level 3 calculus (91579) 2019 - home » nzqa€¦ · you must use calculus and show the results of...

915790

3SUPERVISOR’S USE ONLY

9 1 5 7 9

© New Zealand Qualifications Authority, 2019. 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

TOTAL

Level 3 Calculus, 201991579 Apply integration methods in solving problems

9.30 a.m. Tuesday 26 November 2019 Credits: Six

Achievement Achievement with Merit Achievement with ExcellenceApply integration methods in solving problems.

Apply integration methods, using relational thinking, in solving problems.

Apply integration methods, using extended abstract thinking, in solving problems.

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 – 12 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.

QUESTION ONE

(a) Find 2+ 2x

dx .

(b) Usethevaluesgiveninthetablebelowtofindanapproximationto f x( )2

5

∫ dx , using the TrapeziumRule.

x 2 2.5 3 3.5 4 4.5 5f (x) 0.6 1.1 1.7 2.6 3.2 3.4 2.6

(c) Find cos4x.cos2x dx0

12

.

You must use calculus and show the results of any integration needed to solve the problem.

2

Calculus 91579, 2019


(d) Thegraphbelowshowsthecurve y =63x +1

.

x

y

5

6

10 15

AB

ShowthattheareasofregionsAandBareequal.


3



(e) TherateofchangeofquantityN atanyinstantisgivenbythedifferentialequation:

dNdt

= kN

IfNhaspositivevaluesN1 and N2attimest1 and 2t1respectively,provethat

k = 1

t1lnN2N1

⎛

⎝⎜⎞

⎠⎟


4



QUESTION TWO

(a) Find 1+ 2e4x( )∫ dx.

(b) Thegraphofy = f (x) shown below has the y-axisasalineofsymmetry.

Theareasoftheshadedregionsareshown.

y

x–A

1.2 1.2–B B A

If f (x)dx = 5.8, what is the value of– A

A

∫ f (x)dx ?–B

B

∫

5



(c) Find k such that 82x 5

3

k

dx = 10.


(d) Thediagrambelowshowsthegraphofthefunction 2cos .y x=

y

xπ

Findtheareaoftheshadedregion.


6



(e) Thediagrambelowshowsthegraphsofthefunctions y = ex( )2 and y = 20− ex( )2 .

20

y

x

( )=y ex 2

( )= −y 20 ex 2

Findtheareaoftheregionshadedinthediagram.


7



QUESTION THREE

(a) Find 24 2x −1( )3 dx∫ .

(b) Solvethedifferentialequation dydx

= 4sec2 2x, given that when x = π8, y = 5.

(c) Find x +1+ xx +1

1

4

dx .


8



(d) If dydx

= 4x4x2 − 3

+ x and y(1) = 2, find y(4).

9



Question Three continues on the following page.

(e) Aninvertedrightpyramidhasasquarebaseandaheightof1.5m.

Thepyramidisinitiallyfilledwithwatertoadepthof1m,asshown.

1.5 m1.0 m

0.9 m

TheenergyrequiredtopumpwateroutofatankofheightHisgivenby:

E = 9800 H − h( )

H−d

H

∫ A h( )dh

where Eistheenergyinjoules distheinitialdepthofthewaterinthetank histhedepthofthewaterinthetankatanyinstant A(h)istheareaofthesurfaceofthewateratthisinstant.

Findtheenergyrequiredtopumpthewateroutofthetankshown.


10



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91

57

912



QUESTION NUMBER

Extra paper if required.Write the question number(s) if applicable.

level 3 calculus (91579) 2019 - home » nzqa€¦ · you must use calculus and show the results of...

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