Exam Review: Tests for series
1
Series (11.2 11.7 )
Know: convergent and divergent series, sum of a series, geometric series, p-series, harmonic series, alternating series, absolute and conditional convergence.
Tests for convergence/divergence:
1. Geometr
Math 116Quiz 3 Name:
d
1 Find a derivative of 1nverse cosine function, that 1s (cos 1(93)
dx
(1) Draw a graph of a cosine function and nd a restricted domain to make a cosine
function be onetoone-
(2) (Fill out the blank spaces) (A
Let y = cos 1.(:z
MATH116
Exam2
Name:
Student ID:
Section #:
Instructor:
Problem
1
2
3
4
5
Total
14
8
6
10
12
50
Score
Max
Instructions
1. You will be given exactly 75 minutes for this exam.
2. No graphic calculators, phones, or any electronic devices. Put them away out of
4f .
Math 116Quiz 6 Name:
1. Evaluate the integral
4: . 2 l -
' J [(42"W'm MmMW lamb
CI: MW -9 we= l
_ ' x : 2. o 9
(bk-209949 K - J, 4604*9 S 1+le C 494
/ 4'? / I
( sz9+9$19=1 _ "Jl Malalw'lag/tle
_ 4.5 $49- 1592 949
L
4&9 l C
: .L '1 +
Ll 0. 4? C
MATHius Exmn3 5;42
Name: Student ID:
Section #: _. Instructor:
Instructions
1. You will be given exactly 50 minutes for this exam.
2. No graphic calculators, phones, or any electronic devices. Put them away out of sight.
3. Nothing on your desk
400.
Math 116Quiz 4 Name:
on o
1. Find the limit
argue) /"
.L z 1. .4; ,
[Tm 2.4%th = [M Jodi) = im 9c(1l)L I = in 393(3)
Mac 1 xm 71" T 'x-aao (-. 76700
T , i
oooo"4%m .2. 40" EHoqffal ; Rafe. SW- 0
2. Evaluate the integral
: i3 I du- L9W1C3+
ol
Math 116Quiz 7 ' Name:
1. Find the partial fraction decomposition of the following rational function with num-
bers A, B, 0, as many as you need. You dont need to determine the unknown
consmts' _ a:-74.m3+3n+1_ _ < Smce #49 423W 6?
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Math 216 (Winter 2017) Final Exam Guidelines
Final Exam: 3-6 pm, Tuesday, April 25, in 2063 CB
NO make-up Final Exam will be granted
2.2
Separable ODEs, all related formulas and procedures are required.
2.3 1st-order linear ODEs, including words applicati
MATH 217: Matrix Algebra
Homework #1
Determinant of 4X4 Matrix
Shihab M. Shihab
April 4, 2017
1
Statement of the Problem
Evaluate the determinant of the following 4X4 matrix.
2
0
A=
7
0
2
1
0
4
1 0 2
2 3 5
1 0 3
Description of Solution
First when eval
I NTEGRAL C ALCULUS N OTES
F OR MATHEMATICS 101
Joel F ELDMAN
Andrew R ECHNITZER
T HIS DOCUMENT WAS TYPESET ON T UESDAY 28 TH F EBRUARY, 2017.
Legal stuff
c 2017 Joel Feldman and Andrew Rechnitzer
Copyright
In the near future this will be licensed und
C ALCULUS Q UESTIONS
F OR M ATHEMATICS 101
Joel F ELDMAN
Andrew R ECHNITZER
T HIS DOCUMENT WAS TYPESET ON T UESDAY 7 TH M ARCH , 2017.
Elyse Y EAGER
Legal stuff
c 2016 Joel Feldman, Andrew Rechnitzer and Elyse Yeager
Copyright
In the near future this
Harvard University, Math 101, Spring 2015
Binary relations and Functions
1
Binary Relations
Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two
elements a A is in a relation to an element b B we write a R
Harvard University, Math 101, Spring 2015
Introduction to naive set theory (to be updated after Fridays lecture)
1
Definition of a set
A set is a collection of elements. When an element x belongs to a given set A, we write
x A.
This is the membership rela
Harvard University, Math 101, Spring 2015
Lecture 1 and 2 :
1
Introduction to propositional logic
Logical statements
A statement is a sentence that is either true or false, but not both.
