436 MATH - Exam 2 SHOW ALL WORK
NAME: _________________________
1) a) Apply the Secant method (use newton2v program) for a system of two equations in two unknowns to solve the system
x2 - y2 = 1 and (x + 1)2 + (y - 1)2 = 9
for ALL of its solutions accurate to 10 decimal places. You should also give the graphs of these two curves clearly indicating the solution points. Clearly indicate your INTEGER starting points and the # of iterations you used. b) Give the iteration formula used in the 2 by 2 square system version of Newton's as well as Secant method for this problem.
2) a) Find by hand the intersection point of the 3 spheres
x 2 y 2 z 2 1 , ( x 1) 2 y 2 z 2 2 , x 2 ( y 1) 2 z 2 2 .
b) Give the Iteration formula for Newton's method for this 3 by 3 system of equations. Use your solution of part a as the starting point for Newton's method for this system and tell me what happens when computing the first Iteration. c) In the system version of Newton's method what happens the to Jacobian matrix if you use a solution as starting point? Explain your answer.
9
3) Let y = f(x) =
5 3
3 x 1 1 x 3 on the interval [-3, 5]. 3 x 5
a) Find the 5th Fourier series approximation F 5 ( x) to f(x) on this interval. b) What is the L2 error in approximating f(x) by F 5 ( x) on this interval? c) Give the graphs of both functions on this interval?
2
2 2 2
x 0 x
4) Let y = f(x) =
x 2
0 x 2 2
on the interval [0, 2 ].
a) Find the formulas for ALL of its Fourier coefficients using integration by parts twice. b) Now use the fact that f(x) = on this interval to show that
N
F (x) (its infinite order Fourier series) everywhere
2
1 2 1N
6
. Hint: Evaluate at x = 0.
2
c) Now use Parseval's identity and part b that to show that
N
0
[ f ( x)]2 dx
2 [ 2a 0 K 1
2 aK K 1
2 bK ]
1 4 1N
4
90
.
5) Describe the advantages and disadvantages between using Fourier series versus Taylor polynomials to approximate a function y = f(x) on an interval [a, b].