Analytic Geometry and Calculus II
Section 8024
Grossmont College
Spring 2016
Instructor: Michael J. Lines
Phone: (619) 644-7320
E-mail: [email protected]
Office: 31-383A
Office Hours Monday /Wednesday: 2:00 4:00 p.m.
Tuesday/Thursday: 8:30 - 9:00 a
Partial Fractions (Review Topic from Precalculus)
Procedure for Decomposing a Rational Expression into Partial Fractions
Consider
P( x)
r ( x)
Q( x)
1. Factor
, such that
deg P ( x ) deg Q ( x)
and
P ( x ) , Q ( x)
have no common factors.
into linear fac
Some Basic Identities
Math 280: 7.2 Trignometric Integrals
Working
cos 2 x sin 2 x 1
Strategies for integrating expressions involving:
1 tan 2 x sec 2 x
Powers of sinx and/or cosx
1 cot 2 x csc 2 x
If
sin
m
sin 2 x 2sin x cos x
, and m and/or n is ODD
n
x
Chapter 7 Formula Sheet
Math 280, Working
Derivatives of Inverse Trig Functions:
d
1
sin 1 x
dx
1 x2
d
1
cos 1 x
dx
1 x2
d
1
sec 1 x
dx
x x2 1
d
1
csc 1 x
dx
x x2 1
d
1
tan 1 x
dx
1 x2
d
1
cot 1 x
dx
1 x2
Established Integration Formulas
tan x dx l
7.1 Integration by parts:
u dv u v v du
LIATE rule of thumb (doesnt help EVERY time, but pretty good)
A rule of thumb (proposed by Herbert Kasube of Bradley University):
Whichever function comes first in the following list should be u:
L - Logarithmic fun
Trigonometric Substitution
Integrands involving the sum or difference of squares may sometimes be evaluated by
using trigonometric substitution. The majority of integrals in this section have the following
forms as a part of the integrand.
x 2 a2 ,
a2 x2
Numerical Integration
Left-End Point Approximation:
b
Z
f (x) dx
a
n1
X
f (xk )xk =
k=0
ba
[y0 + y1 + +yn1 ]
n
Right-End Point Approximation:
Z
b
f (x) dx
a
n
X
f (xk )xk =
k=1
ba
[y1 + y1 + +yn ]
n
Midpoint Approximation:
Z
b
f (x) dx
a
n
X
f (mk )xk
Trigonometric Integrals
Powers of Sine and Cosine: Integrals of the form
Z
sinm x cosn x dx
If either m or n is odd, use u-substitution. If m and n are even, use the following power reducing
formulas to reduce the powers on the sine and cosine functions.
Integration by Parts
Recall the product rule for differentiation: If f and g are differential functions then
d
[f (x) g(x)] = f (x) g 0 (x) + g(x) f 0 (x)
dx
which in the notation for indefinite integrals is equivalent to
Z
[f (x) g 0 (x) + g(x) f 0 (x)]
u-Substitution
Recall the chain rule: If f and g are differential functions and F = f g = f (g(x), then F
is differentiable and
F 0 (x) = (f g)0 (x) = f 0 (g(x) g 0 (x)
Examples: Differentiate the following functions:
1. F (x) = (3x2 + 2x 1)6
2. F (x) = c
Mathematics 280
Chapter 11: Sequences and Series
Comparison Test
Direct Comparison Test: Suppose
an and
n=1
0 an bn for all n 1. Then
X
X
bn converges =
n=1
X
X
bn are positive term series and
n=1
X
an converges
n=1
an diverges =
n=1
X
bn diverges
n=1
Loo
Infinite Series
Definition: Consider the infinite series
X
an = a1 + a + a2 + + an +
n=1
We denote the nth partial sum as
Sn =
n
X
ak = a1 + a + a2 + + an
k=1
If the sequence of partial sums cfw_Sn
n=1 converges to a limit S, that is lim Sn = S
n
X
an c
Mathematics 280
Chapter 11: Sequences and Series
Integral Test
Theorem: Let
X
an be a series with positive terms and let f (x) be the function
n=1
that results when n is replaced by x in the formula for f (x). If f is a decreasing and
Z
X
f (x) dx both c
Improper Integrals
Recall the definite Integral
Rb
a
f (x)dx with the following constraints:
1. the interval [a, b] has finite length,
2. f does not have an infinite discontinuity on [a, b].
Definition: A function f is said to have an infinite discontinui
Math 280: 7.5 Strategy for Integration
Working
1. Simplify the integrand, if possible. Do a little algebra first.
Distribute, multiply out (FOIL), etc.
Rationalize the numerator (or denominator)
Change trig functions to sines and cosines
Use a basic t