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King Fahd University of Petroleum & Minerals Department of ......Name. Instructor: Mobiles are not...
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King Fahd University of Petroleum & MineralsDepartment of Mathematics & Statistics
Math 321 Major Exam 2The First Semester of 2017-2018 (171)
Time Allowed: 120 Minutes
Name: ID#:
Instructor: Sec #: Serial #:
• Mobiles are not allowed in this exam.
• Write all steps clear.
Question # Marks Maximum Marks
1 15
2 20
3 20
4 10
5 10
6 25
Total 100
MATH 321 EXAM 1 (Term 171) Page 2 of 8
Q:1 (15) Given a function f defined on [a, b] and a set of nodes
a = x0 < x1 < x2 < x3 = b. State all the conditions that need to be satisfied so that
S will be a quadratic spline interpolant for f .
Q:2 (7+7+3+3) Let f(x) = x ln x and x0 = 1.8, x1 = 1.9, x2 = 2.0, x3 = 2.1, x4 = 2.2.
Compute f ′(2.0) using three–points end–point formula and three-points mid point formula.
Evaluate absolute error for each formula.
Q:3 (6+14) Show that error for the Simpson’s rule is Zero for all polynomials of degree 3 or less.
Use Simpson’s rule to evaluate
5π6∫
π6
sin2(x) dx and compute a bound for
the theoretical error.
Q:4 (10) Write composite Simpson’s rule to evaluateb∫
a
f(x) dx for n = 4.
Q:5 (10) Show that the initial value problem
y′ = t + cos(t2y), 0 ≤ t ≤ 3, y(0) = 0
has a unique solution.
Q:6 (10+15) Consider the IVP y′ =sin(3ty)
t2, 1 ≤ t ≤ 4, y(1) =
1
2.
(a) Show that the IVP is a well–posed problem.
(a) Use Euler’s method to approximate the solution in part (a) with h = 0.5.