90635
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906350
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Level 3 Calculus, 2011
90635 Differentiate functions and use derivatives
to solve problems
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Thursday 24 November 2011
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 room for any answer, use the extra space provided at the back of this booklet.
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.
ASSESSOR’S USE ONLY
Achievement
Differentiate functions and use
derivatives to solve problems.
Achievement Criteria
Achievement with Merit
Demonstrate knowledge
of advanced concepts and
techniques of differentiation and
solve differentiation problems.
Achievement with Excellence
Solve more complex
differentiation problem(s).
Overall level of performance
© New Zealand Qualifications Authority, 2011. All rights reserved.
No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
2
You are advised to spend 50 minutes answering the questions in this booklet.
QUESTION ONE
(a)
Differentiate y = tan2x
You do not need to simplify your answer.
(b) Differentiate y = ln (3x 4)
You do not need to simplify your answer.
(c)Differentiate y =
6x
(1 + x )
2
2
You do not need to simplify your answer.
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3
(d) If a2x + by2 = 0 where a and b are constants, then find an expression for
(e)
Use the definition of the derivative
f ʹ′(x) = lim
h→0
f (x + h) − f (x)
h
to find the derivative of f (x) = 5 – 3x2
Calculus 90635, 2011
dy
in terms of a and b.
dx
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(f)
An oil company is planning to lay an oil pipeline from its offshore oil well to the refinery at
Motunui, which is on the coast.
The oil well is 18 km directly offshore from Sandibay, and the distance from Sandibay to
Motunui is 75 km.
It costs 5 times as much to lay a kilometre of pipeline underwater as it does to lay a kilometre
of pipeline on land.
The plan is to lay the pipeline in a straight line from the oil well to point L on the coast. Then
a further straight stretch of pipeline will be laid from point L to Motunui, as shown in the
diagram below.
Diagram is
NOT to scale
oil well
sea
18 km
Sandibay
L
Motunui
coastline
land
75 km
Assume that the coastline is straight.
How far from Sandibay should L be, if the pipeline is to be laid as cheaply as possible?
(You do not need to prove that your answer is a minimum and not a maximum.)
Give any derivative(s) you need to find when solving this problem.
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6
QUESTION TWO
(a)
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The temperature of a cup of coffee is given by the formula
T = 18 + 60e–0.15t
where T is the temperature of the coffee (oC)
and t is the time in minutes after the coffee is poured.
At what rate is the temperature of the coffee changing 10 minutes after the coffee was poured?
Give any derivative(s) you need to find when solving this problem.
(b) Find the gradient of the tangent to the curve y = sin kx + cos kx at x = 0, where k is a constant.
Give any derivative(s) you need to find when solving this problem.
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(c)
The sum of two positive numbers is S, a fixed number, and their product is P.
Use differentiation to show that P is a maximum when the numbers are equal.
Give any derivative(s) you need to find when solving this problem.
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(d) The graph below shows the function v = f (t), where v is the velocity, in metres per second, of
a particle moving in a straight line at time t seconds.
v
25
20
15
10
5
0
5
10
15
t
–5
–10
For the function v = f (t) above, find all the value(s) of t for which f (t) is not
differentiable.
(i)
(ii) For the function v = f (t) above, find all the times that meet the following conditions:
(1) the acceleration of the particle is zero
(2) the particle is not moving
(3) the particle is decelerating.
(iii) Use the graph to find lim f (t )
t→5
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(e)
A circle that closely fits nearby points on a local section of a curve
can be drawn for any curve.
The radius of curvature of the curve is defined as the radius of the
approximating circle.
The radius changes as we move along the curve.
1+
Radius of curvature =
dy
dx
2
3
2
d2 y
dx 2
Find the radius of curvature of the function y = e–x sin x at the point (0,0).
Give any derivative(s) you need to find when solving this problem.
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10
QUESTION
NUMBER
Extra space if required.
Write the question number(s) if applicable.
Calculus 90635, 2011
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90635