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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 13 Chapter 13 Curve SketchingCurve Sketching
2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics
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9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL ANALYSIS
2007 Pearson Education Asia
• To find critical values, to locate relative maxima and relative minima of a curve.
• To find extreme values on a closed interval.
• To test a function for concavity and inflection points.
• To locate relative extrema by applying the second-derivative test.
• To sketch the graphs of functions having asymptotes.
• To model situations involving maximizing or minimizing a quantity.
Chapter 13: Curve Sketching
Chapter ObjectivesChapter Objectives
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Relative Extrema
Absolute Extrema on a Closed Interval
Concavity
The Second-Derivative Test
Asymptotes
Applied Maxima and Minima
13.1)
13.2)
13.3)
Chapter 13: Curve Sketching
Chapter OutlineChapter Outline
13.4)
13.5)
13.6)
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema13.1 Relative ExtremaIncreasing or Decreasing Nature of a Function
• Increasing f(x) if x1 < x2 and f(x1) < f(x2).
• Decreasing f(x) if x1 < x2 and f(x1) > f(x2).
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
Extrema
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
RULE 1 - Criteria for Increasing or Decreasing Function
• f is increasing on (a, b) when f’(x) > 0
• f is decreasing on (a, b) when f’(x) < 0
RULE 2 - A Necessary Condition for Relative Extrema
existnot does '
or
0'
at
extremum relative
af
af
a
implies
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
RULE 3 - Criteria for Relative Extrema
1. If f’(x) changes from +ve to –ve, then f has a relative maximum at a.
2. If f’(x) changes from -ve to +ve, then f has a relative minimum at a.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
First-Derivative Test for Relative Extrema
1. Find f’(x).
2. Determine all critical values of f.
3. For each critical value a at which f is continuous, determine whether f’(x) changes sign as x increases through a.
4. For critical values a at which f is not continuous, analyze the situation by using the definitions of extrema directly.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
Example 1 - First-Derivative Test
If , use the first-derivative test
to find where relative extrema occur.
Solution:
STEP 1 -
STEP 2 - Setting f’(x) = 0 gives x = −3, 1.
STEP 3 - Conclude that at−3, there is a relative
maximum.
STEP 4 – There are no critical values at which f is not
continuous.
1 for 1
4
x
xxxfy
1 for 1
13
1
32'
22
2
xx
xx
x
xxxf
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.1 Relative Extrema
Example 3 - Finding Relative Extrema
Test for relative extrema.
Solution: By product rule,Relative maximum when x = −2 Relative minimum when x = 0.
xexxfy 2
2' xxexf x
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.2 Absolute Extrema on a Closed Interval13.2 Absolute Extrema on a Closed IntervalExtreme-Value Theorem
• If a function is continuous on a closed interval, then the function has a maximum value and a minimum value on that interval.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.2 Absolute Extrema on a Closed Interval
Procedure to Find Absolute Extrema for a Function f That Is Continuous on [a, b]
1. Find the critical values of f .
2. Evaluate f(x) at the endpoints a and b and at the critical values in (a, b).
3. The maximum value of f is the greatest value found in step 2. The minimum value is the least value found in step 2.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.2 Absolute Extrema on a Closed Interval
Example 1 - Finding Extreme Values on a Closed Interval
Find absolute extrema for over the closed interval [1, 4].
Solution:
Step 1:
Step 2:
Step 3:
542 xxxf
2242' xxxxf
endpoints at of values 54
21
ff
f
4 1, in 2 value critical at of values 12 ff
12 is min and 54 is max ff
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.3 Concavity13.3 Concavity• Cases where curves concave upward:
• Cases where curves concave downward:
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.3 Concavity
• f is said to be concave up on (a, b) if f is increasing on (a, b).
• f is said to be concave down on (a, b) if f is decreasing on (a, b).
• f has an inflection point at a if it is continuous at a and f changes concavity at a.
Criteria for Concavity
• If f’’(x) > 0, f is concave up on (a, b).
• If f”(x) < 0, f is concave down on (a, b).
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.3 Concavity
Example 1 - Testing for Concavity
Determine where the given function is concave up and where it is concave down.
Solution:
Applying the rule,
Concave up when 6(x − 1) > 0 as x > 1. Concave down when 6(x − 1) < 0 as x < 1.
11 a. 3 xxfy
16''
13' 2
xy
xy
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.3 Concavity
Example 1 - Testing for Concavity
Solution:
Applying the rule,
As y’’ is always positive, y = x2 is always concave up.
2 b. xy
2''
2'
y
xy
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.3 Concavity
Example 3 - A Change in Concavity with No Inflection Point
Discuss concavity and find all inflection points for f(x) = 1/x.
Solution:
x > 0 f”(x) > 0 and x < 0 f”(x) < 0. f is concave up on (0,∞) and concave down on (−∞, 0)f is not continuous at 0 no inflection point
0 for 2''
0 for '3
2
xxxf
xxxf
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.4 The Second-Derivative Test13.4 The Second-Derivative Test• The test is used to test certain critical values for
relative extrema.
Suppose f’(a) = 0.
• If f’’(a) < 0, then f has a relative maximum at a.
• If f’’(a) > 0, then f has a relative minimum at a.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.4 The Second-Derivative Test
Example 1 - Second-Derivative Test
Test the following for relative maxima and minima. Use the second-derivative test, if possible.
Solution:
Relative minimum when x = −3.
33218 . xxya
xy
xxy
4''
332'
3 have we,0' When xy
01234'' ,3 When
01234'' ,3 When
yx
yx
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.4 The Second-Derivative Test
Example 1 - Second-Derivative Test
Solution:
No maximum or minimum exists when x = 0.
