MAT 1235
Calculus II
Section 6.4*
General Log. and
Exponential Functions
http://myhome.spu.edu/lauw
Homework and …
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WebAssign HW 6.4*
(7 problems, 30 min.)
Quiz 6.3*, 6.4* (Beginning of the class)
Preview
ln x
log a x
 Properties
 Derivateiv es
 Antideriva tives
e
x
a
x
The Difference….
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Our construction allows us to find the
x
derivatives of ln x (and e ).
Compare to the elementary approach
(6.2 -6.4), one cannot prove the
x
derivative of e .
Recall
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Under our construction, “functions” such
as
2

3
is undefined at this point.
We would like to define functions such
x
as
2
so that it is defined for even irrational
numbers
Preview
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Define general exp. and log. function
We are going to extend the property of
ln(𝑥)
ln(a r )  r ln(a)
to all real numbers 𝑟
(2016) We may skip some of the stories
to save time.
6.3* 𝑎 > 0 Property of Inverse
Function
ae
ln a
a  (e ) , r  rational no.
r
ln a r
e
r ln a
So,…
ae
ln a
a  (e ) , r  rational no.
r
a e
r
ln a r
r ln a
It makes sense to define…
For all x,
a e
x
x ln a
Exp. Function with Base 𝑎 > 0
Extended Property of ln(𝑥)
For all x,
a e
x
x ln a
   ln  e 
ln  a   x ln a
ln a
x
x
x ln a
Extended Property of ln(𝑥)
For all x,
a e
x
x ln a
   ln  e 
ln  a   x ln a
ln a
x
x
x ln a
Law of Exponents
If x and y are real no. and a, b  0, then
1. a
x y
 a a
x y
 a /a
2. a
x
x
3.
a 
4.
 ab 
x
y
y
x
a
y
xy
a b
x
x
Law of Exponents
If x and y are real no. and a, b  0, then
1. a
x y
 a a
x y
 a /a
2. a
x
x
3.
a 
4.
 ab 
x
y
y
x
a
y
a
x y
e
 x  y  ln a

xy
a b
x
x
a e
x
x ln a
Derivatives and Antiderivatives
d x
x
a

a
ln a


dx
d u
du
u
a   a ln a

dx
dx
x
a
x
a
 dx  ln a  C
Derivatives and Antiderivatives
a x  e x ln a
d x
x
a

a
ln a


d x
d x ln a
dx
a   e 

dx
dx
d u
du
u
a

a
ln a


dx
dx

x
a
x
 a dx  ln a  C
a e
x
x ln a
Example 1
Let h( x)  x  4 . Find h( x)
4
x
Example 1
Let h( x)  x  4 . Find h( x)
4
Power
Function
x
exp . function
 
d x
a  a x ln a
dx
Example 2
Let y  5
. Find y.
sin x
 
d u
du
u
a  a ln a
dx
dx
Example 3
Let y  x
cos x
. Find y.
Example 3
Let y  x
cos x
. Find y.
Not
power function
or exp. function
Definition:
Log. Function with Base a
For a  0, a  1, log a x is defined as
the inverse function of a
x
Derivatives
d
1
 log a x  
dx
x ln a
d
1 du

 log a u  
dx
u ln a dx
Example 4
Let f ( x)  log 5 ( x 2  1). Find f ( x)
d
1 du

 log a u  
dx
u ln a dx
Maple Lab 01 Next Monday
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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
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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
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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
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Follow the handout to practice typing
formula.