Introductory maths analysis chapter 12 official

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 12 Chapter 12 Additional Differentiation TopicsAdditional Differentiation Topics


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 develop a differentiation formula for y = ln u.

• To develop a differentiation formula for y = eu.

• To give a mathematical analysis of the economic concept of elasticity.

• To discuss the notion of a function defined implicitly.

• To show how to differentiate a function of the form uv.

• To approximate real roots of an equation by using calculus.

• To find higher-order derivatives both directly and implicitly.

Chapter 12: Additional Differentiation Topics

Chapter ObjectivesChapter Objectives


Derivatives of Logarithmic Functions

Derivatives of Exponential Functions

Elasticity of Demand

Implicit Differentiation

Logarithmic Differentiation

Newton’s Method

Higher-Order Derivatives

12.1)

12.2)

12.3)


Chapter OutlineChapter Outline

12.4)

12.5)

12.6)

12.7)



12.1 Derivatives of Logarithmic Functions12.1 Derivatives of Logarithmic Functions• The derivatives of log functions are:

hx

h x

h

xx

dx

d/

01limln

1ln a.

0 where1

ln b. xx

xdx

d

0 for 1

ln c. udx

du

uu

dx

d



12.1 Derivatives of Logarithmic Functions

Example 1 – Differentiating Functions Involving ln x

b. Differentiate .

Solution:2

ln

x

xy

0 for ln21

2)(ln1

lnln'

3

4

2

22

22

xx

xx

xxx

x

x

xdxd

xxdxd

xy

a. Differentiate f(x) = 5 ln x.

Solution: 0 for 5

ln5' xx

xdx

dxf




Example 3 – Rewriting Logarithmic Functions before Differentiating

a. Find dy/dx if .

Solution:

b. Find f’(p) if .

Solution:

352ln xy

2/5 for 52

62

52

13

x

xxdx

dy

3

4

2

3

1

2

13

141

2

131

1

12'

ppp

ppppf

432 321ln ppppf




Example 5 – Differentiating a Logarithmic Function to the Base 2

Differentiate y = log2x.

Solution:

Procedure to Differentiate logbu

• Convert logbu to and then differentiate.b

u

ln

ln

xx

dx

dx

dx

dy

2ln

1

2ln

lnlog2



12.2 Derivatives of Exponential Functions12.2 Derivatives of Exponential Functions• The derivatives of exponential functions are:

dx

duee

dx

d uu a.

xx eedx

d b.

dx

dubbb

dx

d uu ln c.

0' for '

1 d. 1

11

xff

xffxf

dx

d

dydxdx

dy 1 e.



12.2 Derivatives of Exponential Functions

Example 1 – Differentiating Functions Involving ex

a.Find .

Solution:

b. If y = , find .

Solution:

c. Find y’ when .

Solution:

xe

x

x

xx

e

xe

dx

dxx

dx

de

dx

dy 1

3ln2 xeeyxx eey 00'

xedx

d3

xxx eedx

de

dx

d333

dx

dy




Example 3 – The Normal-Distribution Density FunctionDetermine the rate of change of y with respect to x when x = μ + σ.

221 /

2

1

xe

xxfy

Solution: The rate of change is

e

edx

dy x

x

2

1

12

2

1

2

1

2

/ 221




Example 5 – Differentiating Different Forms

Example 7 – Differentiating Power Functions Again

Find .

Solution:

xexedx

d22

xex

xeexxe

dx

d

xe

xexe

2

2ln2

2

12ln2

1

2ln12

Prove d/dx(xa) = axa−1.

Solution: 11ln aaxaa axaxxedx

dx

dx

d



12.3 Elasticity of Demand12.3 Elasticity of Demand

Example 1 – Finding Point Elasticity of Demand

• Point elasticity of demand η is

where p is price and q is quantity.

dqdp

qp

q

Determine the point elasticity of the demand equation

Solution: We have

0 and 0 where qkq

kp

12

2

q

k

qk

dqdp

qp



12.4 Implicit Differentiation12.4 Implicit DifferentiationImplicit Differentiation Procedure

1. Differentiate both sides.

2. Collect all dy/dx terms on one side and other terms on the other side.

3. Factor dy/dx terms.

4. Solve for dy/dx.



12.4 Implicit Differentiation

Example 1 – Implicit Differentiation

Find dy/dx by implicit differentiation if .