Some examples:
1. 1 is smaller than 2
2. 7 is an odd number
3. 2 is
Integration of sec , csc , sec3 and csc3
These notes show several ways to integrate sec , csc , sec3 and csc3 .
R
sec d by trickery
R
sec d by partial fractions
sec +tan
The standard trick used to integrate sec is to multiply the integrand by 1 = sec
an
Properties of Exponentials
In the following, x and y are arbitrary real numbers, a and b are arbitrary constants that
are strictly bigger than zero and e is 2.7182818284, to ten decimal places.
1) e0 = 1, a0 = 1
2) ex+y = ex ey , ax+y = ax ay
3) ex = e1x
Table of Indefinite Integrals
Throughout this table, a and b are given constants, independent of x
and C is an arbitrary constant.
f (x)
u(x)v (x)
f y(x) y (x)
1
a
x+C
ax + C
xa
1
x
a
g(x) g (x)
sin x
g (x) sin g(x)
cos x
tan x
csc x
sec x
cot x
sec2 x
c
Important Taylor Series
You should know or be able to quickly rederive the following important
Taylor series. (They are also given in Theorem 3.6.6 of the course notes.)
X
xn
e =
n!
n=0
x
1
1
= 1 + x + x2 + x3 +
2
3!
X
1
sin(x) =
(1)n
x2n+1
(2n + 1)!
n=0
Quiz 2
Quiz 2 will be held in class on Thursday, February 2. We will go into quiz mode
at 10:20. Please remain in your seat from then until the quizzes have been collected.
The Quizzes section of the course web page, which is
http:/www.math.ubc.ca/gerg/
Quiz 1
Quiz 1 will be held in class on Thursday, January 19. We will go into quiz mode
at 10:20. Please remain in your seat from then until the quizzes have been collected.
The Quizzes section of the course web page, which is
http:/www.math.ubc.ca/gerg/
Quiz 3
Quiz 3 will be held in class on Thursday, February 16. We will go into quiz
mode at 10:20. Please remain in your seat from then until the quizzes have been
collected.
Quiz 3 covers the material in Units 5 and 6 1.8, 1.9, 1.10 and 1.11 of the
CLP1
Quiz 4
Quiz 4 will be held in class on Thursday, March 9. We will go into quiz mode
at 10:20. Please remain in your seat from then until the quizzes have been collected.
Quiz 3 covers the material in Units 7 and 8
1.12 (improper integrals),
2.1 (work
Unit 3:
Interpolation
Estimating what goes between samples.
Interpolation Page 1
L06: Introduction to Interpolation
Goal: To find out what interpolation is, and how it is useful.
Interpolation in Image Processing
A digital image is an array of pixels (pic
A4. Math 216. Winter 2012.
1. Exercises 1.4. See Problem #6
a. Is this first order IVP linear or separable?
b. For the moment consider the ODE without the initial condition.
i. Is y = 0 a solution of this ODE?
ii. Find another constant solution to this OD
Math 216. Winter 2012.
A1. ODE to Joy.
1. It takes a long time to get there
Let v(t) = 50/(1 + t2), t 0 be a velocity in miles/hour at time t hours of a particle in
linear motion. Let s(t) miles be the position of the particle on a number line at t hours
COURSE GUIDELINES
Math 216.002
Winter Term 2012
M W F 11:30 am - 12:20 pm, 1048 CB
1. Attendance: Students are required to attend each class session and
be on time.
2. Exams and Quizzes: There will be two exams and five quizzes this
term and a final exam
Table of Integrals
Integrals with Logarithms
Basic Forms
Z r
Z
1
x dx =
xn+1
n+1
Z
1
dx = ln x
x
Z
Z
udv = uv vdu
Z
n
(1)
(2)
p
x
dx = x(a + x) a ln x + x + a
a+x
x ax + bdx =
Z
(25)
Z
Z
2
(2b2 + abx + 3a2 x2 ) ax + b (26)
15a2
(42)
ln ax
1
dx = (ln ax)2
Math 216. X1 Fri, Feb 3, 2012, Winter Term 2012.
Some Topics.
1. Order of an ODE. Linear vs. nonlinear ODEs. (See Exercises 1.1 #112.)
2. Determine whether or not a given function (or family of functions) is a
solution of given ODE or IVP. (See Exercises