5.4 Differentiation and Integration of “E” 2012 The Natural Exponential Function The function f(x) = ln x is increasing on its entire domain, and therefore.

5.4

Differentiation and Integration of “E”

2012

The Natural Exponential Function

The function f(x) = ln x is increasing on its entire domain, and therefore it has an inverse function f –1.

The domain of f –1 is the set of all reals, and the range is the set of positive reals, as shown in Figure 5.19.

Figure 5.19

So, for any real number x,

If x happens to be rational, then

Because the natural logarithmic function is one-to-one, you can conclude that f –1(x) and ex agree for rational values of x.


The following definition extends the meaning of ex to include all real values of x.

The inverse relationship between the natural logarithmic function and the natural exponential function can be summarized as follows.


5.4 Exponential Functions:Differentiation and Integration

Definition of the Natural Exponential Function

The inverse of ( ) ln is called the natural exponential

function and is denoted by

f x x

-1 ( ) .xf x eThat is,

ln .xy e y x Solve for x


ln

This inverse relationship

ln( ) and x xe x e x

1Solve 7 xe

1ln 7 ln( )xe ln 7 1x

ln 7 1x 0.946


Solve ln(2 3) 5x

5

5

5

2 3

2 3

13

2

e x

x e

x e

75.707x


THM 5.10 Operations with Exponential Functions

Let and be any real numbers.a b

1. a b a be e e

2. a

a bb

ee

e


Use the Laws of Logarithms to prove: a b a be e e

ln( ) ln ab bae e e

ln lna be e a b

ln( )a be

ln is one-to-onex

Laws of Logarithms

Inverse Functions

Inverse Functions


Properties of the Natural Exponential Function

1. The domain of ( ) is - , , and the range is 0, .xf x e

2. The function ( ) is continuous, increasing, and

one-to-one.

xf x e

3. The graph of ( ) is concave upward on its entire domain.xf x e

4. lim 0 and limx xe e x x


Let be a differentiable function of .u x

1. x xde

xe

d 2. u ud du

e edx dx

ln xe xProve: #1

ln xd de x

dx dx

11

xxde

dxe

x xede

dx

Chain Rule

Identity

Differentiate both sides'u

u


2 1 ?xde

dx

2 1 2xe 2 12 xe 1

3 /

3

3 /2

?

3

x

x

x

de

dxde

dx

ex

3/

2

3 xe

x

1 is a critical #1x xe x

Find the relative extrema of ( ) .xf x xe

'( ) (1) x xf x e ex

11, is a relative min (1 derivativ st. e te )e st

Interval 1x 1 x 1 x Test Value 2x 0x 2x Sign of '( )f x positivenegative positive

Conclusion IncreasingDecreasing Increasing


11, e


THM 5.12 Integration Rules for Exponential Functions

Let be a differential function of .u x

1. 2. x x u ue dx e C e du e C

2. ue du

5.4 Exponential Functions:Differentiation and Integration3 1 ?xe dx

3 1 3u x du dx 1

3ue du

3 11

3xe C

2

5 ?xxe dx 2 2u x du xdx 5

2ue du

25

2xe C

1

3ue C

5.4 Exponential Functions:Differentiation and Integration1/

2?

xedx

x

2

1 1 u du dxx x

ue du 1/ xe C

cossin ?xxe dx cos sinu x du xdx

ue du cos xe C

5.4 Exponential Functions:Differentiation and Integration1

0?xe dx

u x du dx 1 0

0 1

u ue du e du

0

1

ue

0 1e e 1

1e

0.632


0

1cos ?x xe e dx

x xu e du e dx

1

1sin

eu

1sin1 sin e 0.482

1

1cos ?

eu du


Page. 356

33,34, 35 – 51 odd, 57, 58, 65, 69, 85 – 91odd, 99-105 odd

5.4 Differentiation and Integration of “E” 2012 The Natural Exponential Function The function f(x) = ln x is increasing on its entire domain, and therefore.

Documents

x slide

integration slide

positive slide

chain rule slide

exponential functions

natural logarithmic

function fx

inverse function f