Lecturer: Date Approved by: Date

26/4/2023 26/04/2023

.....................................................................................................
Year 2022-2023 Semester 2
Final test Date 30/05/2023
Subject Numerical methods
HCM University of Technology Subject MT1009
Faculty of Applied Science code
Duration 100 mins Exam code 1235
Note
- The question sheet has 2 parts: multiple choice(16 questions) and essay (1 question)
- Documents are allow to used, except laptop, tablet, mobile phone.
- Unless stated otherwise, round all answers to 4 decimal digits.
- Each wrong answer will get a penalty of 0.1 point.

Full name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ID: . . . . . . . . . . . . . . . . . . . . . . . . . . . Invigilator 1: . . . . . . . . . . . . . . . . . . . . . . .
Class: . . . . . . . . . . . . . . . . . . . . . . . . . Invigilator 1: . . . . . . . . . . . . . . . . . . . . . . .

Multiple choice part

(
2.6 + b0 (x − 1) + d0 (x − 1)3 if x ∈ [1, 2.6]
Let S(x) = 2 3
be the natural cubic spline
5.2 + b1 (x − 2.6) + 4(x − 2.6) + d1 (x − 2.6) if x ∈ [2.6, 4.6]
that interpolates a function f (x). Answer the questions 1 →5(L.O.1, L.O.2, L.O.3) .

1. Determine b0 .
A b0 ≈ −0.5083 B b0 ≈ −0.4432 C b0 ≈ −0.2019 D b0 ≈ 0.3330 E b0 ≈ −0.3172

2. Compute the value y1 = f (2.6).

A y1 = 5.2207 B y1 = 5.2625 C y1 = 5.2946 D y1 = 5.2000 E y1 = 5.2245

3. Using polynomial interpolation at nodes x0 = 1, x1 = 2.6, x2 = 4.6, find approximately the value
f (1.4).
A 1.9774 B 1.9700 C 1.9716 D 1.9782 E 1.9709

4. Compute the divided difference f [1, 2.6].

A 1.6534 B 1.6250 C 1.7117 D 1.7230 E 1.6903

5. Apply the least square method to construct the parabola y = A + Bx + Cx2 that fits the data best
at x0 = 1, x1 = 2.6, x2 = 4.6. What is A?
A 7.9844 B 7.9113 C 7.9980 D 7.9702 E 7.9083

A parametric curve is givenpby the equations x(t) = et + t, y(t) = et , t ∈ [2, 3]. We want to find the
length L of this curve(L = x′ (t)2 + y ′ (t)2 dt) Answer the questions 6 →8(L.O.1, L.O.2, L.O.3)
.

6. Using the composite trapezoidal formula with the step size h = 0.25, then the approximate value
of L is:
A 18.7782 B 18.7706 C 18.7778 D 18.7752 E 18.7706
Final test - MT1009 1235 Semester II

7. Using the composite Simpson formula with the step size h = 0.25, then the approximate value of L
is:
A 18.6794 B 18.6792 C 18.6784 D 18.6775 E 18.6869
8. With the method Gauss quadrature, find L for the case n = 2(two points).
A 18.6744 B 18.6731 C 18.6797 D 18.6734 E 18.6816
dN
The growth of a population halibut living in the Pacific is modeled by the logistic equation =
  dt
N
rN 1 − with N (t): the total mass of the members of the population measured in years, K =
K
8 × 107 (kg), r = 0.55(per year). Suppose that at the beginning of 2000, the population is 2 × 107 (kg).
Answer the questions 9 → 12(L.O.1, L.O.2, L.O.3) .
9. With the Euler method and the step size 6 months, find the total mass(kg) of the population at
the beginning of 2003.
A 5.0134×107 B 5.0115×107 C 5.0201×107 D 5.0201×107 E 5.0180×107
10. With the modified Euler method and the step size 6 months, find the total mass(kg) of the
population at the beginning of 2001.
A 2.9308×107 B 2.9410×107 C 2.9254×107 D 2.9352×107 E 2.9447×107
11. From the initial moment, with Runge-Kutta method and the step size 6 months, find the total
mass(kg) of the population after 6 months.
A 2.4400×107 B 2.4453×107 C 2.4417×107 D 2.4598×107 E 2.4466×107
12. With Runge-Kutta method and the step size 6 months, find the rate change of the population after
6 months.
A 0.9480×107 B 0.9358×107 C 0.9370×107 D 0.9400×107 E 0.9327×107
Consider a habitat with 2 species: wolves(predator) and rabbits(prey), let w, r be the numbers of wolves
and rabbits, respectively, their relationship with respect to time t(measured in month) is given by the
following ODE system:
dr dw
= kr − arw = −cw + brw.
dt dt
Given that k = 0.05, a = 0.001, c = 0.02, b = 0.00002 . Answer the questions 13 → 14(L.O.1,
L.O.2, L.O.3) .
13. Taking the step size 7.5(days) on 1 month(30 days), suppose that at some initial moment(t = 0), the
number of wolves is 49 while of rabbits is 995 using the modified Euler method to find the number
of wolves after the first half of month from the initial moment.(Round to the nearest integer).
A 57 B 52 C 49 D 56 E 51
14. Using the information about initial data and step size in question 13, how will rabbit change after
the first half of month from the initial moment?(Round to the nearest integer)
A Increasing at a rate of 4 B Increasing at a rate of 1 C Increasing at a rate of 5
D All the other answers are wrong E Increasing at a rate of 6
 
5 −6
Given the matrix . Answer the questions 15 → 16(L.O.1, L.O.2, L.O.3) .
0 −1
15. When we apply the power method to approximately an eigenvalue of A, we produce a sequence of
real numbers {µk }k∈N , then, to which point below this sequence converges as k → ∞?
A −4 B 8 C 5 D −1 E −3
Full name: ID: Pages 2 / 4
Final test - MT1009 1235 Semester II

16. Also with the power method, after k iterations we produce a vector Xk , then, by which direction
below Xk is?
         
2 1 7 4 3
A . B . C . D . E .
2 0 5 10 4

The essay part

Let m, n be the last two digits in your ID students, for instance, your ID is 212456 then m =
2m + 3n + 4
5, n = 6. Define M := .
10
17. Consider the following boundary value problem:

(M + 1)y ′′ (x) + sin xy ′ (x) + (x2 + 1)y = cos x.

Taking the step size h = 0.25 on [0, 1], knowing that y(0) = M and y(1) = 3.
(a) (0.5 point) Verify that, with the finite difference method, we can find the value of y at other
nodes by solving a linear system.(Prove and write the results for matrix A and B when we
solve the system Ay = B)
(b) (1 point) Write the results for solution y.
(c) (0.5 point) Evaluate y ′′ at all nodes.

———————-THE END————————————–

Solution of the essay part

Full name: ID: Pages 3 / 4

 Final test - MT1009 1235 Semester II

Key answers.

Question Key
1 E
2 E
3 E
4 B
5 A
6 B
7 C
8 E
9 C
10 A
11 A
12 C
13 E
14 D
15 B
16 D
(L.O.1.1, L.O.1.2, L.O.1.3, L.O.2.1, L.O.2.3) ??

Full name: ID: Pages 4 / 4