Chapter 11 - Differentiation
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Transcript of Chapter 11 - Differentiation
INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 11 Chapter 11 DifferentiationDifferentiation
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 Probability10. Limits and Continuity
11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL ANALYSIS
2007 Pearson Education Asia
• To compute derivatives by using the limit definition.
• To develop basic differentiation rules.
• To interpret the derivative as an instantaneous rate of change.
• To apply the product and quotient rules.
• To apply the chain rule.
Chapter 11: Differentiation
Chapter ObjectivesChapter Objectives
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The Derivative
Rules for Differentiation
The Derivative as a Rate of Change
The Product Rule and the Quotient Rule
The Chain Rule and the Power Rule
11.1)
11.2)
11.3)
Chapter 11: Differentiation
Chapter OutlineChapter Outline
11.4)
11.5)
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Chapter 11: Differentiation
11.1 The Derivative11.1 The Derivative• Tangent line at a point:
• The slope of a curve at P is the slope of the tangent line at P.
• The slope of the tangent line at (a, f(a)) is
hafhaf
azafzfm
haz
0tan limlim
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Chapter 11: Differentiation11.1 The Derivative
Example 1 – Finding the Slope of a Tangent Line
Find the slope of the tangent line to the curve y = f(x) = x2 at the point (1, 1).Solution: Slope = 211lim11lim
22
00
h
hh
fhfhh
• The derivative of a function f is the function denoted f’ and defined by
h
xfhxfxz
xfzfxfhxz
0
limlim'
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Chapter 11: Differentiation11.1 The Derivative
Example 3 – Finding an Equation of a Tangent Line
If f (x) = 2x2 + 2x + 3, find an equation of the tangent line to the graph of f at (1, 7).Solution:
Slope
Equation
24322322limlim'22
00
x
hxxhxhx
hxfhxfxf
hh
16
167
xy
xy
62141' f
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Chapter 11: Differentiation11.1 The Derivative
Example 5 – A Function with a Vertical Tangent Line
Example 7 – Continuity and Differentiability
Find . Solution:
xdxd
xh
xhxxdxd
h 21lim
0
a. For f(x) = x2, it must be continuous for all x.b. For f(p) =(1/2)p, it is not continuous at p = 0, thus the derivative does not exist at p = 0.
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Chapter 11: Differentiation
11.2 Rules for Differentiation11.2 Rules for Differentiation• Rules for Differentiation:
RULE 1 Derivative of a Constant:
RULE 2 Derivative of xn:
RULE 3 Constant Factor Rule:
RULE 4 Sum or Difference Rule
0cdxd
1 nn nxxdxd
xcfxcfdxd '
xgxfxgxfdxd ''
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Chapter 11: Differentiation11.2 Rules for Differentiation
Example 1 – Derivatives of Constant Functionsa.
b. If , then .
c. If , then .
03 dxd
5xg
4.807623,938,1ts
0' xg
0dtds
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Chapter 11: Differentiation11.2 Rules for Differentiation
Example 3 – Rewriting Functions in the Form xn
Differentiate the following functions:Solution:a.
b.
xy
xx
dxdy
21
21 12/1
xx
xh 1
2/512/32/3
23
23' xxx
dxdxh
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Chapter 11: Differentiation11.2 Rules for Differentiation
Example 5 – Differentiating Sums and Differences of Functions
Differentiate the following functions: xxxF 53 a.
x
xxx
xdxdx
dxdxF
2115
2153
3'
42/14
2/15
3/1
4 54
b.z
zzf
3/433/43
3/1
4
35
3154
41
54
'
zzzz
zdzdz
dzdzf
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Chapter 11: Differentiation11.2 Rules for DifferentiationExample 5 – Differentiating Sums and Differences of Functions
8726 c. 23 xxxy
7418
)8()(7)(2)(6
2
23
xxdxdx
dxdx
dxdx
dxd
dxdy
2007 Pearson Education Asia
Chapter 11: Differentiation11.2 Rules for Differentiation
Example 7 – Finding an Equation of a Tangent LineFind an equation of the tangent line to the curve when x = 1.
Solution: The slope equation is
When x = 1,
The equation is
xxy 23 2
2
12
23
2323
xdxdy
xxxx
xy
5123 2
1
xdxdy
45
151
xy
xy
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Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change11.3 The Derivative as a Rate of Change
Example 1 – Finding Average Velocity and Velocity
• Average velocity is given by
• Velocity at time t is given by
t
tfttftsvave
t
tfttfvt
0lim
Suppose the position function of an object moving along a number line is given by s = f(t) = 3t2 + 5, where t is in seconds and s is in meters.
a.Find the average velocity over the interval [10, 10.1].b. Find the velocity when t = 10.
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Chapter 11: Differentiation11.3 The Derivative as a Rate of ChangeExample 1 – Finding Average Velocity and Velocity
Solution:a. When t = 10,
b. Velocity at time t is given by
When t = 10, the velocity is
m/s 3.601.0
30503.3111.0
101.101.0
101.010
fffft
tfttftsvave
tdtdsv 6
m/s6010610
tdt
ds
2007 Pearson Education Asia
Chapter 11: Differentiation11.3 The Derivative as a Rate of Change
Example 3 – Finding a Rate of Change
• If y = f(x),
then
xxxx
xfxxfxy
to from interval the over x to respect with y of change of rate average
xrespect toy with
of change of rate ousinstantanelim
0 xy
dxdy
x
Find the rate of change of y = x4 with respect to x, and evaluate it when x = 2 and when x = −1. Solution: The rate of change is .
