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CLS502S - CALCULUS 1 - 2ND OPP - JULY 2023 |
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n Am I BI A u n IVER s I TY
0 F SCIEnCE An D TECHn OLOGY
FACULTYOF HEALTH,NATURALRESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS,STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: Bachelor of science; Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BSAM
LEVEL: 5
COURSECODE: CLS502S
COURSENAME: CALCULUS1
SESSION:JULY 2023
DURATION: 3 HOURS
PAPER:THEORY
MARKS: 100
SUPPLEMENTARY/SECONDOPPORTUNITYEXAMINATION QUESTION PAPER
EXAMINER
Mrs. H.YNKALLE
MODERATOR:
Dr. N. CHERE
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.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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Question 1 [2 Marks]
= Consider the relation P {(1,7), (-1,7), (3,9), (1,3)}. Is Pa function? Justify your answer.
Question 2 [6 Marks]
= Show that h.m -1--cosx 0.
X
Question 3 [8 Marks]
F.In d x1I.-m-+--a--.
Question 4 [S Marks]
= Find the average rate of change of the function f(x) x2 + 4x over the interval [-6, 4).
Question 5 [S Marks]
= = Find the instantaneous rate of change at x b for the function f(x) x2 + 2x.
Question 6 [8 Marks]
= Consider the function f(x) erx . Determine the values of r so that f satisfies the equation
f" (x) + f' (x) - 6f(x) = 0.
Question 7 [S Marks]
= Find the equation of the tangent line to the graph of the function f(x)
(1,1).
Question 8 [7 Marks]
= Differentiate f(x)
24
6x+xS
from
first
principle.
Question 9 [3 Marks]
= Let f(x) c. Find f' (x) using limit definition of derivative.
Question 10 [S Marks]
= Find the range of f(x)
- x2.
Question 11 [4 Marks]
= = Let f(z) In z. Find f"' (z) at z 2.
at the point
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Question 12 [4 Marks]
= Find :~ by using implicit differentiation of xy 1.
Question 13 [5 Marks]
Use logarithmic differentiation to find ddx (x,/3).
Question 14 [2, 2 Marks]
Investigate whether the following functions are odd or even.
(a) f(x) = x3 •
= (b) f(x) cos x.
Question 15 [9, 6 Marks]
= Let f(x) ½x3 + x2 - 15x - 9. Use detailed sign tables in answering the following
questions.
(a) Find the intervals in which f is increasing or decreasing.
= (b) Find the intervals in which the graph of y f(x) is concave upward or downward.
Question 16 [8 Marks]
Air is escaping from a spherical balloon at the rate of 2cm 3 per minute. How fast is the
radius shrinking when the volume is 36rr cm 3 ?
Question 17 [6 Marks]
Find the rate of change of the area A, of the circle with respect to its circumference
C,
.
1. e
dA
de·
End of supplementary/second opportunity examination question paper
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