186 . 34 xxyb
xxy
xxxxy
4872''
1242424'2
223
1 ,0 have we,0' When xy
0'' ,1 When
0'' ,0 When
yx
yx
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes13.5 AsymptotesVertical Asymptotes
• The line x = a is a vertical asymptote if at least one of the following is true:
Vertical-Asymptote Rule for Rational Functions
• P and Q are polynomial functions and the quotient is in lowest terms.
xf
xf
ax
ax
lim
lim
xQ
xPxf
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Example 1 - Finding Vertical Asymptotes
Determine vertical asymptotes for the graph of
Solution: Since f is a rational function,
Denominator is 0 when x is 3 or 1.
The lines x = 3 and x = 1 are vertical asymptotes.
34
42
2
xx
xxxf
13
42
xx
xxxf
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Horizontal and Oblique Asymptotes
• The line y = b is a horizontal asymptote if at least one of the following is true:
Nonvertical asymptote
• The line y = mx +b is a nonvertical asymptote if at least one of the following is true:
bxfbxfxx
lim or lim
0lim or 0lim
bmxxfbmxxfxx
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Example 3 - Finding an Oblique Asymptote
Find the oblique asymptote for the graph of the rational function
Solution:
y = 2x + 1 is an oblique asymptote.
25
5910 2
x
xxxfy
25
312
25
5910 2
x
xx
xxxf
025
3lim12lim
xxxf
xx
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Example 5 - Finding Horizontal and Vertical Asymptotes
Find horizontal and vertical asymptotes for the graph
Solution: Testing for horizontal asymptotes,
The line y = −1 is a horizontal asymptote.
1 xexfy
1101limlim1lim
1lim
x
x
x
x
x
x
x
ee
e
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Example 7 - Curve Sketching
Sketch the graph of .
Solution:Intercepts (0, 0) is the only intercept.
Symmetry There is only symmetry about the origin.
Asymptotes Denominator 0 No vertical asymptote
Since
y = 0 is the only non-vertical asymptote
Max and Min For , relative maximum is (1, 2).
Concavity For , inflection points are
(-√ 3, -√3), (0, 0), (√3, √3).
1
42
x
xy
01
4lim
2
x
xx
22 1
114'
x
xxy
32 1
338''
x
xxxy
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.5 Asymptotes
Example 7 - Curve Sketching
Solution: Graph
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Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima13.6 Applied Maxima and Minima
Example 1 - Minimizing the Cost of a Fence
• Use absolute maxima and minima to explain the endpoints of the domain of the function.
A manufacturer plans to fence in a 10,800-ft2 rectangular storage area adjacent to a building by using the building as one side of the enclosed area. The fencing parallel to the building faces a highway and will cost $3 per foot installed, whereas the fencing for the other two sides costs $2 per foot installed. Find the amount of each type of fence so that the total cost of the fence will be a minimum. What is the minimum cost?
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 1 - Minimizing the Cost of a Fence
Solution:
Cost function is
Storage area is
Analyzing the equations,
Thus,
yxCyyxC 43223
xyxy
10800800,10
x
xx
xxC43200
310800
43
0 since 120
4320030
2
xxxdx
dC
0 ,120 When
86400
2
2
32
2
dx
Cdx
xdx
Cdand
Only critical value is 120. x =120 gives a relative minimum.
720120
432003120 xC
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 3 - Minimizing Average Cost
A manufacturer’s total-cost function is given by
where c is the total cost of producing q units. At what level of output will average cost per unit be a minimum? What is this minimum?
40034
2
qcc
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 3 - Minimizing Average Cost
Solution:Average-cost function is
To find critical values, we set
is positive when q = 40, which is the only relative extremum.
The minimum average cost is
q
q
q
q
cqcc
4003
4
40034
2
0 since 404
16000
2
2
qqq
q
dq
cd
32
2 800
qdq
cd
2340
4003
4
4040 c
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 5 - Economic Lot Size
A company annually produces and sells 10,000 units of a product. Sales are uniformly distributed throughout the year. The company wishes to determine the number of units to be manufactured in each production run in order to minimize total annual setup costs and carrying costs. The same number of units is produced in each run. This number is referred to as the economic lot size or economic order quantity. The production cost of each unit is $20, and carrying costs (insurance, interest, storage, etc.) are estimated to be 10% of the value of the average inventory. Setup costs per production run are $40. Find the economic lot size.
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 5 - Economic Lot Size
Solution: Let q be the number of units in a production run.
Total of the annual carrying costs and setup is
Setting dC/dq = 0, we get
Since q > 0, there is an absolute minimum at q = 632.5.
Number of production runs = 10,000/632.5 15.8
16 lots Economic size = 625 units
2
2
2
4000004000001
400001000040
2201.0
q
q
qdq
dC
q
qC
5.632400000
4000000
2
2
q
q
q
dq
dC
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 7 - Maximizing the Number of Recipients of Health-Care Benefits
An article in a sociology journal stated that if a particular health-care program for the elderly were initiated, then t years after its start, n thousand elderly people would receive direct benefits, where
For what value of t does the maximum number receive benefits?
120 3263
23
tttt
n
2007 Pearson Education Asia
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Example 7 - Maximizing the Number of Recipients of Health-Care Benefits
Solution: Setting dn/dt = 0, we have
Absolute maximum value of n must occur at t = 0, 4, 8, or 12:
Absolute maximum occurs when t = 12.
120 3263
23
tttt
n
8 or 4
32120 2
tt
ttdt
dn
9612 ,3
1288 ,
3
1604 ,00 nnnn