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Sta 2105 Calculus For Statistics Ii Question Paper
Sta 2105 Calculus For Statistics Ii
Course:Bachelor Of Science In Actuarial Science
Institution: Dedan Kimathi University Of Technology question papers
Exam Year:2014
DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
University Examinations 2013/2014
FIRST YEAR SPECIAL/SUPPLEMENTARY EXAMINATION FOR THE DEGREE
OF BACHELOR OF SCIENCE IN ACTUARIAL SCIENCE
STA 2105 Calculus for Statistics II
DATE: TH MARCH 2013 TIME: 2 HOURS
Instructions: Answer QUESTION ONE and any other TWO QUESTIONS.
QUESTION ONE (30 marks) (COMPULSORY)
(a). Find the equation of the tangent and normal line to the curve de ned implicitly
by the function (1 + x2y)3p
y = 4 at the point (1; 1). [6 marks]
(b). Prove that sinh??1 x = ln
h
x +
p
(x2 + 1)
i
[4 marks]
(c). Determine the asymptotes of the curve x(x2 + 2) = y(x2 ?? 5). [4 marks]
(d). Use Simpson''s rule to evaluate the de nite integral
2
Z 6
0
y2 dx, given correspond-
ing values of x and y as:
x 0 1 2 3 4 5 6
y 0 2 2.5 2.3 2 1.7 1.5
[4 marks]
(e). Evaluate the following integrals
(i)
Z 1
0
(tan??1 x)2
1 + x2 dx [4 marks]
(ii)
Z
loge y
p
y
dy [3 marks]
(f). Use partial fractions to solve
Z
4x2 + 1
x(2x ?? 1)2 dx
[5 marks]
QUESTION TWO (20 marks)
(a). Sketch the curve y2 = (x ?? 1)2(2 ?? x) indicating the main features. [8 marks]
(b). Evaluate the integral
Z
2
p
3 ?? 4x2
dx [4 marks]
1
(c). Express
x2 + 4x ?? 14
(x + 2)(x + 5)(x + 8)
in partial fractions. Hence or otherwise, evaluate
Z 4
1
x2 + 4x ?? 14
(x + 2)(x + 5)(x + 8)
dx
[8 marks]
QUESTION THREE (20 marks)
(a). Tabulate, to three decimal places, the values of the function
f(x) =
p
(1 + x2)
for values of x from 0 to 0.8 at intervals of 0.1. Use these values to estimate
Z 0:8
0
f(x)dx
(i). by the trapezium rule, using all the ordinates. [5 marks]
(ii). by Simpson''s rule, using only the ordinates at intervals of 0.2. [5 marks]
(b). Evaluate the following integrals
(i).
Z
1
1 + cos2 x
dx [4 marks]
(ii).
Z
5x + 7
x2 + 4x + 8
dx [6 marks]
QUESTION FOUR (20 marks)
(a). The parametric equations of a function are x = 2 cos3 , y = 2 sin3 . Find the
equation of the normal and tangent at the point for which =
4 . [8 marks]
(b). By rst putting the expression into partial fractions, or otherwise, nd the rst
and second derivatives of the function.
3x ?? 1
(4x ?? 1)(x + 5)
Find the coordinates of any maxima, minima and points of in
exion that the
function may have, and draw a rough sketch of its graph. [6 marks]
(c). Find the second derivative to the curve de ned implicitly by y4 +3x??x2 sin y = 3
at the point (1; 0). [6 marks]
QUESTION FIVE (20 marks)
(a). Using the substitution t = tan
x
2
, evaluate
Z
dx
5 + 4 cos x
(4 marks)
2
(b). Suppose that a community contains 15,000 people who are susceptible to Michaud''s
syndrome, a contagious disease. At time t = 0 the number N(t) of people who
have caught Michaud''s Syndrome is 5000 and is increasing then at the rate of 500
per day. Assuming that N0(t) is proportional to the product of the numbers of
those who have caught the disease and those who have not. How long will it take
for another 5,000 people to contract Michaud''s syndrome? (8 marks)
(c). Upon the birth of their rst child, a couple deposited Ksh.5,000 in a savings account
that pays 8% annual interest compounded continuously. The interest payment are
allowed to accumulate.
(i) How much will the account contain when the child is ready to go to University
at age 19? (3 marks)
(ii) Calculate the e ective annual interest rate. (2 marks)
(iii) How much do the parents need to invest now if they want their child to have
Ksh.150,000 by the time of joining University? (3 marks)
3
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