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

148 APPENDIX A. OLD EXAMS
The normal equations should involve the vector c = [c 0 c 1 c 2 ] ⊤ , which uniquely determines a
function in F.
P10 (9 pnts) Consider the following approximation:
f ′′ (x) ≈
3f(x − h) − 4f(x) + f(x + 3h)
6h 2
• 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 ?) .
P11 (11 pnts) Consider the ODE:
{ x ′ (t) = x(t)g(t)
x(0) = 1
• Use the Fundamental Theorem of Calculus to write a step update of the form:
x(t + h) = x(t) +
∫ t+h
• Approximate the integral using the Trapezoid Rule to get an explicit step update of the
form
x(t + h) ← Q(t, x(t), h)
• Suppose that g(t) ≥ m > 0 for all t. Do you see any problem with this stepping method
if h > 2/m?
Hint: the actual solution to the ODE is
P12 (9 pnts) Consider the ODE:
x(t) = eR t
0 g(r) dr .
{
x ′ (t) = e x(t) + sin (x(t))
x(0) = 0
Write out the Taylor’s series method of order three to approximate x(t + h) given x(t). It is
permissible to write the update as a small program like:
x ′ (t) ← Q 0 (x(t))
x ′′ (t) ← Q 1 (x(t), x ′ (t))
.
t
??
x(t + h) ← R(x(t), x ′ (t), x ′′ (t), . . .)
rather than write the update as a monstrous equation of the form
P13 (15 pnts) Consider the data:
x(t + h) ← M(x(t)).
k 0 1 2
x k −0.5 0 0.5
f(x k ) 5 15 9
• Construct the divided differences table for the data.