34xdxdy
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Chapter 11: Differentiation11.3 The Derivative as a Rate of Change
Example 5 – Rate of Change of VolumeA spherical balloon is being filled with air. Find the rate of change of the volume of air in the balloon with respect to its radius. Evaluate this rate of change when the radius is 2 ft.Solution: Rate of change of V with respect to r is
When r = 2 ft,
22 4334 rr
drdV
ftft1624
32
2
rdr
dV
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Chapter 11: Differentiation11.3 The Derivative as a Rate of Change
Applications of Rate of Change to Economics
• Total-cost function is c = f(q).
• Marginal cost is defined as .
• Total-revenue function is r = f(q).
• Marginal revenue is defined as .
dqdc
dqdr
Relative and Percentage Rates of Change
• The relative rate of change of f(x) is .
• The percentage rate of change of f (x) is
xfxf '
%100'xfxf
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Chapter 11: Differentiation11.3 The Derivative as a Rate of Change
Example 7 – Marginal Cost
If a manufacturer’s average-cost equation is
find the marginal-cost function. What is the marginal cost when 50 units are produced?Solution: The cost is
Marginal cost when q = 50,
qqqc 5000502.00001.0 2
5000502.00001.0
5000502.00001.0
23
2
qqq
qqqqcqc
504.00003.0 2 qqdqdc
75.355004.0500003.0 2
50
qdq
dc
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Chapter 11: Differentiation11.3 The Derivative as a Rate of Change
Example 9 – Relative and Percentage Rates of Change
11.4 The Product Rule and the Quotient Rule11.4 The Product Rule and the Quotient Rule
Determine the relative and percentage rates of change of
when x = 5.Solution:
2553 2 xxxfy
56' xxf
%3.33333.0
7525
55' change %
255565'
ff
f
The Product Rule xgxfxgxfxgxf
dxd ''
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Chapter 11: Differentiation11.4 The Product and Quotient Rule
Example 1 – Applying the Product Rule
Example 3 – Differentiating a Product of Three Factors
Find F’(x).
153412435432
543543'
543
22
22
2
xxxxxx
xdxdxxxxx
dxdxF
xxxxF
Find y’.
26183
432432'
)4)(3)(2(
2
xx
xdxdxxxxx
dxdy
xxxy
2007 Pearson Education Asia
Chapter 11: Differentiation11.4 The Product and Quotient Rule
Example 5 – Applying the Quotient RuleIf , find F’(x).
Solution:
The Quotient Rule
2
''xg
xgxfxfxgxgxf
dxd
1234 2
x
xxF
22
2
2
22
1232122
12234812
12
12343412'
xxx
xxxx
x
xdxdxx
dxdx
xF
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Chapter 11: Differentiation11.4 The Product and Quotient Rule
Example 7 – Differentiating Quotients without Using the Quotient Rule
Differentiate the following functions.
5
6352'
52 a.
22
3
xxxf
xxf
44
33
7123
74'
74
74 b.
xxxf
xx
xf
455
41'
3541
435 c.
2
xf
xx
xxxf
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Chapter 11: Differentiation11.4 The Product and Quotient Rule
Example 9 – Finding Marginal Propensities to Consume and to Save
If the consumption function is given by determine the marginal propensity to consume and the marginal propensity to save when I = 100.
Solution:
Consumption Function
dIdC consume to propensity Marginal
consume to propensity Marginal - 1 save to propensity Marginal
10
325 3
I
IC
2
32/1
2
32/3
1013233105
10
103232105
IIII
I
IdIdII
dIdI
dIdC
536.01210012975
100
IdIdC
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Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule11.5 The Chain Rule and the Power Rule
Example 1 – Using the Chain Rulea. If y = 2u2 − 3u − 2 and u = x2 + 4, find dy/dx.
Solution:
Chain Rule:
Power Rule:
dxdu
dudy
dxdy
dxdunuu
dxd nn 1
xu
xdxduu
dud
dxdu
dudy
dxdy
234
4232 22
xxxxxxdxdy 26821342344 322
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Chapter 11: Differentiation11.5 The Chain Rule and the Power RuleExample 1 – Using the Chain Rule
Example 3 – Using the Power Rule
b. If y = √w and w = 7 − t3, find dy/dt.
Solution:
3
22
32/1
72
323
7
tt
wt
tdtdw
dwd
dtdy
If y = (x3 − 1)7, find y’.Solution:
622263
3173
121317
117'
xxxx
xdxdxy
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Chapter 11: Differentiation11.5 The Chain Rule and the Power Rule
Example 5 – Using the Power Rule
Example 7 – Differentiating a Product of Powers
If , find dy/dx.
Solution:
21
2
xy
22
2112
2
2221
xxx
dxdx
dxdy
If , find y’.
Solution:
452 534 xxy
2425215342
4531053412
453534'
2342
424352
524452
xxxx
xxxxx
xdxdxx
dxdxy