Math 128 Final Exam Syllabus
Be able to:
Evaluate indefinite, definite and improper integrals using recognition (identification as an
antiderivative), substitution, trigonometric substitution, integration by parts, or partial fractions.
Find the volume of
MATH 128 Winter 2016
Worksheet 3
Topics: Strategy for Integration, Improper Integrals, Review of Areas and Volumes
1. Evaluate the following integrals
Z
(a)
x sin2 (x) cos(x) dx
Z
(b)
(c)
(d)
(e)
(f)
2
x
dx
2
2 1 + cos (x)
Z 1 r
2
1+x
dx
1
1x
2
Z
t
dt
4
Math 128 Midterm Syllabus
Here is the topics list for your midterm
Material from weeks 1-5 (up to and including 9.5: First-order Linear DE)
Integration and integration techniques:
indefinite and definite integrals
substitution
integration by parts
trig in
Winter 2014
MATH 128 Online
University of Waterloo
MATH 128 : Calculus 2 for the Sciences
Winter 2014 (Online Section)
Instructor:
Email:
Online Office Hours:
Barbara Forrest
[email protected]
Monday Friday via email; on-campus office hours are not a
Math 128 Dr. Smith Lecture 7
I. Improper Integrals means dealing with
infinities, either in x or in y.
A. Type 1: x goes to (or -)
1.
An area with one infinite dimension
can be finite (Integral converges)
2.
Examples
3.
1 1/xp dx converges if p>1,
diverge
Math 128 Dr. Smith Lecture 3
I. Integration by Parts
A. Product Rule backward: u dv = uv v
du
B. The idea is to make v du an easier integral
to solve than u dv
C. Choose u and dv via LIATE:
1. Logarithmic function
2. Inverse trigonometric function
3. Alge
Math 128 Dr. Smith Lecture 6
I. Integration by Partial Fractions
Given rational function P(x)/Q(x)
A. If deg(P) > deg(Q), divide Q into P and
separate out polynomial from rational remainder
R(x)/Q(x)
B. Factor denominator Q(x)
1. If Q is a product of dist
Math 128 Dr. Smith Lecture 4
I. Trigonometric Integrals
A. Basic formulae
1. Half-angle identities
2. Reduction formula
B. Integrals with powers of sines and cosines
1. Only powers of sines or only powers of
cosines: use angle identity or reduction
formul
Math 128 Dr. Smith Lecture 0
I. Introduction
A. Course Outline
tutorial this week
D. Assignments
1. Due on Wednesdays at noon; first one due
Jan. 14
2. Due at noon; dropbox Box 17, Slots 4 (A
J), 5 (K S), 6 (T- Z)
3. Late work will not be accepted.
E. Te
Math 128 Dr. Smith Lecture 2
Integration by Substitution
I. Chain Rule backward
A. Chain Rule: (f(g(x) = f (g(x) g(x)
B. f(g(x) = (f(g(x) dx = f (g(x) g(x)
dx
C. Want to recognize g(x) = z and g(x) dx
= dz,
D. Find z
1. in parentheses
2. under radical
3.
Math 128 Dr. Smith Lecture 5
I. Trigonometric Substitution
A. Given Use
and
a2 b2x2
a2 + b2x2
b2x2 a2
on
x = a/b sin dx = a/b cos d [-,]
x = a/b tan dx = a/b sec2 d [-,]
x = a/b sec dx = a/b sec tan d [0,]
B. Examples
Area of ellipse, 7.3#1, 5, 21, 7
C. G
MATH 128 Winter 2015
Assignment 1
Topics: Fundamental Theorem of Calculus, Substitution Rule, Integration By Parts
Due: Wednesday, January 14th, 12:00 pm
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and unde
MATH 128 Winter 2015
Assignment 2
Topics: Trig integrals, trig substitution, and partial fractions
Due: Wednesday, January 21st, 12:00 pm
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underline
your last
Reading, Discovering and Writing Proofs
Version 0.4.0.4
c Faculty of Mathematics, University of Waterloo
Contents
I
Introduction to Proof Methods
9
1 In the beginning
1.1 What Makes a Mathematician a Mathematician? . . . . . . . . . . . . . . .
1.2 Why do
MATH 128 Winter 2016
Worksheet 1
Topics: Integration review, integration by parts
1. Evaluate the following.
Z x
e +1
(a)
dx
2ex
Z
(b)
x5 x3 + 7dx
Z
9
(c)
0
Z
1
p
dx
1+ x
5
|x 2|dx
(d)
1
Z
1
(e)
2
ex sin (x5 )dx
1
2. Suppose that a car is decelerating at
Separable: y = f(x) g(y)
1) Exceptional solutions: Values of y such that g(y) = 0
2) Implicit solution: When g(y) 0, integ(1 / g(y)dy = integ(f(x)dx
3) Explicit solution: Isolate the implicit solution for y in terms of x
First order linear: y = f(x)y + g(
MATH 128 : Calculus 2 for the Sciences
SPRING 2015 (Online Section)
Instructor:
Email:
Online Office Hours:
Barbara Forrest
[email protected]
Monday Friday via email; on-campus office hours are not available
Course Description:
This section of Math 1
University of
Mid-term Examination
WATERLOO
Fall 2015
Last Name (Please Print):
First Name:
UW Student Identication Number:
MATH 128
Calculus II for Science
Instructor: Stanley Yao Xiao
Date: Monday, October 26, 2015
Time: 4:00 pm to 5:20 pm
Instructions
University of
Mid-term Examination
WATERLOO
Fall 2015
Last Name (Please Print):
First Name:
UW Student Identication Number:
MATH 128
Calculus II for Science
Instructor: Stanley Yao Xiao
Date: Monday, October 26, 2015
Time: 4:00 pm to 5:20 pm
Instructions
Name:
MATH 128
ID#
Midterm Reection
Due: Monday, November 2, 12:30 PM
This assignment is meant to allow for a more accurate assessment of your knowledge and ability than the midterm exam administered. It consists of three questions, covering some of the t
University of
Practice Mid-term
WATERLOO
Fall 2015
Last Name (Please Print):
First Name:
UW Student Identication Number:
MATH 128
Calculus II for Science
Instructor: Stanley Yao Xiao
Instructions
1. Put your name and ID number at the top
of this page, and
University of
Practice Final
WATERLOO
Fall 2015
Last Name (Please write clearly):
First Name:
UW Student Identication Number:
MATH 128
Calculus II for Science
Instructor:
Stanley Yao Xiao
Instructions
1. Put your name and ID number at the top
of this page
Name:
ID#
MATH 128
Assignment 4
Due: Friday, October 16, 2:30 PM
The assignment is out of 40. The marks for each question are shown in the margins. Please
submit your solutions in the same order as the questions. My suggestion is to print these questions,
Name:
MATH 128
ID#
Assignment 4
Due: Friday, October 16, 2:30 PM
The assignment is out of 40. The marks for each question are shown in the margins. Please
submit your solutions in the same order as the questions. My suggestion is to print these questions,
Name:
ID#
MATH 128
Assignment 3
Due: Friday, October 09, 2:30 PM
The assignment is out of 40. The marks for each question are shown in the margins. Please
submit your solutions in the same order as the questions. My suggestion is to print these questions,
Name:
ID#
MATH 128
Assignment 3
Due: Friday, October 09, 2:30 PM
The assignment is out of 40. The marks for each question are shown in the margins. Please
submit your solutions in the same order as the questions. My suggestion is to print these questions,