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ADC801S - ADVANCED CALCULUS THEORY - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science (Hons)
QUALIFICATION CODE: 08BSHM
COURSE CODE:
ADC8015S
SESSION:
JUNE 2022
DURATION:
3 HOURS
in Applied Mathematics
LEVEL:
8
COURSE NAME: ADVANCED
PAPER:
THEORY
MARKS:
100
CALCULUS
FIRST
EXAMINER:
MODERATOR:
OPPORTUNITY
EXAMINATION QUESTION
DR. DSI ITYAMBO
PROF. OD MAKINDE
PAPER
INSTRUCTIONS
1. Attempt all the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in black or blue inked, and sketches must be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Including this front page)
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Question 1.
Suppose that the equation
y. Fi.nd, aOnZ and BOazg’
re¥*
—
2yy e™*
+
3ze7¥
=
1
defines
z
as
an
implPp icit
function
of x
and
[10]
Question 2.
Find the local extreme values and the saddle points of the function f(x,y) = 4+2°+ y? — 3ay.
[14]
Question 3.
Use the method of Lagrange multipliers to find the minimum and maximum values of the function
f(z, y) = 2x? + y? +2, where x and y lie on the ellipse C' given by 2? + 4y? -4=0.
[15]
Question 4.
Let F = (e*Iny)i+ (< + sin z) j+ (ycosz)k.
a) Determine whether F is a conservative vector field. If it is, find a potential function for F.
b) Evaluate {, F-dr, where C is the curve given by r(¢) = 2costi+2 sin tj+5k, where 0 < t < 2r.
[19,5]
Question 5.
Let f be a differentiable function of x, y and z, and let F(z,y,z) = P(z,y,z)i+ Q(z, y,z)j+
R(x,y, z)k, where P, Q and R are differentiable functions of x, y and z. Prove that div(fF) =
fdivF + F- Vf.
[10]
Question 6.
Evaluate f, C zy* dS, where C is the upper half of the circle x? + y? = 16 in the counter clockwise
direction.
[9]
Question 7.
Use Green’s Theorem to evaluate ¢ y° dz — x° dy, where C is the positively oriented circle of
radius 2 centred at the origin.
Cc
[10]
Question 8.
Evaluate the integral /I/ 8xyzdV over the box B = [2,3] x [1,2] x [0,1].
[8]
B