Level 3 Calculus, 201290635 Differentiate functions and use derivatives
to solve problems
2.00 pm Monday 26 November 2012 Credits: Six
Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.
You should attempt ALL the questions in this booklet.
Show ALL working.
Make sure that you have the Formulae and Tables Booklet L3–CALCF.
If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.
Check that this booklet has pages 2 – 10 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
906350
3SUPERVISOR’S USE ONLY
9 0 6 3 5
© New Zealand Qualifications Authority, 2012. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
ASSESSOR’S USE ONLY Achievement Criteria
Achievement Achievement with Merit Achievement with ExcellenceDifferentiate functions and use derivatives to solve problems.
Demonstrate knowledge of advanced concepts and techniques of differentiation and solve differentiation problems.
Solve more complex differentiation problem(s).
Overall level of performance
You are advised to spend 50 minutes answering the questions in this booklet.
QUESTION ONE
(a) Differentiate y = 3x2 + x
You do not need to simplify your answer.
(b) Differentiate =−
yxe
1
x
2
You do not need to simplify your answer.
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(c) Differentiate = + +y x x x( ) ln(2 3)2
You do not need to simplify your answer.
(d) Find yx
dd for the curve defined parametrically by
x = 4 sin(t2 + 1) y = 3 cos(2t – 3)
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(e) The graph below shows the function y = f (x).
–8
–8–7–6–5–4–3–2–1
12345678
y
x–7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8
For the function f (x) above:
(i) What is the value of →f xlim ( )
x 4?
State clearly if the value does not exist.
(ii) Find all the value(s) of x that meet the following conditions:
(1) f '(x) = 0
(2) f '(x) > 0
(3) f ''(x) > 0
(iii) Find all the value(s) of x where f (x) is continuous but not differentiable.
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(f) A cylinder of height h cm and radius r cm is inscribed inside a sphere of radius 10 cm.
10
y
h
r
x10
–10
–10
Find the value of r that maximises the volume of the cylinder.
You do not need to prove that the volume is a maximum and not a minimum.
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QUESTION TWO
(a) An object is moving in a straight line.
The velocity of the object is given by =+
v tt
2
362
where t is time in seconds
and v is velocity in metres per second.
Find the time(s) at which the object’s acceleration is zero.
(b) Find the equation of the normal to the curve y = ln x at the point on the curve where x = 2.
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(c) The light intensity, I lux, at a distance x m below the surface of a lake is given by:
I = Ioe–k x, where Io and k are constants.
The light intensity at the surface of the lake is 8000 lux.
The light intensity 5 metres below the surface of the lake is 2000 lux.
At what rate, in lux per metre, is the light intensity decreasing 5 metres below the surface of the lake?
(d) Find the gradient of the curve x2 – y2 – 6x = 4y – 5 at the point (–2, 3) on the curve.
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(e) The volume of a ‘cap’ of a sphere is given by the formula
V h R hπ
3
2= −
where R is the radius of the sphere
and h is the height of cap.
h
R
A large hemispherical wok has a diameter of 60 cm. It is being filled with water at a constant rate of 50 cm3 per second.
h
r
60 cm
At what rate is the radius of the surface of the water increasing when the height of the water is 10 cm?
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QUESTION NUMBER
Extra paper if required.Write the question number(s) if applicable.
