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Unformatted text preview: 8014 SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE MATH 1 111
INTRODUCTION TO CALCULUS June 2009 LECTURER: C Cresswell TIME ALLOWED: Two hours THIS EXAMINATION CONSISTS OF 6 PAGES, NUMBERED FROM 1 TO 6. THERE ARE 10 QUESTIONS, NUMBERED FROM 1 TO 10. All questions may be attempted. The paper will be marked out of 100; the marks allocated for each question are shown.
A formula sheet is provided on the last two pages. Calculators Will be supplied; no other calculators are permitted. PAGE 1 OF‘ 6 8014 SEMESTER 1 2009 PAGE 2 OF‘ 6 Answer these questions in the answer book(s) provided.
Ask for extra books if you need them. 1. (a) Solve 2(11: + 3) — 5(a: — 1) = 3(2m + 1) for 11:. (1 mark)
—— 2 — 1
(b) Express :1: 2 3 — a: 5 as a single fraction in its simplest form. {1 mark)
(0) Find the equation of the line with slope 2 that passes through the point (0,7).
(I mark)
((1) What is the natural domain of the function f (:5) = x 1 2? (1 mark)
(e) What is the range of the function f (9) = sin(6)? (1 mark)
(f) Factorise x2 + x — 6. {1 mark)
(g) Solve ext1 = 1 for x. (1 mark)
. . ea:(e:z)4
(h) Simplify e2“ . (1 mark)
(i) Find the derivative of f(a:) = 7r. (1 mark)
(3') Let f(a:) = x2 + 2 and g(x) = ﬂ. Find g(f(:n)). (1 mark)
2. (3.) Find the derivatives of the following functions.
(1') y = x4 + sinx. (2 marks)
(ii) 3) = (238 + 21:)101. (2 marks)
(iii) 3) 2 e12” + Inst. (2 marks)
(2'21) y = 382(33 + e”)9. (2 marks)
2:2 _ . d
(b) Let f (at)  m Find one value of a: for which d5; — 0. {4 marks)
3. Find the following indeﬁnite integrals.
(a) /2x2dsc. (I mark)
(b) /sin6d¢9. (1 mark)
1 I2
(0) /; +xe dx. (3 marks)
25c + 1
(d) f x2 + xdx. (3 marks) (e) /3x2 cos(:1:3 + 1)dx. (3 marks) 8014 SEMESTER 1 2009 PAGE 3 op 6 4. (a) Sketch the region enclosed by the straight line y = 2x + 3 and the parabola y = r2. (2 marks)
(b) Find the area of the region described in part (a). (4 marks) 5. The following diagram shows the graph of the derivative of a function f(.'17), for —4 S :c S 4.
(That is, the graph shown is that of y = f’($).) (a) For which values of :0 is f (as) increasing? {1 mark)
(b) For which values of x does f (to) have a critical point? (2 marks)
(0) Draw a graph of the function y = f (at), given that f (0) = 0. (4 marks) 6. An open box is to be made by cutting a square from each corner of a 12cm by 12 cm
piece of metal and then folding up the sides. What size square should be cut from each
corner in order to produce a box of maximum volume? ( 6 marks) 7. Let f(x) 2 2326—7”. (a) Find, and classify, all the stationary points of f (:5) ( 6 marks)
(b) Find all the points of inflection of f (cc). (4 marks)
(c) Describe the behaviour of f(:1:) as a: > oo. (2 marks)
(d) Sketch the graph of y = f(:c). {2 marks) 8. Show that the ellipsoid 3$2+2y2+22 = 9 and the sphere x2+y2+22~8x~6y~8z+24 = 0
are tangential to one another at (l, 1,2). (10 marks) 8014 SEMESTER 1 2009 PAGE 4 or 6 9. Consider the function f (3:) = A1133 — A3: + 1, with A > O. (a) Show that, when 0 < A < 191;; the graph of y = f (9:) cuts the w—axis exactly once. (8 marks)
(b) Show that, when 0 < A < ﬁg, f (:13) = 0 does not have a solution in the interval
—1 g x _<_ 1. (3 marks) (0) Use the result from part (b) to show that the equation 2sin3(:c) — 2sin(:r) + 1 = 0
has no solutions. (3 marks) 10. A ship is to sail from a wharf in a certain harbour to the harbour entrance, and then
out to sea. The depth of the water, Cl metres, at the Wharf is given by 47rt
d—7+3cos (EB—>’ where t is the number of hours after high tide. At the harbour entrance, the depth of water is given by the same formula as at the
wharf, but high tide occurs one hour earlier than it does at the wharf. An overhead cable obstructs the ship’s exit from the wharf. The ship can only leave if
the water depth at the wharf is less than or equal to 8.5m. In order for the ship to be able to sail through the shallow harbour entrance, the water
level at the harbour entrance must be at least 2m above low tide level. The ship takes
20 minutes to sail from the wharf to the harbour entrance. On the morning the ship is to sail, high tide at the wharf occurs at 2 am, and the ship
must be out at sea by 7 am. Find the earliest possible time and the latest possible time
that the ship can leave the wharf. (10 marks) 8014 SEMESTER 1 2009 o Quadratic formula ax2+bx+c=0givesx= o Derivatives —(constant) PAGE 5 OF 6 Notes —b i vb2 — 4ac 2a 0 Integration Techniques /kf(x )dx=k/f(a:)dx j'UH+Mxnm=/fm Wm+/mm Integration by recognition If F is an indeﬁnite integral for f ffoendewx=Feen+c dz + [email protected] yda: da:
dy dz
22!; ~ yin? /sina:dx= —cos:z:+C /cosa:d2: =sinx+C Integration by substitution Let u = g(x). Then /[email protected]@»[email protected]) dac=/f(u)du 8014 SEMESTER 1 2009 PAGE 6 OF 6 o Emotions of Two Variables Tangent plane to the surface 2 =2 f (33,31) at the point (a, b):
2 = f(a, 5) + sza, b)($  a) + fy(a,b)(y ‘ b) The differential of a function z = f (9:, y) at the point (a, b):
df = fI(a,b)dx + fy(a,b)dy. THIS IS THE LAST PAGE OF THE QUESTION PAPER. ...
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