Math 119 Spring 2012
Assignment #1
Due Wednesday, May 16th
Hand in the following: 1. (a) Find L2 (x) for f (x) = 2x3 +5x-1 (that is, find the linear approximation to f (x) at x0 = 2). (b) Find L1 (x)
7
Integration of Scalar Fields
We now turn our attention to the question of how integral calculus might be generalized for
functions of more than one variable. It might have occurred to you that there
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
7
Evaluation of Double Integrals in Polar Coordinates
For ordinary single integrals, the most important tool available is the method of substitution.
For double integrals
f (x, y ) dA, we can do somet
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
Part II
Multivariate Calculus
1
Introduction
So far we have only dealt with functions of one variable. We represent this dependence by
writing y = f (x), and you might also have seen it diagrammed as
Assignment #3
Math 119
Winter 2015
Due: Monday, February 2nd
There is a CalcPortal portion to this assignment. However, we recommend reading
problem #4 on this written assignment before trying the fir
MATH 119
Assignment #2
Winter 2015
1. Find a polynomial of degree 6 passing through the points (2, 3), (3, 0), (4, 3), (5, 1), (6, 2), (7, 6), (8, 0).
2. Let f (x) = cos(2x). Let P (x) denote the degr
MATH 119
Assignment #1
Winter 2015
due Monday, January 19th, in class
Please include your tutorial section on your front page.
Note: There is an online Assignment #1 as well, on CalcPortal.
1. In the
A Taylor series for the function arctan
The integral If we invert y = arctan(x) to obtain x = tan y, then, by
differentiating with respect to y, we find dx/dy = sec2 y = 1 + tan2 y = 1 + x2 .
Thus we
1
Approximation Methods
1.1
Linear Approximation
Can form a tangent line of f(x) at a with the derivative f(x)
L(x) = f (a) + f 0 (a)(x a)
1.1.1
(1)
Solution Methods
Bisection Method
1. Determine inte
Convergence of Infinite Series
Definitions:
We define the convergence of a series
1
X
ak using the sequence of partial sums, which is defined such that S0 = a0 ,
k=0
S1 = a0 + a1 , S2 = a0 + a1 + a2
5
Other Forms of the Chain Rule
In our introduction to the multivariate chain rule, we considered the situation in which z =
f (x, y ), with x and y each being dependent upon a third variable t. Howev
6
Optimization Techniques
Part A: Unconstrained Optimization
We now turn to the problem of identifying maxima and minima of functions of two variables.
First, we need to dene them:
A function f (x, y
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
Assignment #2
Spring 2012
Due: Wednesday, May 23rd
1. Find the required Pn,a (x) for the following functions f (x) :
f (x) = ln(x) P3,1 (x)
2
g (x) = x P3,9 (x)
h(x) = 10x P4,0 (x)
2. a) F
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
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 + . .
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
9
Convergence of Innite Series
In the previous section we made an attempt to discover when it might be true that f (x) =
f (k ) ( x 0 )
(x x0 )k (that is, when a function might be equal to its Taylor
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
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
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
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
Operations on power series
1. Term by term differentiation and integration
<x
If at) :2 Z an (a: c) for cfw_:6 cl < R, then
n20
a) f'(gc) = E nan (a: - c)"1 for lac ~ cl < R,
n21
b)(:1:/f d$=n
University of Waterloo
MATH 119 Calculus 2 for Engineering
Midterm Examination
Monday, March 1st, 2010.
Winter 2010
Duration: 7:00 - 8:30 p.m.
Name (print):
I.D. Number:
Signature:
Please indicate you
University of Waterloo
Math 119 Calculus II for Engineering
Midterm Examination
Winter 2013
Wednesday February 27th , 2:30 - 4:20pm
Name (print):
I.D. Number:
Signature:
Please indicate your section (
Math 119 Winter 2013
Assignment #1
Due Monday, January 21st / Tuesday, January 22nd
Hand in the following:
1. (a) Find L2 (x) for f (x) = 2x3 +5x 1 (that is, nd the linear approximation
to f (x) at x0