Solution:

73 xyy

2

2

3

31

1

013

7

ydx

dydx

dyy

dx

dydx

dxyy

dx

d



12.4 Implicit Differentiation

Example 3 – Implicit Differentiation

Find the slope of the curve at (1,2).

Solution:

223 xyx

2

7

2

443

223

2,1

2

32

22

223

dx

dy

xy

xxyx

dx

dy

xdx

dyxy

dx

dyx

xydx

dx

dx

d



12.5 Logarithmic Differentiation12.5 Logarithmic DifferentiationLogarithmic Differentiation Procedure

1. Take the natural logarithm of both sides which gives .

2. Simplify In (f(x))by using properties of logarithms.

3. Differentiate both sides with respect to x.

4. Solve for dy/dx.

5. Express the answer in terms of x only.

xfy lnln



12.5 Logarithmic Differentiation

Example 1 – Logarithmic Differentiation

Find y’ if .

Solution:

4 22

3

1

52

xx

xy

xx

xx

xxxy

xx

xy

21

1

4

1ln252ln3

1ln52lnln

1

52lnln

2

4 223

4 22

3




Example 1 – Logarithmic Differentiation

)1(

2

52

6

1

)52('

)1(2

2

52

6

)2)(1

1(

4

1)

1(2)2)(

52

1(3

'

24 22

3

2

2

xx

x

xxxx

xy

x

x

xx

xxxxy

y

Solution (continued):




Example 3 – Relative Rate of Change of a Product

Show that the relative rate of change of a product is the sum of the relative rates of change of its factors. Use this result to express the percentage rate of change in revenue in terms of the percentage rate of change in price.

Solution: Rate of change of a function r is

%100'

1%100'

%100'

%100'

%100'

'''

p

p

r

r

q

q

p

p

r

r

q

q

p

p

r

r



12.6 Newton’s Method12.6 Newton’s Method

Example 1 – Approximating a Root by Newton’s Method

Newton’s method: ,...3,2,1

'1 nxf

xfxx

n

nnn

Approximate the root of x4 − 4x + 1 = 0 that lies between 0 and 1. Continue the approximation procedure until two successive approximations differ by less than 0.0001.



12.6 Newton’s Method

Example 1 – Approximating a Root by Newton’s Method

Solution: Letting , we have

Since f (0) is closer to 0, we choose 0 to be our first x1.

Thus, 44

13

' 3

4

1

n

n

n

nnn x

x

xf

xfxx

25099.0 ,3 When

25099.0 ,2 When

25.0 ,1 When

0 ,0 When

4

3

2

1

xn

xn

xn

xn

144 xxxf

21411

11000

f

f

44'

143

4

nn

nnn

xxf

xxxf



12.7 Higher-Order Derivatives12.7 Higher-Order DerivativesFor higher-order derivatives:



12.7 Higher-Order Derivatives

Example 1 – Finding Higher-Order Derivatives

a. If , find all higher-order derivatives.

Solution:

b. If f(x) = 7, find f(x).

Solution:

26126 23 xxxxf

0

36'''

2436''

62418'

4

2

xf

xf

xxf

xxxf

0''

0'

xf

xf




Example 3 – Evaluating a Second-Order Derivative

Example 5 – Higher-Order Implicit Differentiation

Solution:

.4 when find ,4

16 If

2

2

xdx

yd

xxf

3

2

2

2

432

416

xdx

yd

xdx

dy

16

1

4

2

2

x

dx

yd

Solution:

y

x

dx

dydx

dyyx

4

082

.44 if Find 222

2

yxdx

yd




Example 5 – Higher-Order Implicit Differentiation

Solution (continued):

32

2

3

22

2

2

4

1

16

4

get to 4

ateDifferenti

ydx

yd

y

xy

dx

yd

y

x

dx

dy

Introductory maths analysis chapter 12 official

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