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The solutions to the math 174 final exam held on december 5, 2005. The exam covers topics such as polynomial interpolation, error terms, differential equations, linear systems, and numerical integration. Students are required to show their work to receive credit.
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December 5, 2005
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p(x) = x
3
− 5 x
2
q(x) = x
2
interpolate the data
x 1 2 3
y 4 9 16
and explain why this does not violate the unique-
ness of the Lagrange interpolating polynomial.
′ (x) is approximated by the formula φ(h) such that
f
′
(x) = φ(h) + K 3 h
3
6
+....
Use the Richardson extrapolation idea to obtain a formula involving φ(h) and φ(h/2)
with error O(h
6 ).
{
x
′
= y + t
y
′ = x
with initial conditions x(0) = −1 and y(0) = 0 at t = 1/2 using one step of second
order Runge-Kutta,
1
= h
F (t,
2
= h
F (t + h,
1
x(t + h) ≈ x(t) +
1
2
x
. Approximate
∫ 4
0
f (x) dx by the trapezoid rule,
∫ b
a
f (x) dx ≈
h
[f (x 0 ) + 2f (x 1 ) +... + 2f (xn− 1 ) + f (xn)],
using partition points 0, 2, and 4. Repeat by using partition points 0, 1, 2, 3, and 4.
Now apply Romberg extrapolation,
R(i, j) = R(i, j − 1) +
j
− 1
[R(i, j − 1) − R(i − 1 , j − 1)],
R(i, 0) = trapezoid rule with 2
i equal subintervals,
to obtain a better approximation.
3 / 4 using one step of
Newton’s method,
x n+
= x n
f (x n
f
′ (x n
with x 0
