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Unformatted text preview: 8006A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY
SCHOOL OF MATHEMATICS AND STATISTICS MATH 1011
LIFE SCIENCES CALCULUS
June 2009 LECTURERS: E Carberry, R Hewlett and N Saunders
TIME ALLOWED: One and a half hours
N ame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SID , , _ , _ , _ _ , , _ _ _ Seat Number: This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination;
there are 30 questions; the questions are of equal value. 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; each question is worth 20 marks;
the marks for each part of each question are shown. Calculators will be supplied; no other calculators are permitted. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE ‘
EXAMINATION ROOM. PAGE 1 OF 16 8006A SEMESTER 1 2009 PAGE 7 OF 16 Extended Answer Section There are three questions in this section, each with a number of parts.
in the space provided below each part. If you need more space there are end of the examination paper. Write your answers
eztra pages at the 1. (a) Write down a formula for sinusoidal function f with mean value 3, amplitude 4, period 71' and f(0) = 3. {4 marks)
(b) For What values of :1: is the function f = stem—9”2 increasing? ( 5 marks) 8006A SEMESTER 1 2009 PAGE 8 OF 16 (c) A researcher believes that variables a: and y are related by an equation of the form
y : A331” for some constants A and b. After applying the transformation Y = lny
and X = lncr to experimental data it is found that Y = 0.2X — 0.7. What are the values of A and 6?. marks)
‘?
................................................................................................................................................. (d) A cylindrical compost bin With radius 7" metres and height h metres is to be con—
structed. Its total surface area, including base, lid and curved surface, is to be 24
square metres. By expressing the volume V as a function of 7", find the maximum possible value of V, correct to two decimal places. (8 marks) 8006A SEMESTER 1 2009 PAGE 9 OF 16 2. (a) Let f(x,y) = (x — x2)(y — 3/2), and let S be the square in the ($,y)~plane with
vertices (0,0), (1,0), (1,1) and (0,1).  Show that f (as, y) is zero on the ed ges of S and f(x, y) 2 0 for all points (as, y)
in the region enclosed by S. {4 marks) (ii) Let f and S be as in . Find the maximum value of f (
enclosed by S.  as, 3/) on the region (’7 marks) 8006A SEMESTER 1 2009 PAGE 10 OF 16 (b) Let S be the surface 2 = 11:2 + 2mg +3.7;2 — 233+ 63/ +4. Find all points on S at which
the tangent plane to S is horizontal. {6 marks)
(C) Differentiate xlnz: and use your answer to ﬁnd /ln1: 03:13. (3 marks) 8006A SEMESTER 1 2009 PAGE 11 OF 16 3. (a) An object is projected vertically upwards with initial velocity 196 metres per second. Let 210$) be its velocity after t seconds, and suppose that it has constant (negative)
vertical acceleration 7/05) 2 —9.8 metres per second per second. (2') Find a formula for v(t) and hence ﬁnd the value of t for Which 11(t) = O. (4 marks)
(ii) By integrating the velocity, ﬁnd the distance the object travels in the ﬁrst 20
seconds. {5’ mar/cs) 8006A SEMESTER 1 2009 PAGE 12 OF‘ 16 (b) You are given that the following formula is valid: 28in(%6) cos(k¢9) = sin((/c + — sin((k; — (You are not required to prove this formula.)
Use the formula above to ﬁnd a formula for 2 sin(%t9) (cos(6) + cos(26) + cos(36) +    + cos(n6)). (5 mar/ts)
(c) Use Part (b) and your calculator to evaluate
cos(1) + cos(2) + cos(3) +    + cos(50)
correct to three decimal places. {3 marks) 8006A SEMESTER 1 2009 PAGE 13 OF 16 (d) Find the indeﬁnite integral /cos(sin($)) cos(a:) d3: [2 marks) (ii) Find the indeﬁnite integral /Cos(sin(sin(a:))) cos(sin(x)) cos(:I:) dx
(3 marks)
The space below may be used if you need more space for your answers. ...
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This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.
 Three '11
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