Homework 5
Math 15300 (section 35), Winter 2016
Due Wednesday, January 27
You may cite results from class as appropriate. Unless otherwise stated, you must provide a complete explanation for your solutions, not simply an answer. You are encouraged to work
9. (a) Prove that / sec soda: : 111 | tancc+seczt| +0. Hint: One way to start
is by doing the following steps:
1 cos :1: cos :1:
sec mdm I d2: 2 2 : ~.2d3:-
cosz: cos m 1 3111 m
(b) Compute f sec4 atdm and / sec3 sedan.
_1_
10. (a) Consider the sequen
MATH 153, Section 56 Fall Quarter 2015
Review for Final Exam
Theory Questions
1. Proof of the Limit Comparison Test.
2. Proof of the absolute convergence test.
3. Proof of the theorem: If lim an r: L and lim b = M 1 then lirn (an +
n+oo nwo new
an) = L +
/ 3. Compute the radius of convergence and the interval of convergence of each
of the following power series:
mic
(a) Z (2k)!
117633.79
0: 31:11
(c) 2k 111]: 23k
4. Compute the following integrals:
(a) / sing mdm
(b) f eos3 xdx
I (e)/% 93+?)
$2+433+10d$
Problem 3 (14 = 7 + 7' points):
2 4
1. Doane Kproof ofthe limit lim .
nroo +3
2- Let f : IR. + R be a continuous function. We consider an inductive sequence with formula
33n+1 = f (mm) and 81 : 0. If 2: converges to a limit L, show that L is a xed point o
Math 153 (56) Midterm 1
Instructor: Evangelia Gazaki I Date: October 22, 2015
0 Please write your answers in the blue books provided, indicate clearly the appropriate
question numbers to which the answers correspond.
0 Do not forget to Write your name on
Final
Laurence Field Math 153, Section 32 June 11, 2013
Instructions: This exam has 18 questions for a total of 100 points. The value of each part of
each question is stated. Please always show your working and reasoning. You have 2 hours.
Techniques of I
Midterm I
Laurence Field Math 153, Section 32 April 25, 2013
Instructions: This midterm has 9 questions for a total of 100 points. The value of each part of
each question is stated. Please show your working. You have 50 minutes.
In questions 15, nd the in
Midterm II
Laurence Field Math 153, Section 32 May 23, 2013
Instructions: This midterm has 6 questions for a total of 100 points. The value of each part of
each question is stated. Please show your working. You have 50 minutes.
1. Answer true or false:
10
1
MATH 153, Section 43 Spring Quarter 2014
Homework 1(due Monday, 04/07/14)
1. From section 7.7 from the book do the problems: 6, 9, 12, 14, 21, 23, 36,
45, 51, 53, 58, 59.
2. From section 7.8 from the book do the problems: 5, 8, 12, 30, 31.
3. Compute th
1
MATH 153, Section 43 Spring Quarter 2014
Review for Midterm 2
Theory Questions
The material includes all chapter 11 and sections 12.1, 12.2, 12.3 (only the
integral test).
1. K definition of the limit of a sequence.
2. Proof of the theorem: Every conver
MATH 153, AUTUMN 2015
Assignment 7, Due November 16 in class
(a)
(b)
(c)
(d)
Do
Do
Do
Do
problems
problems
problems
problems
28, 40 and 46 from Exercises 12.4 of the textbook.
18, 24 and 34 from Exercises 12.5 of the textbook.
14, 16, 18, 30, 40 and 44 fr
MATH 153, AUTUMN 2015
Assignment 6, Due November 9 in class
(a)
(b)
(c)
(d)
Do
Do
Do
Do
problems
problems
problems
problems
32, 35 and 36 from Exercises 12.2 of the textbook.
37, 38 and 50 from Exercises 12.3 of the textbook.
4, 10, 12, 13, 16, 18 and 36
MATH 153, AUTUMN 2015
Assignment 3, Due October 19 in class
(a)
(b)
(c)
(d)
(e)
Do
Do
Do
Do
Do
problems
problems
problems
problems
problems
34 and 38 from Exercises 11.2 of textbook.
56 and 57 from Exercises 11.3 of textbook.
22 from Exercises 11.4 of tex
MATH 153, AUTUMN 2015
Assignment 2, Due October 12 in class
(a)
(b)
(c)
(d)
Do problems 12, 15, 18, 26 and 42 from Exercises 11.2 of textbook.
Do problems 1, 16, 19, 38 and 40 from Exercises 11.3 of textbook.
Do problems 2, 4, 7, 12, 20 and 48 from Exerci
MATH 153, AUTUMN 2015
1. Assignment 1, Due October 5 in class
(a) Prove the following using K denition (Denition 11.3.1 of the textbook):
1. lim
n
1
n
n2
=
2
n n +1
1
lim n = 0
n 2
2. lim
3.
1
=1
1
(b) Do problems 1, 2, 3, 6, 9, 12, 19, 30 from Exercises
FINAL EXAM INFORMATION
The Final Exam will take place Thursday 4-6pm in our usual classroom. Since
the exam covers a wide range of topics, you may bring a single standard (8.5
by 11.0 inches) page of notes. It is fine to type them and to use both sides.
Y
EXAM 1 INFORMATION
There may be some true/false questions, you may be asked to write some definitions, and the rest will be short problems that are similar to the homework
problems. The exam will cover Sections 11.1-11.3, 11.7, and 12.1-12.3.
Beyond the H
Math 15300, Section 47
Homework 2
Due Wednesday, October 15, 2014
Indeterminate Forms and LH
opitals rule
1. The limits below are of the indeterminate form 00 . Use LHopitals rule to evaluate
them.
(a)
(c)
log x
lim
x1 1 x
log (cos x)
x0
x2
lim
(d)
(b)
x
1
MATH 153, Section 43 Spring Quarter 2014
Homework 6(due Friday, 05/09/14)
1. From section 11.3 from the book do the problems: 34.
2. From section 11.4 from the book do the problems: 1, 3, 7, 8, 17, 25, 29,
37, 43.
Hint for exercise 37: Feel free to use
Midterm
No hard integrals. Know first 8 integrals in integral table
1a) State epsilon-n definition of (5 points)
Equation 11.3.1
2) Given a sequence and limit
Provide epsilon-n proof
Example proof) Let epsilon be greater than zero be given. Need to find N
CALCULUS 153: MIDTERM 1
Please answer all questions in a blue book that's provided to you (even the true/false). Don't forget to write your name. There are two sides to this exam. Problem 1 (16 points). Determine the least upper bound and greatest l
CALCULUS 153: SAMPLE MIDTERM 2 SOLUTIONS
Problem 1 (28 points). Compute the following integrals. (1) sin3 x cos2 x dx (2) e-x sin x dx (3) x2 dx 1 + x2 (4) arcsin x dx Solution (1) sin3 x cos2 x dx = (1 - cos2 x) cos2 x sin x dx.
Let u = cos x so t
CALCULUS 153: SAMPLE MIDTERM 2
Problem 1 (28 points). Compute the following integrals. (1) sin3 x cos2 x dx (2) e-x sin x dx (3) x2 dx 1 + x2 (4) arcsin x dx Problem 2 (18 points). (1) Find the general solution of xy - 4y = 2x and the particular sol