3.1 Extrema on an Interval
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Definition of Extrema
1. f(c) is the minimum of f on I if f(c) f(x) " x in I.
2. f(c) is the maximum of f on I if f(c) ! f(x) " x in I.
If f is continuous on a closed interval [
7 .2
V o lu m e :
T h e D is c M e t h o d
T h e a rea u n d er a cu rv e
is t h e s u m m a t io n o f a n
in f in it e n u m b e r o f
r e c t a n g le s .
I f w e t a k e t h is r e c t a n g le
a n d r e v o lv e it a b o u t a
lin e , w e h a v e a d
7.1
Area of a Region
Between Two Curves
f
g
Area of region
between f and g
f
=
=
g

Area of region
under f(x)
b
a
a
g
b
b
[ f ( x )  g( x )] dx = f ( x)dx
f

Area of region
under g(x)
g ( x)dx
a
Ex. Find the area of the region bounded by the graphs
o
5.8
Inverse Trigonometric
Functions: Integration
and Completing
the Square
du
u
= arcsin + C
a2  u2
a
du
1
u
a 2 + u 2 = a arctan a + C
u
du
1
u u 2  a 2 = a arc sec a + C
dx
x
= arcsin + C
2
2
4 x
Ex.
Ex.
dx
=
2
2 + 9x
1
=
3
( 2)
du
2
+ ( u)
( 2)
5.7
Inverse Trigonometric
Functions
Definition of the Inverse Trig. Functions
1.
arcsin x
quadrants I and IV
2.
arccos x
quadrants I and II
3.
arctan x
quadrants I and IV
p
p
 y
2
2
0 y p
p
p
 < y<
2
2
3
Find the values of :
1
arcsin 
2
sin
1
3
2
I
5.6
Growth and Decay
Law of Exponential Growth and Decay
y = Ce
kt
C = initial value
k = constant of proportionality
if k > 0, exponential growth occurs
if k < 0, exponential decay occurs
Ex. 1 A sample contains 1 gram of radium. How
much radium will rema
5.3
Inverse Functions
Definition of Inverse: A function g is the inverse of the
function f if f(g(x) = x and g(f(x) = x.
Domain of f = Range of g
Domain of g = Range of f
Ex. Show that the following are inverses of each other.
f ( x) = 2 x  1
3
g ( x) =
5.2
The Natural Logarithmic
Function and Integration
Log Rule for Integration
1
1. dx = ln x + C
x
1
2. du = ln u + C
u
u'
3. dx = ln u + C
u
Ex.
2
1
2 ln x + C = ln x 2 + C
x dx = 2 x dx =
Ex.
1
2 x  1 dx =
1 du
u 2 =
11
= du
2u
1
1
= ln u + C = ln 2
5.1
Logarithmic,
Exponential, and Other
Transcendental Functions
The graph and some properties of y = ln x.
D : (0, ) R : (, )
Continuous, increasing, 1 1,
and concave downward
1.
2.
3.
4.
ln 1 = 0
ln (ab) = ln a + ln b
ln (an) = n ln a
ln (a/b) = ln a l
7.3
Volume:
The Shell Method
h = height of the rectangle
w = width of the rectangle
r = distance between the axis of revolution and the
center of the rectangle
h
w
r
Note: If this shell is cut apart and flattened, it forms
a rectangular prism.
h
w = dx
l