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CLS502S- CALCULUS 1- JAN 2020 |
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FACUOL F HTEAYLTH FIAMIBIA UNIVERSITY AND APPLIED SCIENCES
OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science; Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BAMS
LEVEL: 5
COURSE CODE: CLS502S
COURSE NAME: CALCULUS 1
SESSION: JANUARY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
Dr N. Chere and Mrs Y.Shaanika-Nkalle
MODERATOR:
Prof Gunter Heimbeck
INSTRUCTIONS
1. Answer 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 blue or black ink 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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SECTION A: [Short answer questions] 2[ - marks for each question]
QUESTION 1 [25]
1.1. Suppose that lim, f(x) = 12, lim, g(x) = —3.
x=
xXxo--
1.1.1. lim (/3f(+ %g()x) =
Xxo-
1.1.2. li® m ((g(X))° 2 +x) =————---—
Then find
1.1.3. lim (=e) 7
x2-2\\ f(x)
1.1.4. lim (2x + (f(9))?). = se
1.2. Determine the following derivatives.
1.2.1. 2 (sin (:)) inne
1.2.2, a= (e°S*) = COSX) — ___e
1.2.3. If y == In(sinIn(<j x), then a| ss
ee
1.3. Suppose that f and g are continuous functions such that g (4) = 2
and lim (2f(x) + 3g(x)) = 20. Then the value of f (4) = ----------------------
x-
1.4. The domain of the function f(x) = V4 — 9x? is equal to ---------------------
1.5.
lim
fS(ux)pp=os-e-—--a --f-u-n—c-t-i-o-—n
f has
the
property
that
for
all
real
numbers
x,
1-x?
<
f(x)
<
cosx.
Then
x70
SECTION B [Workout Problems]
QUESTION 2 [75]
2.1. Let f(x) = V2+x2. Then
2.1.1. find a formula for f~ +(x).
[5]
2.1.2. state the range of f~1.
[2]
2.2. Evaluate the following limits if it exists.
9394. lim =4 A2
[4]
x9—0o X24+2x341
2.2.2. lim Bet+3 2)
[6]
x2-1 Xt1
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2.2.3, limate8
(5]
x2
X-2
2.2.4. li.m x—X*-2x
[5]
2.3. Let f(x) = x? — x. Find f’(x) by using the limit definition of derivative.
[5]
2.4. Use the precise definition of limit to prove that lim (2x +3) =9.
[7]
x>
2.5. Use chain rule to find < ify = sin(In2x)
[5]
2.6. Ifx+y =2xy? Then
2.6.1. Use implicit differentiation to solve and express = in terms of x and y.
2.6.2. Use the result in (2.6.1) to find an equation of a tangent line to the curve x+y =2xy?
at (-1, -1).
[3]
2.7. Suppose f(x) =—2x? —x +3. Then
2.7.1. find (f~*)'(x)
[5]
2.7.2. use (3.6.2) to find (f~*)’(0)
[3]
2.8. Let f(x) =2x? — 3x? — 12x.
2.8.1. find the local maximum and local minimum value of f if there are any.
[5]
2.8.2. the intervals on which f is increasing and when where it is decreasing.
[4]
2.8.3. the open intervals on which the graph of f is concave upward and on which the graph
off is concave down ward.
[4]
2.8.4. the inflection point(s).
[2]
END OF EXAMINATION