Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

Appendix A
Old Exams
A.1 First Midterm, Fall 2003
P1 (7 pnts) Clearly state Taylor’s Theorem for f(x + h). Include all hypotheses, and state the
conditions on the variable that appears in the error term.
P2 (7 pnts) Write out the Lagrange Polynomial l 0 (x) for nodes x 0 , x 1 , . . . , x n . That is, write the
polynomial that has value 1 at x 0 and has value 0 at x 1 , x 2 , . . . , x n .
P3 (7 pnts) Write the Lagrange form of the polynomial of lowest degree that interpolates the
following values:
x k −1 1
y k 11 17
P4 (14 pnts) Use the method of Bisection for finding the zero of the function f(x) = x 3 − 3x 2 +
5x − 5 2 . Let a 0 = 0, and b 0 = 2. What are a 2 , b 2 ?
P5 (21 pnts) • Write out the iteration step of Newton’s Method for finding a zero of the function
f(x). Your iteration should look like
x k+1 = ??
• Write the Newton’s Method iteration for finding √ 5.
• Letting x 0 = 1, find the x 1 and x 2 for your iteration. Note that √ 5 ≈ 2.236.
P6 (7 pnts) Write out the iteration step of the Secant Method for finding a zero of the function
f(x). Your iteration should look like
x k+1 = ??
P7 (21 pnts) Consider the following approximation:
f ′ (x) ≈ f(x + h) − 5f(x) + 4f(x + h 2 )
3h
• Derive this approximation using Taylor’s Theorem.
• Assuming that f(x) has bounded derivatives, give the accuracy of the above approximation.
Your answer should be something like O ( h ?) .
• Let f(x) = x 2 . Approximate f ′ (0) with this approximation, using h = 1 3 .
P8 (21 pnts) Consider the data:
k 0 1 2
x k 3 1 −1
f(x k ) 5 15 9
145