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.