Further Calculus. Each correct answer in this section is worth two marks.
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1 Multiple Choice Questions Further Calculus Each correct answer in this section is worth two marks..differentiate2(4 x) 2withrespecttox. A. (4 x) B. (4 x) C. (4 x) 3 2 D. (4 x) 3 2 C 3.2 C NC C2, C2 HSN 73 2.Whatisthegradientofthetangenttothecurvewithequationy =cos2xatthe pointwherex = π 4? A. 2 B. C. 0 D. 2 A 3.2 C NC C4, C20, T3 HSN 27 hsn.uk.net Page
2 3.Find (2x 4 +cos5x)dx. A. 2 5 x 5 5sin5x +c B. 2 5 x 5 + 5sin5x +c C. 2 3 x 3 + 5sin5x +c D. 2 3 x 3 5sin5x +c C 3.2 C 0 0 CN C3, C P Q9 4.Differentiate3cos ( 2x π ) 6 withrespecttox. A. 3sin(2x) B. 3sin(2x π 6 ) C. 6sin(2x π 6 ) D. 6sin(2x π 6 ) C 3.2 C NC C20 HSN 096 hsn.uk.net Page 2
3 5.Giventhatf(x) =3cos(2x),whatisthevalueoff ( ) π 6? A. 3 B. 3 3 C. 3 D B 3.2 C NC C20 HSN 07 6.Giventhatf(x) =4sin3x,findf (0). A. 0 B. C. 2 D. 36 C 3.2 C 0 0 NC C20, T3 20 P Q3 hsn.uk.net Page 3
4 7.Giventhatf(x) = 2 sin2 x,whatisthevalueoff ( π 3 )? A. 2 B. C. D D 3.2 C NC C20, C2 HSN 08 8.Giventhatf(x) = (4 3x 2 ) 2onasuitabledomain,findf (x). A. 3x(4 3x 2 ) 2 B. 2 (4 6x) 3 2 C. 2(4 3x 3 ) 2 D. 3x(4 3x 2 ) 3 2 D 3.2 C 0 0 NC C P Q8 hsn.uk.net Page 4
5 9.Differentiate (6x 2 ) 5 withrespecttox. A. 60x 9 B. 5(6x 2 ) 4 C. 30(6x 2 ) 4 D. 60x(6x 2 ) 4 D 3.2 C CN C2 HSN 25 0.Afunctionfisdefinedforx 4byf(x) = (8 2x) 3 2. Whatisthevalueoff (2)? A. 24 B. 6 C. 3 D. 8 B 3.2 C NC C2 HSN 34 hsn.uk.net Page 5
6 .Ify =3cos 4 x,find dy dx. A. 2cos 3 xsinx B. 2cos 3 x C. 2cos 3 xsinx D. 2sin 3 x C 3.2 C 0 0 NC C2, C P Q6 2. Find (2x ) 2dxwherex > 2. A. 3 (2x ) 3 2 +c B. 2 (2x ) 2 +c C. 2 (2x ) 3 2 +c D. 3 (2x ) 2 +c A 3.2 C 0 0 NC C P Q4 hsn.uk.net Page 6
7 3. Find (2x 5) 4 dx. A. 8(2x 5) 3 +c B. 4(2x 5) 3 +c C. 5 (2x 5) 5 +c D. 0 (2x 5) 5 +c D 3.2 C NC C22 HSN 4 4.Whatisthevalueof A. 2 π 0 sinxdx? B. 0 C. D. 2 D 3.2 C NC C23, C5 HSN 65 [END OF MULTIPLE CHOICE QUESTIONS] hsn.uk.net Page 7
8 Written Questions 5. (a) C NC A6 993 P2 Q (a) 2 A/B NC A6 (b) C NC C, C2 (b) 6 A/B NC C, C2 hsn.uk.net Page 8
9 6. Findtheequationofthetangenttothecurvey =2sin(x π 6 )atthepointwhere x = π C CN C5,C20 y = 3x + π P2Q6 pd: findderivative 2 ss: know derivative at x =... represents grad. 3 pd: findcorrespondingy-coordinate 4 ic: stateequationoftangent dy dx =2cos(x π 6 ) 2 m = 3 3 y x= π 3 = 4 y = 3(x π 3 ) 7. Thegraphsofy=f(x)andy=g(x)are y shown in the diagram. 7 y = f( x) f(x) = 4cos(2x) +3andg(x)isofthe formg(x) =mcos(nx). 3 (a)writedownthevaluesofmandn. (b) Find, correct to one decimal place, 0 π x the coordinates of the points of intersection of the two graphs in the 3 y = g( x) interval0 x π. 5 (c) Calculate the shaded area. 6 (a) C CN T4 m =3andn =2 2009P2Q5 (b) 5 C CR T6 (0 6, 3), (2 6, 3) (c) 6 B CR C7, C ic: interpretsgraph 2 ss: knowshowtofindintersection 3 pd: startstosolve 4 pd: findsx-coordinstquadrant 5 pd: findsx-coordin2ndquadrant 6 pd: findsy-coordinates 7 ss: knowshowtofindarea 8 ic: stateslimits 9 pd: integrate 0 pd: integrate ic: substitutelimits 2 pd: evaluatearea m =3andn =2 2 3cos2x = 4cos2x +3 3 cos2x = x =0 6 5 x =2 6 6 y = 3, 3 7 ( 4cos2x +3 3cos2x ) dx sin2x 0 3x 7 2 sin2x ( sin5 2) ( sin 2) hsn.uk.net Page 9
