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Matematika Bisnis
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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
2007 Pearson Education Asia
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 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
2007 Pearson Education Asia
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)
2007 Pearson Education Asia
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
h
afhaf
az
afzfm
haz
0
tan limlim
2007 Pearson Education Asia
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 = 2
11lim
11lim
22
00
h
h
h
fhfhh
• The derivative of a function f is the function denoted f’ and defined by
h
xfhxf
xz
xfzfxf
hxz
0
limlim'
2007 Pearson Education Asia
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
24
322322limlim'
22
00
x
h
xxhxhx
h
xfhxfxf
hh
16
167
xy
xy
62141' f
2007 Pearson Education Asia
Chapter 11: Differentiation11.1 The Derivative
Example 5 – A Function with a Vertical Tangent Line
Example 7 – Continuity and Differentiability
Find .
Solution:
xdx
d
xh
xhxx
dx
dh 2
1lim
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.
2007 Pearson Education Asia
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
0cdx
d
1 nn nxxdx
d
xcfxcfdx
d'
xgxfxgxfdx
d''
2007 Pearson Education Asia
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 1 – Derivatives of Constant Functions
a.
b. If , then .
c. If , then .
03 dx
d
5xg
4.807623,938,1ts
0' xg
0dtds
2007 Pearson Education Asia
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 3 – Rewriting Functions in the Form xn
Differentiate the following functions:
Solution:
a.
b.
xy
xx
dx
dy
2
1
2
1 12/1
xx
xh1
2/512/32/3
2
3
2
3' xxx
dx
dxh
2007 Pearson Education Asia
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 5 – Differentiating Sums and Differences of Functions
Differentiate the following functions: xxxF 53 a.
x
xxx
xdx
dx
dx
dxF
2
115
2
153
3'
42/14
2/15
3/1
4 5
4 b.
z
zzf
3/433/43
3/1
4
3
5
3
154
4
1
5
4'
zzzz
zdz
dz
dz
dzf
2007 Pearson Education Asia
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 5 – Differentiating Sums and Differences of Functions
8726 c. 23 xxxy
7418
)8()(7)(2)(6
2
23
xx
dx
dx
dx
dx
dx
dx
dx
d
dx
dy
2007 Pearson Education Asia
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 7 – Finding an Equation of a Tangent Line
Find an equation of the tangent line to the curve when x = 1.
Solution: The slope equation is
When x = 1,
The equation is
x
xy
23 2
2
12
23
2323
xdx
dy
xxxx
xy
5123 2
1
xdx
dy
45
151
xy
xy
2007 Pearson Education Asia
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
tfttf
t
svave
t
tfttfv
t
0
lim
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.
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 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 3.60
1.0
30503.311
1.0
101.10
1.0
101.010
fffft
tfttf
t
svave
tdt
dsv 6
m/s6010610
tdt
ds
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 3 – Finding a Rate of Change
• If y = f(x),
then
xxxx
xfxxf
x
y
to from interval
the over x to respect with
y of change of rate average
xrespect toy with
of change of rate ousinstantanelim
0 x
y
dx
dyx
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 .34x
dx
dy
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 5 – Rate of Change of Volume
A 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,
22 433
4rr
dr
dV
ft
ft1624
32
2
rdr
dV
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 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 .
dq
dc
dq
dr
Relative and Percentage Rates of Change
• The relative rate of change of f(x) is .
• The percentage rate of change of f (x) is
xf
xf '
%100'
xf
xf
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 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 qqdq
dc
75.355004.0500003.0 2
50
q
dq
dc
2007 Pearson Education Asia
Chapter 11: Differentiation
11.3 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
75
25
5
5' change %
255565'
f
f
f
The Product Rule
xgxfxgxfxgxfdx
d''
2007 Pearson Education Asia
Chapter 11: Differentiation
11.4 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
xdx
dxxxxx
dx
dxF
xxxxF
Find y’.
26183
432432'
)4)(3)(2(
2
xx
xdx
dxxxxx
dx
dy
xxxy
2007 Pearson Education Asia
Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 5 – Applying the Quotient Rule
If , find F’(x).
Solution:
The Quotient Rule
2
''
xg
xgxfxfxg
xg
xf
dx
d
12
34 2
x
xxF
22
2
2
22
12
32122
12
234812
12
12343412'
x
xx
x
xxx
x
xdxd
xxdxd
xxF
2007 Pearson Education Asia
Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 7 – Differentiating Quotients without Using the Quotient Rule
Differentiate the following functions.
5
63
5
2'
5
2 a.
22
3
xxxf
xxf
4
4
33
7
123
7
4'
7
4
7
4 b.
xxxf
xx
xf
4
55
4
1'
354
1
4
35 c.
2
xf
xx
xxxf
2007 Pearson Education Asia
Chapter 11: Differentiation
11.4 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
dI
dC consume to propensity Marginal
consume to propensity Marginal - 1 save to propensity Marginal
10
325 3
I
IC
2
32/1
2
32/3
10
13233105
10
103232105
I
III
I
IdId
IIdId
I
dI
dC
536.012100
12975
100
IdI
dC
2007 Pearson Education Asia
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 Rule
a. If y = 2u2 − 3u − 2 and u = x2 + 4, find dy/dx.
Solution:
Chain Rule:
Power Rule:
dx
du
du
dy
dx
dy
dx
dunuu
dx
d nn 1
xu
xdx
duu
du
d
dx
du
du
dy
dx
dy
234
4232 22
xxxxxxdx
dy26821342344 322
2007 Pearson Education Asia
Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule
Example 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
3
2
3
7
t
t
w
t
tdt
dw
dw
d
dt
dy
If y = (x3 − 1)7, find y’.
Solution: 622263
3173
121317
117'
xxxx
xdx
dxy
2007 Pearson Education Asia
Chapter 11: Differentiation
11.5 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:
2
12
x
y
22
2112
2
2221
x
xx
dx
dx
dx
dy
If , find y’.
Solution:
452 534 xxy
2425215342
4531053412
453534'
2342
424352
524452
xxxx
xxxxx
xdx
dxx
dx
dxy