett/Ziegler/Byleen Business Calculus 12e
#3-2A: Pg 150
(3-15 mult. of 3,17, 19, 31-35 odd, 39, 43, 45, 61, 65)
Chapter 3
Limits and the Derivative
Section 2
Infinite Limits and Limits at Infinity
(Part 2)
20Barnett/Ziegler/Byleen Business Calculus 12e
Objectives for Section 3.2 Infinite Limits and Limits at Infinity
The student will be able to solve applications involving limits.
2 โ & >
21
Application: Business
T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100.
A. Assuming the total cost per day is linearly related to the number of boards made per day, write an equation for the cost function.
B. Write the equation for the average cost function.
C. Graph the average cost function:
D. What does the average cost per board approach as production increases?
Barnett/Ziegler/Byleen Business Calculus 12e
22
Application: Business
T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Assuming the total cost per day is linearly related to the
number of boards made per day, write an equation for the cost function.
Barnett/Ziegler/Byleen Business Calculus 12e
๐ฆ=๐๐ฅ+๐5100=๐(20)+300๐=240
h๐ ๐๐๐๐ ๐ก ๐๐ข๐๐๐ก๐๐๐๐๐ :๐ถ (๐ฅ )=240 ๐ฅ+300
23
Application: Business
T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Write the equation for the average cost function.
Barnett/Ziegler/Byleen Business Calculus 12e
๐ถ (๐ฅ )=240๐ฅ+300๐ฅ
๐ถ (๐ฅ )=๐ถ (๐ฅ)๐ฅ
24
Application: Business
T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Graph the average cost function:
Barnett/Ziegler/Byleen Business Calculus 12e
Number of surfboards
Average cost per
day
25
Application: Business
T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. What does the average cost per board approach as
production increases?
Barnett/Ziegler/Byleen Business Calculus 12e
๐ถ (๐ฅ )=240๐ฅ+300๐ฅ
As the number of boards increases, the average cost approaches $240 per board.
Number of surfboards
Average cost per
day
26
Application: Medicine
A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patientโs bloodstream t hours after the drip was started is modeled by the equation:
A. What is the drug concentration after 2 hours?
B. Evaluate and interpret the meaning of the limit:
Barnett/Ziegler/Byleen Business Calculus 12e
๐ถ (๐ก )=5 ๐ก (๐ก+50 )๐ก 3+100
lim๐กโ โ
๐ถ (๐ก)
27
Application: Medicine
A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patientโs bloodstream t hours after the drip was started is modeled by the equation:
What is the drug concentration after 2 hours?
Barnett/Ziegler/Byleen Business Calculus 12e
๐ถ (๐ก )=5 ๐ก (๐ก+50 )๐ก 3+100
๐ถ (2 )=5 (2)(2+50 )
23+100โ 4.8
After 2 hours, the concentration of the drug is 4.8 mg/ml.
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Application: Medicine
A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patientโs bloodstream t hours after the drip was started is modeled by the equation:
Evaluate and interpret the meaning of the limit:
Barnett/Ziegler/Byleen Business Calculus 12e
๐ถ (๐ก )=5 ๐ก (๐ก+50 )๐ก 3+100
lim๐กโ โ
๐ถ (๐ก)
lim๐กโ โ
5 ๐ก (๐ก+50 )๐ก 3+100
=0
As time passes, the drug concentration approaches 0 mg/ml.
29
Homework
Barnett/Ziegler/Byleen Business Calculus 12e
#3-2B: Pg 150(2-8 even, 11, 13, 18,
34, 36, 37, 41, 49, 63, 67, 69, 73, 76)