10 8. A point moves in a straight line such that its acceleration a is given by a =2(4 t) 2,0 t 4. Ifitstartsatrest,findanexpressionforthevelocity vwherea = dv dt. 4 4 C NC C8,C22 V = 4 3 (4 t) P2Q8 ss: knowtointegrateacceleration 2 pd: integrate 3 ic: useinitialconditionswithconst. of int. 4 pd: processsolution V = (2(4 t) 2)dt statedorimplied by (4 t) =2 (4 0) c 4 c = Thegraphofy =f(x)passesthroughthepoint ( π 9, ). Iff (x) =sin(3x)expressyintermsofx. 4 4 A/B NC C8,C23 y = 3 cos(3x) PQ8 ss: knowtointegrate 2 pd: integrate 3 ic: interpret ( π 9,) 4 pd: process y = sin(3x)dx statedorimpliedby cos(3x) 3 = 3 cos(3π 9 ) +corequiv. 4 c = Acurveforwhich dy ( dx =3sin(2x)passesthroughthepoint 5π 2, ) 3. Findyintermsofx. 4 4 A/B CN C8,C23 y = 3 2 cos(2x) P2Q0 pd: integratetrigfunction 2 pd: integratecompositefunction 3 ss: usegivenpointtofind c 4 pd: evaluate c 3sin(2x)dx statedorimpliedby cos(2x) 3 3 = 3 2 cos(2 5 2π) +c 4 c = 4 3( 0 4) hsn.uk.net Page 0
11 2. Differentiatesin2x + 2 withrespecttox. 4 x 2 C NC C3 989PQ0 2 A/B NC C Giventhatf(x) = (5x 4) 2,evaluatef (4). 3 5 C CN C P2Q8 2 A/B CN C2 pd: differentiatepower 2 pd: differentiate2ndfunction 3 pd: evaluatef (x) 2 (5x 4) f (4) = Givenf(x) =cos 2 x sin 2 x,findf (x). 3 C NC C2 999 P Q9 2 A/B NC C2,C20 hsn.uk.net Page
12 24. Giventhatf(x) =5(7 2x) 3,findthevalueoff (4). 4 4 A/B NC C2 99PQ3 25. Differentiate2x2 3 +sin 2 xwithrespecttox. 4 C NC C2 992 P Q 3 A/B NC C2 26. Findthederivative,withrespecttox,of x 3 +cos3x. 4 4 A/B NC C2 994PQ0 hsn.uk.net Page 2
13 27. (a) A/B CN CGD 995 P2 Q (b) C CN C2 (b) 4 A/B CN C2 (c) C CN C (c) 2 A/B CN C hsn.uk.net Page 3
14 28. Iff(x) =cos 2 x 2 3x 2,findf (x). 4 2 C NC C2,C 990PQ9 2 A/B NC C2,C 29. Differentiate4 x +3cos2xwithrespecttox. 4 2 C NC C2,C 993PQ9 2 A/B NC C2,C 30. Differentiatesin 3 xwithrespecttox. Hence find sin 2 xcosxdx. 4 C NC C2,C9 994PQ7 3 A/B NC C2,C9 hsn.uk.net Page 4
15 3. Find dy dx giventhaty = +cosx. 3 3 A/B NC C2,C20 996PQ3 32. Givenf(x) = (sinx+) 2,findtheexactvalueoff ( π 6 ). 3 3 A/B NC C2,C20,T2 998PQ6 hsn.uk.net Page 5
16 33. (a)acurvehasequationy = (2x 9) 2. Showthattheequationofthetangenttothiscurveatthepointwherex=9 isy = 3 x. 5 (b)diagramshowspartofthecurveandthetangent. Thecurvecutsthex-axisatthepointA. y O y = 3 x Diagram A 9 y = (2x 9) 2 x Find the coordinates of point A. (c) Calculate the shaded area shown in diagram 2. 7 y y = 3 x y = (2x 9) 2 O A 9 x Diagram 2 (a) 5 B CN C2, C24 proof 200 P2 Q6 (b) C CN A6 ( 9 2,0) (c) 7 A CN C7, C =4 2 =4 5 ss: knowtoandstarttodifferentiate 2 pd: completechainrulederivative 3 pd: gradientviadifferentiation 4 pd: obtainy curve atx=9 5 ic: stateequationandcomplete 6 ic: obtaincoordinatesofa 7 ss: strategyforfindingshadedarea 8 ss: knowtointegrate (2x 9) 2 9 pd: startintegration 0 pd: completeintegration hsn.uk.net ic: limitsx A and9 Page 6 2 pd: substitutelimits 3 pd: evaluate area and complete strategy 2 (2x 9) y 3 = 3 (x 9)andcomplete 6 ( 9 2,0) 7 Shadedarea =areaoflarge areaundercurve 8 (2x 9) 2dx 9 (2x 9)3 2 Questions marked c SQA and9 2 (8 9) 3 2 0
