M348 (55775)
Midterm #1
EID:
Name:
There are ve questions on this exam, which are worth 20 points each. You must show all your work to
receive full credit. You have 50 minutes. You are permitted a non-programmable calculator and a single
8.5x11 double-sid
Math 348, Prof. Arbogast
Exam 2, November 8, 2013
NAME:
You may use a calculator and a page of notes on this exam. Where appropriate, show your work!
1. [20 pts] For h > 0, approximate f 0 (x1 ) using only the values of f at x1 h, x1 , and x1 + 3h.
(a) De
Math 348, Prof. Arbogast
Exam 1, October 4, 2013
NAME:
You may use a calculator and a page of notes on this exam. Where appropriate, show your work!
8
50
1. [25 pts] Consider finding a root of f (x) = x2 using g(x) =
.
x
(x + 3)2
i
h
100
8
.
Hints: f (2)
CURRICULUM VITAE
Todd James Arbogast
August 25, 2015
Personal Data
Current Address:
Phones:
Electronic Mail:
The University of Texas at Austin
Department of Mathematics
2515 Speedway, Stop C1200
Austin, Texas 78712-1202
512-471-0166 512-475-8628
arbogast@
/
/ BISECTION METHOD
/
/ Solves the problem
/
f(x) = 0
/ using the bisection algorithm.
/
/ state bisection(double a, double b, double& x,
/
double tolerance, int maxIteration, int debug)
/
/ Inputs:
/
a,b
The initial bounding interval, with a root betwee
M348 (55775)
Midterm #2
Solutions
1. Consider the approximation formula
f (x0 )
1
[f (x0 h) 2f (x0 ) + f (x0 + h)] ,
h2
which has an error of the form Kh2 + O(h4 ).
Use Richardson extrapolation to generate a O(h4 ) accurate approximation of f (x0 ) that
M348 (55775)
Midterm #2
EID:
Name:
There are ve questions on this exam, which are worth 20 points each. You must show all your work to
receive full credit. You have 50 minutes. You are permitted a non-programmable calculator and a single
8.5x11 double-sid
M348 (55775)
Midterm #1
Solutions
1
1
1. Let f (x) = x + 1 x and g(x) = 1+x .
(a) Show that g(x) has a xed point at p = 0 if f (x) has a root at p.
Solution.
Let p be a root of f so that
f (p) = p + 1
1
= 0.
p
Multiply through by p:
p2 + p 1 = p(p + 1) 1