UNIVERSITY OF WATERLOO
FINAL EXAMINATION
WINTER 2016
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COURSE NUMBER
Math 119
COURSE TITLE
Calculus 2 for Engineering
COURSE SECTIONS
001 002 003
University of Waterloo
Math 119 Calculus II for Engineering
Midterm Examination
Winter 2015
Time: 12:30 - 2:20pm
Monday, February 23rd.
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4
Polynomial Interpolation
The goal of the rst half of this course will be to discuss how we can improve on the idea of
the tangent line approximation, but in order to do that we need one more digress
7
Approximation of Integrals using Taylor Polynomials
Suppose we wish to evaluate
0.5
2
2
et dt. It is known that et does not possess a nice an-
0
tiderivative, so there is no way to evaluate this exa
5
Taylor Polynomials
Weve now discussed two ideas which are both credited to Isaac Newton:
For a smooth function f (x), a tangent line to the graph at a point (x0 , f (x0 ) can be
dened by considerin
Math 119
Calculus II for Engineering
Lecture Notes
David Harmsworth
Department of Applied Mathematics
University of Waterloo
2013
Chapter 1: Approximation Methods
1
Introduction
In Math 117 you were i
6
The Remainder Theorem for Taylor Polynomials
We now turn to the question of accuracy. How good is the approximation f (x) Pn,x0 (x)?
The magnitude of the error is |f (x)
Pn,x0 (x)|. How large can th
3
Root Finding
As we discussed last time, part of the motivation for the rst half of this course is the inconve
2
nient fact that we cannot evaluate integrals such as
ex dx exactly. In fact, though, t
10
Power Series
We now return to our discussion of Taylor series. More generally, a power series centered at
x0 is any series of the form
k=0
c k ( x x0 ) k = c 0 + c 1 ( x x0 ) + c 2 ( x x0 ) 2 + . .
8
Innite Series
So far weve been suggesting that the accuracy of our Taylor polynomial approximations should
improve when we incorporate more terms. This certainly appears to be true in the examples
w
11
The Big-O Order Symbol
To motivate the concept were about to introduce, consider the following situation: suppose
youve shown that the 4th -order Maclaurin polynomial for a certain function is, say
2
Limits and Partial Derivatives
2.1
Multivariate Limits (optional section)
Scalar elds can exhibit some perplexing behaviour near discontinuities. Consider the following
examples:
2xy
have a limit as
MATH 119
Page 11 of 14
Final Examination - Winter 2016
Formulas (I)
(you may detach this page, and discard it at the end of the examination period)
Integration Formulas:
R
dx = x + C
R n
n+1
x dx = xn
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2014
Mathematics
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MATH 119
Page 10 of 13
Final Examination - Winter 2015
Formulas (I)
(you may detach this page, and discard it at the end of the examination period)
Integration Formulas:
R
xdx = x + C
R n
n+1
x dx = x
MATH 119
Assignment #1
Spring 2017
due by Wednesday, May 10th, 8:30am
(to be submitted to Crowdmark)
Only Problems #1 and #2 are for submission. Problem #3 is included for practice.
Note: There is an
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2012
Mathematics
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3
Taylor Series: the Two-Variable Case
3.1
Finding a Formula
In single-variable calculus, the most basic application of the derivative is in the construction
of a tangent line. For the calculus of fun
4
Introduction to Vector Functions: Parametric Representations of Curves
Early in Math 117 we dened the cosine and sine functions as the coordinates of a point
moving around the unit circle; in Figure