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CLS502S - CALCULUS 1 - 1ST OPP - NOV 2022 |
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nAm IBl A un IVERSITY
OF SCIEnCE AnDTECHnOLOGY
FACULTY OF HEALTH, NATURAL RESOURCES AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science; Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BOSC; 07BSAM LEVEL:
5
COURSE CODE:
CLS502S
COURSE CODE: CALCULUS 1
SESSION:
NOVEMBER 2022 PAPER:
THEORY
DURATION:
3 HOURS
MARKS:
100
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER:
DR. DSI IIYAMBO
MODERATOR:
DR. N CHERE
INSTRUCTIONS
l. 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 3 PAGES (Including this front page)
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Question 1.
The functions f, g and hare defined by, f(x) = ---;:2:x=+==1==, g(x) = x 2 +3 and h(x) = 2x+a.
+5x+4
a) Find the domain off.
[6]
b) Given that (go h)(x) = 4x 2 - 8x + 7, where x =J0, calculate the value of a.
[5]
Question 2.
2.1 Find the following limits, if they exist.
a)
11. 111
v'4+/1-
h
2
.
[7]
b)
lim
2
xIx --
4
21
[6]
c) lim (ex + x) ¼
[8]
.
1
d) 11111(3 - X ) 2 .
[4]
2.2 Using the Precise definition (the c - 8 method), prove that lim (14 - 5x) = 29.
[9]
Question 3.
a) Use the definition (first principle) to find the derivative of f(x) = Jx+T.
[10]
b) Find the equation of the tangent line to the graph off at the point where x = 3.
[5]
c) Find g1( x) for each of the following functions.
(i) g(x) = cos2 (cosx)
[5]
(ii) g(x) = 3xex
[4]
Question 4.
Consider the function f (x) = X - m if X < 3;
{ 1 - mx if x 2: 3.
a) Find the value of m for which f is a continuous function at x = 3.
[9]
b) With the value of m you found in a), is f differentiable at x = 3 or not? Justify your answer.
[5]
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Question 5.
Let f(x) = X31 (2x + 7) and g(x) = 2x - 3x32.
a) 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.
b) Find the intervals on which the graph of y = g(x) is concave upwards and on which it is
concave downwards.
[7]
END OF EXAMINATION QUESTION PAPER
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