Download LESSON 8.1 – E Theorem 8.1: Derivative of Natural Exponential

NOTES 8: EXPONENTIAL AND
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LOGARITHMIC FUNCTIONS AND CALCULUS
Date:________________Period:_________
Mrs. Nguyen’s Initial:_________________
LESSON 8.1 – EXPONENTIAL FUNCTIONS: DIFFERENTIATION AND INTEGRATION
Theorem 8.1:
Derivative of
Natural
Exponential
Function
Let u be a differential function of x.
1.
d x
e   e x
dx
2.
d u
e   eu u '
dx
Examples:
Practice Problems: Find f '( x) for the following functions.
1.
3.
f ( x)  e1 x
f ( x) 
e
x
2
2.
f ( x )  e3
4.
f ( x)  1  e  x 
x
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 1
x
2
Review:
Guidelines
for finding
relative
extrema
using the
first
derivative
test
1. Find the CN: a, b, c
2. Set up the intervals using the CN
3. Choose a test value in each interval
4. Find f '(TV ) in each interval
(, a ) (a, b)
(b, c)
(c,  )
Review:
Guidelines
to find
Points of
Inflection
and
Concavity
1.
2.
3.
4.
f '(TV )  0
f '(TV )  0
Inc
Dec
RMax  a, f (a) 
f '(TV )  0
f '(TV )  0
Dec
Inc
RMin  c, f (c) 
Find the 1st derivative
Find the 2nd derivative
Set f ''( x)  0
Solve for x to obtain POSSIBLE POINTS OF INFLECTION:
 a, f (a)  ,  b, f (b)  ,  c, f (c) 
5. Set up the intervals using the PP of I
(, a )
( a, b)
f ''(TV )  0
f ''(TV )  0
Upward
Downward
P of I at  a, f (a) 
(b, c)
(c,  )
f ''(TV )  0
f ''(TV )  0
Downward
Upward
P of I  c, f (c) 
Practice Problem 5: Find the extrema and the points of inflection for f ( x)  xe x .
Step 1:
f '( x) 
Step 2:
Critical numbers
Step 3:
Set up intervals using CN
Intervals

,
x


,

x
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 2

,
x


,
x

Test Values
f '(TV )
Conclusions
Identify
Relative
Extrema
Step 4:
f ''( x) 
Step 5:
Possible points of inflection
Step 6:
Set up intervals using PP of I
Intervals

,
x


,

x
Test Values
f ''(TV )
Conclusions
Identify Points
of Inflection
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 3

,
x


,
x

Practice Problem 6: Find the extrema and the points of inflection for
e x  e x
.
f ( x) 
2
f '( x) 
Step 1:
Step 2:
Critical numbers
Step 3:
Set up intervals using CN

Intervals
,
x


,
x


,
x




,
x

Test Values
f '(TV )
Conclusions
Identify
Relative
Extrema
Step 4:
f ''( x) 
Step 5:
Possible points of inflection
Step 6:
Set up intervals using PP of I
Intervals

,
x


,
x

Test Values
f ''(TV )
Conclusions
Identify Points
of Inflection
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 4

,
x
,
x

Theorem 8.2:
Integration
Rules for
Exponential
Functions
Review
Problems:
Evaluate the
following
Let u be a differential function
of x.
1.
 e
2.
 e
1.
2
 x 4  9 x dx 
x
dx  e x  C
u
du  eu  C
Examples:
2.
  x  1
2
1
2  xdx 
Practice Problems: Integrate the following functions and state which integration
rule is being applied.
7.

1
0
e 2 x dx 
8.
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 5

1  e x  dx 
2
9.
11.


e2 x
1  e 
2x 2
dx 
e 2 x  2e x  1
dx 
x
e
10.
12.
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 6
1
ex
0
1 e


x
dx 
e x  e x
e e
x
x
dx 
LESSON 8.2 – LOGARITHMIC FUNCTIONS AND DIFFERENTIATION
Theorem 8.3:
Derivative of
Natural
Logarithmic
Function
Let u be a differential function of x.
1.
d
1
ln x   , x  0
dx
x
2.
d
1 du u '
ln u     , u  0
dx
u dx u
Examples: Find f '( x)
given f ( x)  ln  2 x 
Practice Problems: Find f '( x) for the following functions.
1.
f ( x)  ln  x 2  1
2.
f ( x)  x ln x
3.
f ( x)   ln x 
4.
f ( x)  ln x  1
3
1/ 4
 2  x2  4 
Practice Problem 4: Find f '( x) given f ( x)  ln 



x


Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 7
Practice Problem 5: Locate the relative extrema of y  ln  x 2  2 x  3
Step 1:
f '( x) 
Step 2:
Critical numbers
Step 3:
Set up intervals using CN
Intervals

