MAT 1235Calculus II

Section 6.4*

General Log. and Exponential Functions

http://myhome.spu.edu/lauw

Homework and …

WebAssign HW 6.4*(7 problems, 30 min.)

Quiz 6.3*, 6.4* (Beginning of the class)

Preview

xln

xe

xalog

tivesAntideriva

esDerivateiv

Properties

xa

The Difference….

Our construction allows us to find the derivatives of (and ).

Compare to the elementary approach (6.2 -6.4), one cannot prove the derivative of .

ln x xe

xe

Recall

Under our construction, “functions” such as

is undefined at this point. We would like to define functions such

as

so that it is defined for even irrational numbers

32

2x

Preview

Define general exp. and log. function We are going to extend the property of

to all real numbers ln( ) ln( )ra r a

6.3* Property of Inverse Function

ln

ln

ln

( ) , rational no.

a

r a r

r a

a e

a e r

e

So,…

ln

ln

ln

( ) , rational no.

a

r a r

r r a

a e

a e r

a e

It makes sense to define…

ln

For all ,

x x a

x

a e

Exp. Function with Base

Extended Property of

ln

ln

For all ,

ln ln

ln ln

x x a

x x a

x

x

a e

a e

a x a

Extended Property of

ln

ln

For all ,

ln ln

ln ln

x x a

x x a

x

x

a e

a e

a x a

Law of Exponents

If and are real no. and , 0, then

1.

2. /

3.

4.

x y x y

x y x y

yx xy

x x x

x y a b

a a a

a a a

a a

ab a b

Law of Exponents

If and are real no. and , 0, then

1.

2. /

3.

4.

x y x y

x y x y

yx xy

x x x

x y a b

a a a

a a a

a a

ab a b

axx ea ln

lnx y ax ya e

Derivatives and Antiderivatives

Ca

adxa

dx

duaaa

dx

d

aaadx

d

xx

uu

xx

ln

ln

ln

Derivatives and Antiderivatives

Ca

adxa

dx

duaaa

dx

d

aaadx

d

xx

uu

xx

ln

ln

ln

ln

ln

x x a

x x a

a e

d da e

dx dx

axx ea ln

Example 1

)( Find .4)(Let 4 xhxxh x

Example 1

)( Find .4)(Let 4 xhxxh x

Function

Powerfunction .exp

lnx xda a a

dx

Example 2

. Find .5Let sin yy x

lnu ud dua a a

dx dx

Example 3

. Find Let cos y.xy x

Example 3

. Find Let cos y.xy x

function exp.or

function power

Not

Definition: Log. Function with Base a

x

a

a

xaa

offunction inverse the

as defined is log ,1,0For

Derivatives

dx

du

auu

dx

dax

xdx

d

a

a

ln

1log

ln

1log

Example 4

)( Find ).1(log)(Let 25 xfxxf

1log

lna

d duu

dx u a dx

Maple Lab 01 Next Monday

You are supposed to know Maple at the level of calculus I. Review tutorials.

I will take points off from you if both you and your partner do not know how to use Maple.

All “new” students should partner with someone who was in 1234 last quarter.

Maple and Equation Builder Tool

2 persons per lab report. All computations are done on Maple. All lab reports need to be typed. All Formulas are entered using “Equation

Builder Tool” in Word (Window based PC).

It is kind of similar to WebAssign.

Maple

If you are new to Maple, you can learn the basics by doing the Maple tutorials.

You can find them in my Web Pages. Maple is a very powerful tool. You will

use it in other classes in your major.

Equation Builder Tool

Follow the handout to practice typing formula.