Numerical Analysis/Exam #1/Spring 2013
NAME:
Sign the following pledge:
On my honor, I have neither received nor given any unauthorized assistance on this examination.
SIGNATURE:
DATE:
For full credit, you must show your work. Answer 8 out of 10 questions

Numerical Analysis/Quiz #1/February 7, 2012
SOLUTIONS
Let f (x) = x3 3x.
1. f (1) = 2 < 0, f (3) = 18 > 0, so by the Intermediate Value Theorem, f
has a root in (1, 3).
2. f 0 (x) = 3x2 3 = 3(x2 1) > 0 for x (1, 3), so f is increasing on [1, 3],
and the r

Numerical Analysis/Exam #1/Spring 2013
SOLUTIONS
1. (a) f (0) = 3 > 0, f (2) = 1 < 0, so by the Intermediate Value Theorem,
f has a root in [0, 2].
(b) f 0 (x) = 3x2 6 = 3x(x 2) < 0 on (0, 2). Thus f is decreasing on
[0, 2] and the root is unique.
(c)
2.

473/SPRING 2013/HOMEWORK #2: NEWTON
Apply Newtons method to the following problems. In each case, you
are given a function f and an initial estimate p0 to a root p of f . You
need to find p1 , p2 , p3 , . . . , using the iteration formula
f (pn )
pn+1 = p

473/SPRING 2013/HOMEWORK #3: LAGRANGE
(1) Find the Lagrange interpolating polynomial for the data
i x i yi
0 0 1
1 1 3
2 2 7
(2) Use Lagrange interpolation to guess the next number in this sequence:
1, 2, 4, 8.
(3) Give an example of n + 1 points (xi , yi

473/SPRING 2013/HOMEWORK #4: NUMERICAL
INTEGRATION ERROR ESTIMATES
Z
(1) (a) Use Simpsons rule with n = 4 to estimate
0
2
1
dx.
x+4
(b) Give a bound for the error in this estimate.
(c) Find the minimum value of n such that using Simpsons rule results
in e

473/SPRING 2013/HOMEWORK #1: BISECTION
(1) Do 3 iterations of the bisection method for f (x) = x cos x
on [0, 1].
(2) Use the bisection method to find solutions accurate to within
102 for f (x) = x3 7x2 + 14x 6 = 0
(a) on [0, 1]
(b) on [3.2, 4].
(3) Use t

Math 473/Numerical Analysis/Spring 2013
The Heat Equation
The heat equation is the partial differential equation
ut = uxx ,
2
u
u
where ut =
, uxx =
. We will solve the equation on the region
t
x2
0 x L for some L, t 0, with
Boundary conditions: u(0, t)

Numerical Analysis/Exam #2/Spring 2013
NAME:
Sign the following pledge:
On my honor, I have neither received nor given any unauthorized assistance on this examination.
SIGNATURE:
For full credit, you must show your work. Answer 5 out of 7 questions. Cross

Numerical Analysis/Exam #2/Spring 2013
SOLUTIONS
1. Calculated parameters are
T = 0.5, u = 1.10517, v = 0.904837,
p = 0.6,
S:
V:
122.14
100.00
81.87
15.57 22.71 32.14
0.00 5.85 10.00
0.00 0.00 0.00
q = 0.4.
The current value of the option is approximately

Math 473/Spring 2013/Discrete Fourier Transform
Digital signals. A continuous signal is a function from some interval in R to C.
The domain is regarded as time, and f (t) is the amplitude of the signal at time t.
A digital signal s = (s0 , s1 , . . . , sn

(1) (1]
= (1+1xisk1r)+%(eosk1r+l)= 2(100111911)
.1: 3,: my 2:2 k
0 2 0 0 E 1:11 odd
_ _ _ k
1 1 1 1 SUM2,6k0f0rk0,adElkcfw_U haven
2 2 4 4 Alternatively, note that any function f can be written as f (1:) = g(:e)+
3 3 24 9 11(2), where g is 111m and 11.

A data set has n 2 30. 232.; = -51.
2.0?
Wekncwutthedvueofaregcssicnnefmayvmvneufx isgivmby.
The data selhasthe followingpalameners,
3:30
3
Exf=67_ll
H
N
21:13:22.7
L:
30
2x? =ssu
H
3
2y: 40.600
:4
so
2 xiyi = w3-8!
1-:
Nuw,ndteva1neaof3,a andlemedvahlewh

473/Spring 2013/Homework #6: Options
(1) A stock has a current price of $100 per share. I anticipate that the stock
will go up, and have $2000 to invest.
(a) Suppose I buy 20 shares of the stock, and sell them at the end of the
month. Using the table belo

Math 473/Spring 2013/Project #1/Due 2/18/13
Solutions
(1) The function f (x) = x2 x 1 has two roots. You need to find a starting
value for each root, and then apply Newtons method. You can do this in
several ways, for example:
(a) [1, 0] and [1, 2] are br

Numerical Analysis/Spring 2013/Review #2
Sample questions/Solutions.
(1) There is no question (1).
(2) The calculated parameters are
T = 1,
u = 1.10517,
v = 0.904837,
(r 1 2 ) t
p = 0.6,
2
(a) p = 21 +
= 0.5 + (0.10.02)0.5
= 0.5 +
2
0.4
(b) S:
100.00 110.

Numerical Analysis/Spring 2013/Review #3
Exam is on Monday, May 13, 2013, 11:452:45, Smith 241. Bring a calculator. You may also bring in a single sheet of 8 21 1100 paper with anything you
want written on it.
You will be asked questions similar to those

Numerical Analysis/Spring 2013/Review #1/Solutions
(1) (a) f (1) = 2 < 0, f (2) = 2 > 0, so by the Intermediate Value Theorem,
f has a root in (1, 2).
(b) f 0 (x) = 3x2 3 = 3(x2 1) = 3(x 1)(x + 1) > 0 for x (1, 2), so f
is increasing on [1, 2], and the ro

Numerical Analysis/Spring 2013/Review #2
Exam is on Monday, April 22, 2013. Bring a calculator. You may also bring
in a single sheet of 8 12 1100 paper with anything you want written on it.
You will be asked questions similar to those on quizzes, homework

Numerical Analysis/Quiz #1/February 7, 2012
NAME:
Let f (x) = x3 3x.
1. Show that f has a root in the interval [1, 3].
2. Show that this root is unique.
3. Suppose the bisection method is applied to f with initial interval [1, 3].
What is the minimum numb