MA/CS 371 Exam I, February 20, 1996
Directions: This examination is closed book and notes. You may
have only one 8.5 (inches) by 11 (inches) sheet of paper containing
any review material (front and back) in addition to the examination
sheet and solution sheets.
To receive credit, you must show all your work. If you find it necessary
to make assumptions, please explicitly state them. You have 75 minutes to
complete this examination. Show all your work on the solution sheets
provided (you may keep the exam sheet) and make sure all the solution sheets
remain stapled together before handing them in.
- [10 Pts] Convert the base 10 floating-point number
to base 2 (i.e. to binary form).
- [10 Pts] Suppose you have a decimal machine with 4 decimal digits
allocated to the mantissa. Also, assume that such a computer
correctly rounds to the nearest machine-representable number.
What positive real numbers x have the property
fl(1.0+x) = 1.0?
- [10 Pts] Show how log x - log (1/x) can be computed
without a serious loss of significant figures. Note: the logarithm
base does not matter.
- [20 Pts] Suppose you apply Newton's method to solve the nonlinear
x + 3x = 6x - 1 .
- (a) Compute the first two Newton iterates which approximate
a solution to the nonlinear equation using the initial guess
x = 0 .
- (b) Would Newton's method converge to a solution of the nonlinear
equation if the initial guess is
x = 1 ?
Explain why or why not.
Turn page over ...
- [15 Pts] Using Lagrange interpolation, determine the polynomial
of least degree that assumes the following table of values.
- [15 Pts] How accurately can one compute cos x by
quadratic interpolation on the closed interval [0,1]
using a table of known values (equally) separated by
- [10 Pts] Compute an approximation to the definite integral
using lower sums with stepsize h=1/2.
- [10 Pts] Complete the following Matlab function trap.m which
could be used to approximate the definite integral of an arbitrary function
f(x) on the closed interval [a,b] using the
Trapezoid Rule with constant stepsize h. You may
assume the existence of a Matlab function func.m defined
by the declaration
function [f] = func(x),
which can be used to evaluate the function f(x) at any real
[I] = trap(a,b,n)
% a is the left endpoint of the interval
% b is the right endpoint of the interval
% n is the no. of subintervals in the mesh
% I is the approximation to the definite
Total points possible on this exam : 100 Points
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