5-1 The Natural Logarithmic Function and Differentiation
5-3 Inverse Functions
p. 322: 45-59 odd,63, 65, 69, 71, 73, 77, 87, 89
p. 340: 71, 73
I. Properties of Logarithmic Functions
If a and b are positive numbers and n is rational, then
A. ln 1 
B. ln( ab) 
C. ln( a n ) 
a
D. ln   
b
E. domain =
Range =
II. Derivative of the Natural Logarithmic Function
d
ln x  
A.
dx
d
ln u  
B.
dx
d
ln u  
C.
dx
III. Definition of the Natural Logarithmic Function.
x1
ln x   dt , x  0
1 t
IV. Definition of e.
e1
ln e   dt
1 t
V. The Derivative of an Inverse Function.
Let f be a function that is differentiable on an interval I. If g ( x)  f

f 1 ( x)  g x  
1
( x) , then:
 
Find the derivative of each function.


1. y  ln x 2  3
2. h( x)  ln 4  3x 2
3. y  ln ln 5x
4.
 x2  4 
2 
 6 x  5 
g  x   ln 
5. y  ln(sin 3 x)
6. Find an equation of the tangent line to the graph of y  x 2  ln 4 x  7 at the point (2,4).
7. Find y  by using implicit differentiation.
x
y 2  ln    4 x  3
 y
dy
by using logarithmic differentiation.
dx
2
3
y   x  1  x  2
8. Find
  a
9. Find f 1
for the function f and real number a.
f x   2  x  x 3 , a  8
5-2 The Natural Logarithmic Function and Integration.
p. 330: 1-13 odd, 19-23 odd, 27-43 odd,
61-64 all, 67, 69, 77, 79, 81
I. Logarithmic Rule for Integration
Let u be a differentiable function of x.
1
A.  dx 
x
1
B.  du 
u
II. Integrals of the Six Basic Trig Functions
A.  sin udu 
B.
C.
D.
E.
F.
 cos udu 
 tan udu 
 cot udu 
 sec udu 
 csc udu 
Find each indefinite integral.
3x 2  1
dx
1.  3
2x  2x  5
1
dx
4. 
sin 3 x
2.
1
 x ln x dx
sin 2 x
dx
5. 
cos x
3.

x  32 dx
x
x 2  5x  6
dx
6. 
x 1
7. Solve the differential equation.
dy 5
; 2,0
dx 5  x
8. Find the area of the region bounded by the graphs of each equation.

y  2 tan x , y  0 , x  0 , x 
4
9. Find
10. Find the average value of the function over the given interval.
;
5-4 Exponential Functions: Differentiation and Integration
p. 347: 37-61 odd, 78b, 87-111 odd, 113b, 115
I. The Natural Exponential Function
A. y  e x is the inverse of y  ln x
B. Domain of y  e x is  , , Range is 0, 
C. It is continuous, increasing, concave up on its entire domain.
D. lim e x 
and lim e x 
x 
x 
II. The derivative of the Natural Exponential Function
d x
e  
A.
dx
d u
e  
B.
dx
III. Integration Rules of Exponential Functions
A.  e x dx 
B.
e
u
du 
1. Find an equation of the tangent line to the graph of y  xe
Find the derivative of each.
2. f x   e 5 x ln x

3. y  ln sin e x
1
x3
 ln 2  x 2 at the point 1, e

4. e xy  x 2  y 2  0
Evaluate each integral
5.
6
3  ex
dx
e3x
7.
cot e 2 x
 e 2 x dx
5-5 Bases Other than e
p. 357: 41-67 odd
I. Definition of Exponential Function to Base a
If a is a positive real number, then a x  e x ln a
II. Properties of Exponential and Logarithmic Functions to Base a
A. a 0 
B. a x a y 
ax
C. y 
a
 
y
D. a x 
E. a loga x 
F. log a a x 
G. log a x 
III. Derivatives for Bases Other than e
d x
a  
A.
dx
d u
a  
B.
dx
d
log a x  
C.
dx
d
log a u  
D.
dx
IV. Integration for Bases Other than e
A.
B.
Find the derivative of each function.
1. f x   5 x
2. g x   12 3 x
3
2
3. y  log 3
2x  5
3x  8
5. y  2 x  1
4. y  x 3 3 x
Evaluate each indefinite integral.
6.
x
 2 dx
7.
 
2
x
 4x 5 dx
3
x
8.
7x
 7 x  4 dx
5-6 Differential Equations: Growth and Decay
p. 366: 1-9 odd, 15, 17, 25
5-6 Worksheet: 1, 4, 7, 10
I. Separable Differential Equations
dy
 ky
dt
II. Exponential Growth and Decay
If y is a differentiable function of t such that y > 0 and y   ky for some constant k, then
y  Ce kt
y=
C=
t=
k=
III. Newton’s Law of Cooling
d
k  y  c
dt
y  e kt e b  c
y=
C=
k =
Solve the differential equation.
12 x
1. y  
y


2. 5  x 3 y   3x 2 y  0
3. A bacteria culture starts with 500 bacteria and after 3 hours there are 8000 bacteria. Find an expression for
the number of bacteria after t hours. Find the number of bacteria after 4 hours. When will the population
reach 30,000?
4. Polonium-210 has a half life of 140 days. If a sample has a mass of 200mg, find a formula for the mass that
remains after t days. Find the mass after 100 days.
5. An object takes 30 minutes to cool from 125 F to 100 F when placed in a room kept 75 F. What will be
its temperature at the end of 1 hour of cooling? When will its temperature be 80 F?
5-8 Inverse Trigonometric Functions and Differentiation
p. 386: 41-51 odd, 56
Differentiation Review Worksheet
I. Inverse Trigonometric Functions
-

 y
2
2
1
B. y  cos x iff cos y  x for  1  x  1 and 0  y  
-

C. y  tan 1 x iff tan y  x for    x   and
 y
2
2
1
D. y  cot x iff cot y  x for    x   and 0  y  
A. y  sin 1 x iff sin y  x for  1  x  1 and
E. y  sec 1 x iff sec y  x for x  1 and 0  y   , y 
F. y  csc 1 x iff csc y  x for x  1 and

2
-

 y y0
2
2
II. Derivatives of Inverse Trigonometric Functions
Let u be a differentiable function of x.
d
d
arcsin u  
arccos u  
A.
D.
dx
dx
d
d
arctan u  
arc cot u  
B.
E.
dx
dx
d
d
arc sec u  
arc csc u  
C.
F.
dx
dx
Find the derivative of the function.
1.
4.
2.
3.
5.
6. Evaluate the limit as a derivative of a function at the indicated value.
2
2
32  h   32
lim
h 0
h
5-9 Inverse Trigonometric Functions and Integration
p. 393: 1-27 odd, 31-41 odd, 47, 49,
51b, 55
I. Integrals of Inverse Trigonometric Functions.
Let u be a differentiable function of x and let a>0.
du
1. 

2
a  u2
du
2.  2

a  u2
du
3. 

u u2  a2
II. Integration Tips for Quotients
A. Make the integrand conform to the power rule
B. Let the entire denominator represent the u for
1
 u du  ln u  c
C. Divide improper quotients
D. Apply Inverse Trig Integration rules.
Evaluate the integral.
1
dx
1. 
4  25 x 2
4.
.
x
 x 2  36 dx
2.
x
1
9x 2  1
5.

dx
ex
49  e 2 x
3.
dx
x5
 x 4  1dx
6.
x
2
7.
x
2
8.

dx
 4x  9
2x
dx
 8 x  20
1
 x 2  6x
dx