Midterm Exam
Math 128A (Section 3) - Fall 2002
Problem 1) Can Newton method be used to solve f(x)=(x−3)1/2= 0 given an initial approximation
p0= 4? Explain why.
Problem 2) Consider function f(x) = sin(x) on the interval [0,1] on equally spaced nodes. If his the
spacing between the nodes Determine the size of hyou need so that the cubic Lagrange
interpolating polynomial approximates f(x) with an accuracy of 10−5, i.e. |E3(x)| ≤ 10−5for
any x∈[0,1].
Problem 3) Let g(x) = 3x4−8x3+ 6x2.
Part a) Show that p= 0 and ˆp= 1 are fixed points of the function g(x).
Part b) Assume that you used fixed point iteration with some initial approximation p0which
was sufficiently close to p= 0 so that the sequence of approximations p0,p1, p2, ... you
generated converges to the fixed point p= 0. Derive what will be the order of convergence
of this sequence.
Part c) Assume that you used fixed point iteration with some initial approximation ˆp0which
was sufficiently close to ˆp= 1 so that the sequence of approximations ˆp0,ˆp1,ˆp2, ... you
generated converges to the fixed point ˆp= 1. Derive what will be the order of convergence
of this sequence.
Problem 4) Show that any third-degree polynomial f(x) = a0+a1x+a2x2+a3∗x3is its own clamped cubic
spline on any closed interval [a, b]. Hint: use uniqueness of a clamped cubic spline interpolating
a function f(x) on a given set of nodes.
Problem 5) Suppose real numbers are represented with a 5-bit mantissa and a 3-bit characteristic.
Part a) How many positive real numbers such representations system contains?
Part b) What is the magnitude of a maximum number that can be represented with this system?