MATH 128A-2 SUMMER 2009: MIDTERM 2 PRACTICE PROBLEMS
Questions
(1) Fill in the table as accurately as possible using two-point formulas (including the centered difference
formula) for f0:
f(−1) = 6 f(0) = 1 f(1) = 2
f0(−1) = f0(0) = f0(1) =
(2) Do you remember the definitions of the following quadrature methods? Newton-Cotes, composite
trapezoidal, Romberg, adaptive Simpson, Gaussian.
(3) Find coefficients a,b, and cso that the following quadrature rule has the best possible degree of
precision:
Z1
0
f(x)dx ≈af(−1) + bf(0) + cf (+1).
What is the resulting rule’s degree of precision?
(4) Use an easy substitution to transform the improper integral R∞
1
1
x2+1 dx into an integral of the
form R1
0f(u)du. Transform the result into an integral which can be estimated with the following
quadrature rule:
Z1
−1
g(t)dt ≈f(−1) + f(+1).
(5) Use Euler’s method with h= 1 to estimate the solution to y0= 2t+y/t, 1 ≤t≤3, y(1) = 1.
(6) Find a Lipschitz constant Lfor the above problem.
(7) Find y00 in terms of tand (if necessary) y, and compute an upper bound Mon |y00 (t)|.
(8) Find a bound on the local truncation error τwhen using Euler’s method to solve this problem.
(9) Find a bound for the absolute error in estimating y(3), using the error bound hM
2L(eL(ti−t0)−1).
(10) Write down a step (wi+1 as a function of ti,wi,h) of the second-order Taylor method for solving
the IVP from problem 5.
(11) Based on your answers to problem 5, estimate y(1.5) based on y(1), y0(1), y(2), and y0(2).
(12) Express the following Runge-Kutta method in one line (i.e., as a single formula for wi+1):
k1=hf(ti, wi)
k2=hf(ti+h, wi+k1)
wi+1 =wi+k2
(13) What is the order of the Runge-Kutta method in problem 12?
(14) (Skip if confused.) Consider Romberg integration with the following notation:
•h1=h, h2=h, . . . , hi=h/2i−1, . . .;
•Ri,1is the composite trapezoidal estimate of Rb
af(x)dx with step-size hi;
•Ri+1,j+1 is the value of Rb
af(x)dx extrapolated from Ri,j and Ri+1,j , assuming Ri,j has error
O(h2j
i).
Show that Ri+1,j+1 is equal to Ri+1,j +h2
i
h2
i−j−h2
i
(Ri+1,j −Ri,j).
Date: Monday 7/26.
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