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CLS502S - CALCULUS 1 - 2ND OPP - JANUARY 2024 |
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nAm I BIA UnlVERSITY
OF science AnOTECHnOLOGY
Facultyof Health, Natural
ResourcesandApplied
Sciences
School of Natural and Applied
Sciences
Department of Mathematics,
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13Jacl<SonK."lujeuaStreet
Private B.:l~13388
Windhoek
NAMIBIA
T: •264 61 207 2913
E: msas@nust.na
W: www.nust.na
QUALIFICATION:BACHELOROFSCIENCEIN APPLIEDMATHEMATICASNDSTATISTICS
QUALIFICATIONCODE: 07BSAM
LEVEL:
5
COURSE:
CALCULUS1
COURSECODE:
CLSS02S
DATE:
JANUARY2024
SESSION:
1
DURATION:
3 HOURS
MARKS:
100
SECONDOPPORTUNITY/SUPPLEMENTARYEXAMINATION: QUESTION PAPER
EXAMINER:
MODERATOR:
Dr. David liyambo and Mrs. Yvonne Nkalle
Dr. Nega Chere
INSTRUCTIONS(add other relevant instructions):
1. Attempt all the questions in the booklet provided.
2. Please write neatly and legibly using a black or blue inked pen, and sketches must be done in
pencil.
3. Do not use the left side margin of the answer script. This must be allowed for the examiner.
4. No books, notes or other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
6. Show clearly all the steps used in the calculations.
PERMISSIBLEMATERIALS:
1. Non-programmable calculator without a cover.
ATTACHMENTS:
None
This paper consists of 3 pages including this front page
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Question 1.
= .
The functions
f,
g and hare
.
defined
by, J(x)
k(x) = 4x 2 - 3; x 0.
2x+ 1
·>
, g(x) = x-,, + 3, h(x) = 2x + a and
+5x+4
1.1 Find the domain off.
[6]
1.2 Given that (go h)(x) = 4x 2 - 8x + 7, where x-/- 0, calculate the value of a
[4]
1.3 Determine whether k- 1 exists. If it does, find it.
[9]
Question 2.
Find the following limits, if they exist.
2.1
I1. m
J4+7i-2
h
.
[6J
h--+0
2.2
lim
x--+2-
-xI-x2 --421
[6J
1
2.3
lim
x--+3
(3
-
)".
X-
[4J
2.4
lim
x--+oo
x 2 sin
(~4).x-
[8J
Question 3.
3.1 Use the definition (first principle) to find the derivative off (x) = y'x+l.
[8J
3.2 Find the equation of the tangent line to the graph off at the point where x = 3.
[4J
3.3 Find g' (x) for each of the following functions.
a) g(x) = cos2 (cosx)
[SJ
b) g(x) = 3xex
[4J
c) g(x) = sin (tan- 1(1nx))
[SJ
= = 3.4 If the equation x2y + xy2 6 determines a differentiable function f such that y f (x ), find the
equation ofthe normal line to the graph of this equation at the point P(2, 1).
[7J
Question 4.
Consider the function
f (x) = {
x -m
1- mx
ifx < 3;
ifx 3.
Find the value of m for which f is a continuous function at x = 3.
[9J
1
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Question 5.
Let f(x) = x:1i(2x + 7) and g(x) = 2x - 3x:2i.
5.1 Find the intervals on which f is increasing and on which it is decreasing, and hence state the local
extreme values off. If you answer is not a whole number, round it correct to 2 decimal places.
[9]
= 5.2 Find the intervals on which the graph of y g(x) is concave upwards and on which it is concave
downwards.
[6]
2