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Examination
University of Waterloo
Final
Final Examination
Math Winter 2012
118
(Calculus
Times: Monday 2012-04-16 at
UNIVERSITY OF WATERLOO
FINAL EXAMINATION
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COURSE NUMBER
COURSE TITLE Calculus 2 for Engineering
COURSE SECTION(S) 001 002 003 004
DATE OF EXAM April 14th, 2011
TIME PERIO
Math 118 Spring 2016: Practice Problems 10
1. The geometric series
1
X
xn (for |x| < 1) is particularly useful, as its sum,
1
, can be
1
x
n=0
related to many other functions. Using this, find series representations of the following, and
state for which x
Math 118 Spring 2016: Practice Solutions 7
1. Use the integral test to determine the convergence of the following series:
(a)
1
X
n=1
1
(n + 1)
(b)
4
3
1
X
n=2
1
p
n ln(n)
(c)
1
1
X
2n
n=1
n2
2. Use the comparison test to determine the convergence of the
Series
Geometric Series.
Harmonic Series.
Properties
Binomial Series
Testing Convergence
Divergence Test.
Limit Comparison Test
Integral Test.
Alternating Series Tes
P-Series
P-Series.
Absolute Convergence Thm
Comparison Test.
The Comparison Test
Absolute
Math 118 Spring 2016: Practice Problems 6
1. Write the number 2.317 = 2.3171717. as a ratio of integers. Hint: Treat 0.0171717. as
an infinite sum 0.017 + 0.00017 + 0.0000017 + .
2. (a) Is the series
22n 31
n
convergent or divergent? If it is convergent,
Wednesday, January 6 Lecture 2 : Integration by parts
Students who have practiced the techniques presented in this lecture should be able to: Recognize those
integrals which are best integrated by the technique "Integration by parts", apply the technique
Math 118 Spring 2016: Practice Problems 9
1. (a) Approximate f (x) = (1 + x)5 by finding T5 (x) (the 5th degree Taylor polynomial of
f ) centered at 0.
(b) Find the remainder R5 (x) for the above approximation and use this to show T5 (x) is not
actually a
Last name (Print):
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First name (Print):
University of Waterloo
Final Examination
Math 118
(Calculus 2 for Engineering )
Instructor: See table below
Date: Thursday, April 15, 2010
Term: 1101
Number of exam pages: 11
(including cove
Monday, February 6 - Lecture 17 : Integral test, p-series. (Refers to Section 8.3 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: State the Integral test and apply it to determine
Tuesday, January 24 - Lecture 11 : Linear differential equations (Refers to Section 10.5 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: Recognize a linear first order differentia
Wednesday, February 1- Lecture 16 : Infinite series: Divergence test (Refers to Section 8.2 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: State and apply the Divergence test (nt
Tuesday, February 7 - Lecture 18: Tests for convergence of series : Comparison test. (Refers to Section 8.4 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: Apply the comparison te
Tuesday, January 31 - Lecture 15 : Infinite series (Refers to Section 8.2 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: Define an infinite series, define a partial sum of a seri
Math 118 Spring 2016: Practice Problems 11
(for practice with the last few topics)
1. Sketch the curve of the parametric equations
x = sec2 t
y = tan t
1
for 0 t /3
2. Sketch the curve of the parametric equations
x = t2 + t
y = t2 t
for 2 < t < 2. (It may
Wednesday, January 13 Lecture 5 : Still More Integration Methods : Completing
the square and rational functions.
After having practiced the problems associated to the concepts of this lecture the student
should be able to: Solve integrals containing expre
Friday, January 15 Lecture 6 : Rational functions : Partial fractions
After having practiced the problems associated to the concepts of this lecture the student
should be able to: Solve integrals containing rational functions by partial fractions method.
Monday, January 18 Lecture 7 : Improper integrals.
After having practiced the problems associated to the concepts of this lecture the student
should be able to: Recognize the two different types of improper integrals,
distinguish between an improper integ
Wednesday, January 20 Lecture 9 : Error estimation for numerical integration.
After having practiced the problems associated to the concepts of this lecture the student should
be able to: Find error's bounds when applying either the midpoint rule, the tra
Friday, January 22 Lecture 10 : Differential equations: Definitions Separable
DEs.
After having practiced the problems associated to the concepts of this lecture the student should
be able to: Recognize a solution to a differential equation and an initial
Wednesday, January 20 Lecture 8 : Numerical integration.
After having practiced the problems associated to the concepts of this lecture the student should
be able to: Apply the Trapezoidal and the Simpson's rule to approximate the value of a definite
inte
Math 118 Spring 2016: Practice Problems 8
1
X
1
converges. Use the remainder estimate
2
n(ln
n)
n=2
for the integral test to find the smallest integer N which guarantees that |S SN | 1.
1. Using the integral test, prove that S =
Note: Since the series sta
Wednesday, January 25 Lecture 12 : Second-order Differential equations. (Not
explicitly discussed in your text)
After having practiced the problems associated to the concepts of this lecture the student should
be able to: Solve second-order differential e
Thursday, January 26 - Lecture 13 : Infinite sequences of numbers. (Refers to Section 8.1 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be able to: Find the limit of simple sequences.
13.1