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.
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