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Unformatted text preview: 8003A SEMESTER 2 2009 THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE MATH1003
INTEGRAL CALCULUS AND MODELLING November 2009 TIME ALLOWED: One and a half hours This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination;
there are 25 questions; the questions are of equal value;
all questions may be attempted. Answers to the Multiple Choice questions must be entered on
the Multiple Choice Answer Sheet. The Extended Answer Section is worth 50% of the total examination;
there are 3 questions; the questions are of equal value;
all questions may be attempted;
working must be shown. Calculators will be supplied; no other calculators are permitted.
There is a table of integrals after the last question in this booklet. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE
EXAMINATION ROOM. PAGE 1 OF 28 LECTURERS: F. Cirstea, C. Cosgrove, A. Crisp, T. Gagen, C. Macaskill MARKEE’S USE
ONLY 8003A SEMESTER 2 2009 PAGE 16 OF 28 Extended Answer Section There are three questions in this section, each with a number of parts. Write your answers
in the space provided below each part. If you need more space there are extra pages at the
end of the examination paper. MARKS 1 1. F' d th t'alf t' d 't' f—————
(a) in e par 1 rac 1011 ecompos1 1011 o (a: _ BXI + 1) , and hence evaluate 2 1
/1 (a: — 3)(z + 1) d” (b) Determine the general solution of the diﬁerential equation, (0) Find the volume of the solid that is generated when the area enclosed by the curve
y = sin(2m) and the w—axis on the interval [0, is rotated about the m—axis.
[Use the disk method] 3 More space is available on the next page. 8003A SEMESTER 2 2009 PAGE 17' OF 28 More space is available on the next page. 8003A SEMESTER 2 2009 PAGE 18 OF 28 More space is available on the next page. 8003A SEMESTER 2 2009 PAGE 19 OF 28 Question 2 starts on the next page. 8003A SEMESTER 2 2009 PAGE 20 OF 28 MARKS
2. Let n be an integer greater than 1. We denote by n! the product of all positive integers
less than or equal to n (that is, n! = 1 X 2 X >< (a) Draw the graph of the function f = lnm on the interval [1, 2
(b) Solve the indeﬁnite integral / lnm (12: by using integration by parts. 2 (0) Approximate the area under the graph of f on the interval [1, n] by an upper Rie
mann sum with equal subintervals of length l and show that 4 TL
lnnl>f lnzclm.
1 (d) Hence show that for every integer n > 1, we have 2 n TL
nl>e(——) .
e More space is available on the next page. 8003A SEMESTER 2 2009 PAGE 21 OF 28 More space is available on the next page. 8003A SEMESTER 2 2009 PAGE 22 OF 28 Question 3 starts on the next page. 8003A SEMESTER 2 2009 PAGE 23 OF 28 3. (a) Find a particular solution yp(:r) of the differential equation, d2?! 5
E + 43/ = 24:13 ,
by postulating that at least one such solution takes the form of an odd polynomial of degree ﬁve. (b) Show that, in the notation of part (a), w = y — yp(x) satisﬁes a homogeneous
differential equation of the second order, and deduce the general solution of the
differential equation in part (a). (c) [Harden] Consider the second—order inhomogeneous linear differential equation,
ﬂ+PWW+QWM=RWl BM Let yo be any nonzero solution of the corresponding homogeneous equation, that
is, yo(:r) satisﬁes %+H®%+M®m=0 9% (i) Deﬁne the auxiliary function Mr) 2 yoy’ — gay. Find a ﬁrst—order differential
equation satisﬁed by Find the general solution of the differential equation satisﬁed by W
(@ Hence, or otherwise, obtain a ﬁrst—order linear differential equation in stan—
dard form for the original variable y having one constant of integration in its
coefﬁcients. [This equation is an example of a “ﬁrst integral” of equation (3.1); do not solve it.] More space is available on the next page. MARKS 8003A SEMESTER 2 2009 PAGE 28 OF 28 Table of Standard Integrals $n+1
1./$”dz= +C’ (n7é—1) 9./se02$dz=tan$+0
n+1
dz: 2
2. —=1n93]+0 10. cosec asdaz=—cotz+C
a:
3. /e$dz=em+0 11. /secxdx=lnseca:+tana:+0
4. /sinxdm=—cosx+0 12. /coseca3da:zlnlcosecm—cotasl+C’
5. foosxdm=sinm+0 13. f5inhmdm=coshas+0
6. ftanmdm=—lnlcosm[+0 14. /Coshwd$=sinh$+0 7 t d —1 ' 0 15 L—sin—1(£)+C (mm)
. co 3: a:— nlsmm‘ +  m— a 8./ d5” 21—tan“1(E)l—C' 16./ d2: =—1—ln\a+$‘+0 a2+$2 a a aZ—mz 2a a—z =sinh_1(£> +0 = h1<m+W> +6” CL 17 /__d_$_
' V$2+CL2
= cosh—1(5) +C’ =ln(a:+\/932 ~a2) +0” (3: > a) a lgfi
"m Linearity: /()\f($) + #9(CC)) dz: = A / f(a:) da: + p/g(:1:) d3: Integration by substitution: / do: = / f(u)du Integration by parts: /f(:r;)g’(cc) d3: = f(ac)g(x) — /f’(m)g(a:) d1; End of Extended Answer Section THIS IS THE LAST PAGE OF THE QUESTION PAPER. ...
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