,
x


,
x


,
x


,
x
Test Values
f '(TV )
Conclusions
Identify
Relative
Extrema
Logarithmic
To differentiate the function y  u , use the following steps:
Differentiation
1. Take ln of each side: ln y  ln u
2. Expand ln u completely
3. Differentiate implicitly:
4. Solve fore y '  y 
y' d
  ln u 
y dx
d
ln u 
dx
5. Substitute for y and simplify: y '  u 
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 8
d
ln u 
dx

Example: Find
f '( x) given
Method 1: Using Product
Rule
f ( x)  x x 2  1
Method 2: Using Logarithmic
Differentiation
1.
2.
3.
4.
5.
Practice Problem 6: Find f '( x) using Logarithmic Differentiation given
y
x  x  1
3/ 2
x 1
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 9
Theorem 8.4:
If u is a differential function of x
Derivative
such that u  0 , then
1 du u '
involving Absolute d
ln u     .
Value
dx
u dx u
Example:
Practice Problems: Find f '( x) for the following functions.
1.
f ( x)  ln x 2  1
2.
f ( x)  ln
x5
x
Theorem 8.5: Let a be a positive real number  a  1 and let u be a differential
Derivatives for function of x.
Bases other
d x
d
1
 a    ln a a x
1.
3.
log a x  
than “e”
dx
dx
 ln a  x
d u
d
u'
 a    ln a a u u '
2.
4.
log
u



a
dx
dx
 ln a u
c. y  log 2  x 2  1
Examples:
Find f '( x)
a. y  2 x
b. y  23 x
Theorem 8.6:
The Power
Rule for Real
Exponents
Examples:
Find f '( x)
Let n be any real number and let u be a differential function of x.
d n
d n
 x   nx n1
u   nu n1 u '
1.
2.
dx
dx
a. y  ee
b. y  e x
c. y  x e
d. y  x x
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 10
LESSON 8.3 – LOGARITHMIC FUNCTIONS AND INTEGRATION
Review
Concepts
1.
d
1
ln x  
dx
x
2.
d
u'
ln u  
dx
u
Let u be a differential function of x.
Theorem 8.7:
Log Rule for
Integration
1.
1
 x dx  ln x  C
2.
1
 u du  ln u  C
Practice Problems: Evaluate and state which rule is being applied.
1.
3.
5.
5
 x dx 

4
0

x2
dx
3
3 x
x
9 x
2
dx 
2.
1
 3x  2 dx 
4.
3x 2  1
 x3  x dx 
6.
x2  4
 x dx 
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 11
7.
2 x2  7 x  3
 x  2 dx 
8.
9.
1
 x ln x dx 
10.
11.
x
3
 dx 
  x  1

2
dx 
1
dx 
x 1
1
x 1
12.
Summary of
Power Rule
all the Rules
for Integration
u n1
n
 u du  n  1  C
Examples:
Integrate the
following
2x

x
9 x
2
dx 
e
 ( x  1)2 dx
Logarithmic Rule
Exponential Rule
1
 u du  ln u  C
 e
x
 9  x 2 dx 
3x
 xe dx 
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 12
u
du  eu  C
2
LESSON 8.4 – DIFFERENTIAL EQUATIONS: GROWTH AND DECAY
2x
Solve the
Example: Given y ' 
, find the
differential
y
equation using equation.
the strategy
2x
y' 
called
y
“Separation of
Variables”
Practice Problem 1: xy  y '  100 x
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 13
Remember: y ' 
dy
dx
Practice Problem 2: The rate of change of N is proportional to N. When t  0 ,
N  250 and when t  1, N  400 . What is the value of N when t  4 ?
Use: y  aebt
Practice Problem 3: Find the exponential function y  aebt that passes through
1, 1 and  5, 5
Practice Problem 4: Suppose that 10 grams of the plutonium isotope Pu-239 was
released in the Chernobyl nuclear accident. How long will it take for the 10 grams
to decay to 1 gram? The half-life of Pu-239 is 24,360 years.
Mrs. Nguyen – Honors PreCalculus – Chapter 8 Notes – Page 14