17 34. Find +3xdxandhencefindtheexactvalueof +3xdx A/B NC C22 993PQ6 35. Find (7 3x) 2dx. 2 2 A/B CN C22,C4 +c 3(7 3x) 2000P2Q0 pd: integratefunction 2 pd: dealwithfunctionoffunction (7 3x) Evaluate 0 3 ( 2x +3) 2dx. 4 4 C NC C22,C5 996PQ5 hsn.uk.net Page 7
18 37. (a) C NC T 998 P Q5 (b) C NC C6 (b) 3 A/B NC C (a) Evaluate π 2 0 cos2xdx. 3 (b) Draw a sketch and explain your answer. 2 (a) 3 A/B NC C P Q4 (b) C NC T, C6 (b) A/B NC T,C6 hsn.uk.net Page 8
19 39. (a)showthat (cosx +sinx) 2 = +sin2x. (b) Hence find (cosx+sinx) 2 dx. 3 (a) C NC T8 993 P Q9 (b) 3 A/B NC C Find ( ) 6x 2 x+cosx dx. 4 4 C NC C23 995PQ3 4. Thecurvey =f(x)passesthroughthepoint ( π 2,)andf (x) =cos2x. Findf(x). 3 3 A/B NC C23 997PQ5 hsn.uk.net Page 9
20 42. (a)bywritingsin3xassin(2x +x),showthatsin3x =3sinx 4sin 3 x. 4 (b) Hence find sin 3 xdx. 4 (a) 2 C NC T8,T8 995P2Q9 (a) 2 A/B NC T8,T8 (b) 4 A/B NC C23 hsn.uk.net Page 20
21 43. (a) 4 C NC CGD 989 P2 Q8 (b) 2 C NC C23, C5 (b) 3 A/B NC C23, C5 (c) 2 C NC T (d) 2 A/B NC CGD hsn.uk.net Page 2
22 44. 2 C NC C23,C6 996P2Q5 4 A/B NC C23,C6 45. (a) Findthederivativeofthefunctionf(x) = (8 x 3 ) 2,x <2. 2 (b) Hence write down x 2 dx. (8 x 3 ) 2 (a) 2 A/B CN C2 3 2 x2 (8 x 3 ) PQ0 (b) A/B CN C (8 x3 ) 2 +c pd: processdifferentiation 2 pd: usethechainrule 3 ic: interpretanswerfrom(a) 2 (8 x3 ) x f(x)or 2 3 (8 x3 ) 2 hsn.uk.net Page 22
23 46. (a)theexpression3sinx 5cosxcanbewrittenintheformRsin(x +a)where R >0and0 a<2π. CalculatethevaluesofRanda. 4 (b)hencefindthevalueoft,where0 t 2,forwhich t 0 (3cosx +5sinx)dx =3. Part Marks Level Calc. Content Answer U3 OC4 (a) 4 C CN T3 R = 34,a = P2Q6 (b) 7 B CN C23,T3,T6 t =0 6 7 ss: usecompoundangleformula 2 ic: comparecoefficients 3 pd: processr 4 pd: processa 5 pd: integrategivenexpression 6 ic: substitutelimits 7 pd: processlimits 8 ss: knowtousewaveequation 9 ic: writeinstandardformat 0 ss: starttosolveequation pd: completeandstatesolution Rsinxcosa +Rcosxsina 2 Rcosa =3andRsina = (accept5 8) (accept5 3) 5 3sinx 5cosx 6 (3sint 5cost) (3sin0 5cos0) 7 3sint 5cost sin(t +5 3) +5 9 sin(t +5 3) = t +5 3 =3 5,5 9 t =0 6 hsn.uk.net Page 23
24 47. Part Marks Level Calc. Content Answer U3 OC4 (a) 4 C CR C6 992 P2 Q0 (b) 2 C CR A6 (c) 2 C CR T6, C23, C7 (c) 8 A/B CR T6,C23,C7 [END OF WRITTEN QUESTIONS] hsn.uk.net Page 24
Exam Revision 3. Find the equation of the straight line which is parallel to the line with equation 2x +3y =5andwhichpassesthroughthepoint (2, 1).
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