Download 9 Problems on Maximum Absolute Error - Examination 2 | MATH 375 and more Exams Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Millersville University Name
Department of Mathematics
MATH 375, Numerical Analysis, Test 2
November 09, 2006
Please answer the following questions. Your answers will be evaluated on their correctness, com-
pleteness, and use of mathematical concepts we have covered. Please show all work and write out
your work neatly. Answers without supporting work will receive no credit. The point values of
the problems are listed in parentheses. You may use your textbook, calculator, and notes. Un-
less otherwise indicated all numerical approximations should be carried out to at least six decimal
places.
- (4 points each) Consider the function
f (x) = e
− 2 x
(a) Use the 3–point centered difference approximation formula for f
′ (x) with h = 0.05 to
approximate f
′ (0.7).
(b) Find the maximum absolute error in the approximation calculated in part (1a).
(c) Approximate f
′′ (0.3) using h = 0.01.
(d) Find the maximum absolute error in the approximation calculated in part (1c).
- (4 points each) Consider the following definite integral.
0
(x
2
Note that
d
3
dx^3
(x
2
= π
3 (x
2
- 3x + 5) sin(πx) − 6 π sin(πx) − 3 π
2 (2x + 3) cos(πx).
(a) Approximate the definite integral using the Trapezoidal rule.
(b) Determine the maximum absolute error for the approximation in part (3a).
(c) Approximate the definite integral using Simpson’s rule.
(d) Determine the maximum absolute error for the approximation in part (3c).
(e) Approximate the definite integral using the Midpoint rule.
(f) Determine the maximum absolute error for the approximation in part (3e).
- (7 points each) Use Gaussian Quadrature with the indicated value of n to approximate the
following definite integrals.
(a)
− 1
e
√ x^2 + dx, n = 3
(b)
0
x
3
x^2 + 1
dx, n = 2
- (10 points) Determine the LU factorization of matrix A where
A =
- (5 points) Find all numbers c such that the following matrix is positive definite.
0 1 c
