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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:2013

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: 09TH JULY 2013 TIME: 11.00AM?? 1.00PM
Instructions: Answer QUESTION ONE and any other TWO QUESTIONS.
QUESTION ONE (30 MARKS) (COMPULSORY)
(a). Find the asymptotes of the curve f (x) =
2x
1 ?? x. (2 marks)
(b). Evaluate the following integrals
(i)
Z
3x2 ex3dx. (3 marks)
(ii)
Z
x2 ln x dx. (4 marks)
(c). If z = 2 + 3i nd jzj and arg z. (4 marks)
(d). Find all the rst and second partial derivatives of the function f (x, y) = x cos y+yex.
(6 marks)
(e). Find the indenite integral
Z 7x + 1
(x + 3)(x ?? 1)
dx. (5 marks)
(f). Oil is leaking out of a ruptured tanker at the rate of r(t) = 50e??0.02t thousand liters
per minute.
(i) At what rate, in liters per minute, is oil leaking out at t = 0? At t = 60?
(ii) How many liters leak out during the rst hour? (6 marks)
QUESTION TWO (20 marks)
(a) Throughout much of the 20th century, the yearly consumption of electricity in the
US increased exponentially at a continuous rate of 7% per year. Assume this trend
continues and that the electrical energy consumed in 1900 was 1.4 million megawatt-
hours.
1
(i). Write an expression for yearly electricity consumption as a function of time, t,
in years since 1900. (2 marks)
(ii). Find the average yearly electrical consumption throughout the 20th century.
(3 marks)
(iii). During what year was electrical consumption closest to the average for the cen-
tury?
(3 marks)
(iv). Without doing the calculation for part (iii), how could you have predicted which
half of the century the answer would be in? (2 marks)
QUESTION THREE (20 marks)
(a) Express
23 ?? x
(x ?? 5)(x + 4)
as the sum of its partial fractions. Hence nd
Z 23 ?? x
(x ?? 5)(x + 4)
dx.
(6 marks)
(b) Use De Moivre''s Theorem to express cos 4q and sin 4q in terms of cos q and sin q.
(4 marks)
(c) Evaluate the integral
Z dx
p
16 + 6x ?? x2
. (6 marks)
(d) Find
¶z
¶x if the equation yz ?? ln z = x + y (implicity), where z = z(x, y). (4 marks)
QUESTION FOUR (20 marks)
(a) Sketch the curve of the polynomial f (x) = x5 +2x4 + x3 indicating its main features.
(8 marks)
(b) Approximate
Z 2
0
x
1 + x2 dx using the Trapezoidal Rule with n = 4 sub-intervals.
(6 marks)
(c) For what value of x does the function y = sinh x +2 cosh x have its minimum value?
(6 marks)
QUESTION FIVE (20 marks)
(a) Evaluate the following integrals
(i)
Z 4
2
3x2 + 24
x3 + x2 + 1
dx. (4 marks)
(ii)
Z 2
0
x ex2 dx. (3 marks)
(iii)
Z p
9 ?? 4x2 dx. (5 marks)
(b) If z1 = ??3 + 5i and z2 = 2 ?? 3i nd the complex numbers (i) z1 + z2, (ii) z1 ?? z2
(iii) jz1z2j (iv) z1/z2. (8 marks)
2

Sta 2105 Calculus For Statistics Ii Question Paper

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