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
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
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
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
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
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
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
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
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
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
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
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
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
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 = xn+1 + C (for n 6= 1)
R
sin xdx = cos x + C
R
sec2 xdx =
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 (
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
The Big-O Order Symbol: Another Way to Think
of Error
To motivate the concept we're about to introduce, consider the following situation: suppose you've shown that the 4
3
1x +
certain function is, say,
calculate that the 5
th
1 4
10 x .
th
-order Maclaur
1.2 Application of the Linear Approximation: Newtons 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 as
Math 119
Assignment #2: Solutions
1. Find the required Pn,a (x) for the following functions f (x) :
(a) f (x) = tan(x);
P3,0 (x)
Take derivatives:
f 0 (x) = sec2 x f 00 (x) = 2 sec2 x tan x f 000 (x) = 2 sec4 x + 4 sec2 x tan2 x
(details omitted)
Thus,
P3
Math 119
Assignment #1: Solutions
1. (a) Find L1 (x) for f (x) = 3x2 2x + 1 (that is, find the linear approximation to f (x)
at x0 = 1).
By definition L1 (x) = f (1) + f 0 (1)(x 1). In this case, f 0 (x) = 6x 2 meaning
f 0 (1) = 4 and since f (1) = 2 we g
Math 119
Assignment #3: Solutions
1. Consider the function f (x) =
3
1
.
1+x
(a) Find the Taylor polynomial P2,0 (x) for f (x).
f (x) = (1 + x)1/3 f (0) = 1
1
1
f 0 (x) = (1 + x)4/3 f 0 (0) =
3
3
4
4
f 00 (x) = (1 + x)7/3 f 00 (0) =
9
9
f 00 (0) 2
1
2
x
Math 119 Winter 2014
Assignment #2
Due Friday, January 24th
Hand in the following:
1. Find the required Pn,a (x) for the following functions f (x) :
(a) f (x) = tan(x); P3,0 (x)
(b) g(x) = x; P2,4 (x)
(c) h(x) = 2x ;
P3,0 (x)
2. Use Taylor polynomials der
Math 119 Winter 2014
Assignment #3
Due Friday, January 31st
Hand in the following:
1. Consider the function f (x) =
3
1
.
1+x
(a) Find the Taylor polynomial P2,0 (x) for f (x).
p
(b) Use P2,0 (1/3) to write a fraction that estimates f (1/3) = 3 3/4. Then