INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia 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


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


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


• 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


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 OutlineChapter Outline

11.4)

11.5)



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


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'



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



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.



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''



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




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




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




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




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



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.



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




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




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




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




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




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''



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




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




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




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



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



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



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

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INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 11 Differentiation