level 3 calculus (91579) 2017 - nzqa · 2017. 10. 23. · 9 calculus 91579, 2017 assessor’s use...

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Level 3 Calculus, 201791579 Apply integration methods in solving problems

9.30 a.m. Thursday 23 November 2017 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 – 16 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 4sec2 2x dx.

(b) Useintegrationtofindtheareaenclosedbetweenthecurve y = x2 + xx

andthelines y = 0, x = 1, and x=4(theareashadedinthediagrambelow).

–1

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6x

y

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

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(c) Anobject’saccelerationismodelledbythefunction

a(t) = 1.2 t whereaistheaccelerationoftheobject,inms–2

and tisthetimeinsecondssincethestartoftheobject’smotion.

Iftheobjecthadavelocityof7ms–1after4seconds,howfardidittravelinthefirst9secondsofmotion?

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

(d) Findthevalueofkif 3e2x dx = 40

k

∫ .

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

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(e) Themeanvalueofafunctiony = f (x)fromx = a to x = bisgivenby

Mean value =

f (x)dxa

b

∫b − a

Findthemeanvalueofy = sin2xbetweenx = 0 and x=π.

Partofthegraphofy = sin2xisshownbelow.

y

1

x�

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

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QUESTION TWO

(a) Find ∫ −xx6

2 1d .

(b) Find 2x − 5( )∫4dx .

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(c) Thediagrambelowshowsthecurvey = –x2 + 3x+10,andtheliney = –x+14,whichisthetangenttothecurveatthepoint(2,12).

x

y

(2,12)

5 14

Calculatetheshadedarea.

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

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(d) Partofthegraphofy = sin 3x cos2xisshownbelow.

x

y

4 3

Findtheareaenclosedbetweenthecurvey = sin 3x cos2x andthelinesy = 0, x = 0, and

x =

π4 .

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

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(e) Theaccelerationofanobjectismodelledbythefunctiona(t) = 20 ln tt

.

whereaistheaccelerationoftheobjectinms–2

and tisthetimeinsecondssincethestartoftheobject’smotion.

Theobjectwasmovingwithavelocityof12ms–1whent = 4.

Findthevelocityoftheobjectafter10seconds.

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

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QUESTION THREE

(a) Find 9x4 + 8e4 x dx.

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Question Three continues on the following page.

(b) Juliawantstofindanapproximationoftheareaofapavedcourtyardthatshewishestoconstructonherproperty.Shetakessomemeasurementsandtheseareshownonthediagrambelow.

2 m 2 m 2 m 2 m 2 m

6 m 8 m 10 m 11 m 12 m

Usingthesemeasurements,andtheTrapeziumrule,findanapproximationoftheareaofpavedcourtyard.

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(c) Julia’sfriendSarahbelievesthattheequationofthecurvedborderofthepavedcourtyardcan

bemodelledbythefunction y =15x −15x + 2

.

1

2

x

y

4

6

8

10

12

2 3 4 5 6 7 8 9 10 11 12

Useintegrationtofindtheareaofthecourtyard,showninthediagramabove.

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

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(d) Solvethedifferentialequation dydx

= yx,giventhatwhenx=4,theny = 1.

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

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(e) y and tsatisfythedifferentialequation dydt

= k cos0.5t × esin0.5t , 0 ≤ t ≤ 5.

Giventhatwhent = 0, y=8,andthatwhent = 2, y=12,findthevalueofywhent = 5.

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

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QUESTION NUMBER

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

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QUESTION NUMBER

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

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QUESTION NUMBER

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