1.2 Application of the Linear Approximation: Newton's Method Consider the problem of finding the point where the curves y 1 x and y sin x intersect:
This means solving the equation sin x 1 x , or, equivalently, sin x x 1 0 . We refer to solving f ( x) 0 a
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 the input pair (x, y ) ap+ y2
proaches the origin? If
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 ) has a local maximum at (x0 , y0 ) if f (x0 , y0 ) f (
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. However, we may encounter
problems in which x and y are them
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 should such a
thing as partial integration, as a count
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 introduced to the basic tools of calculus, and you have
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 something similar; we can change the coordinate
R
system (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, these
intractable integrals arent the rst examples of un
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
x f y.
We could also write f
: R R, although this impli
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 first of the online questions.
1
Z
sin(3t2 ) dt by using t
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 degree 10 Taylor polynomial of f at x = 0. Calculate
P (x).
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 special theory of relativity, a particle
q with the res
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 1 the the point labelled (x, y ) has coordinates given
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 functions of two variables, the corresponding tool is the
Math 191 Supplement: Linear Approximation
Linear Approximation
Introduction
By now we have seen many examples in which we determined the tangent line to the graph of a function f (x) at a point x = a. A linear approximation (or tangent line approximation)
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) for g(x) = cos(x) + x ln(x) - x 2. Estimate the value o
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) Find the third degree Taylor polynomial of 3 x centered
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 digression. We hope
that youll nd this material to be useful i
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 exactly. There are numerical methods available
(in fact yo
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 + . . . .
(1)
Aside: Theres really no dierence between a Tay
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, 1
x3 +
14
10 x .
Suppose also that you go ahead and
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 series). We were
k!
k=0
able to use Taylors Inequality
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 considering the secant line joining (x0 , f (x0 ) to a second poi
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 this be? We wont be able to
calculate it exactly, so inst
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
weve discussed so far, and in fact for some functions it
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 your section:
MATH 119 - 001
MATH 119 - 002
MATH 119 -
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 (this is worth 1 mark!):
Math 119-001
B.N.A. Chalmers
University of Waterloo
Math 119 Calculus II for Engineering
Midterm Examination
Spring 2012
Tuesday, June 12th .
Time: 5:30 - 7:20pm
Name (print):
I.D. Number:
Signature:
Please indicate your section (this is worth 1 mark!):
Math 119-001
P.Stechlinski
M
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 = 2).
(b) Find L1 (x) for g(x) = sin(x) + 2x ln(x) + 1
f (x, y) =
2xy
x2 + y 2
(x, y)
y
x
x
f (x, y) = 0
y
y
(x, y) = (0, 0),
x
y = x
f (x, y) = f (x, x) =
f (x, y) =
2x2
=1
2x2
(x, y) = (0, 0),
2x2 y
.
+ y2
x4
y=0
x0
f 0.
x=0
y0
f 0.
y=x
f (x, y) = f (x, x) =
f 0.
y = kx,
2x3
2x
= 2
,
4 + x2
x
x +1
f (x, y)