ACalculusRefresher - mathcentre.ac.uk

1 A Calculus Refresherv1. March 2003 This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike Unported License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, 2003 mathcentreContentsForeword2 Preliminary work2 How to use this booklet2 Reminders3 Tables of derivatives and integrals41. Derivatives of basic functions52. Linearity in differentiation73. Higher derivatives94. The product rule for differentiation105. The quotient rule for differentiation116. The chain rule for differentiation137. Differentiation of functions defined implicitly158. Differentiation of functions defined parametrically 169. Miscellaneous differentiation exercises1710. Integrals of basic functions2011. Linearity in integration2112. Evaluating definite integrals2313. Integration by parts2414. Integration by substitution2615. Integration using partial fractions2916.

2 Integration using trigonometrical identities3317. Miscellaneous integration exercises35 Answers39 Acknowledgements461 ForewordThe material in this refresher course has been designed to enable youto cope better with your university mathematics programme starts you will find that the ability to differentiate andintegrate confidently will be invaluable. We think that thisis so impor-tant that we are making this course available for you to work througheither before you come to university, or during the early stages of workYou are advised to work through the companion bookletAn AlgebraRefresherbefore embarking upon this calculus revision to use this bookletYou are advised to work through each section in this booklet in or-der. You may need to revise some topics by looking at an AS-level orA-level textbook which contains information about differentiation should attempt a range of questions from each section, and checkyour answers with those at the back of the booklet.

3 The more questionsthat you attempt, the more familiar you will become with these vitaltopics. We have left sufficient space in the booklet so that youcan doany necessary working within it. So, treat this as a you get questions wrong you should revise the material andtry againuntil you are getting the majority of questions you cannot sort out your difficulties, do not worry about this. Youruniversity will make provision to help you with your problems. This maytake the form of special revision lectures, self-study revision material ora drop-in mathematics support material has been prepared for students who have completed anA-level course in mathematics2 RemindersUse this page to note topics and questions which you found help with these from your tutor or from other universitysupport services assoon as following tables of common derivatives and integrals are provided for revisionpurposes. It will be a great advantage to know these derivatives and integrals becausethey are required so frequently in mathematics of derivativesf(x)f (x)xnnxn 1lnkx1xekxkekxaxaxlnasinkxkcoskxcoskx ksinkxtankxksec2kxTable of integralsf(x)Zf(x) dxxn(n6= 1)xn+1n+ 1+cx 1=1xln|x|+cekx(k6= 0)ekxk+csinkx(k6= 0) coskxk+ccoskx(k6= 0)sinkxk+csec2kx(k6= 0)tankxk+c41.

4 Derivatives of basic functionsTry to find all the derivatives in this section without referring to a table of derivatives of these functions occur so frequently thatyou should try to memorisethe appropriate rules. If you are really stuck, consult the table on page each of the following with respect tox.(a)x(b)x6(c)6(d) x(e)x 1(f)x1/7(g)1x3(h)x79(i) (j)13 x(k)x 5/3(l) each of the following with respect to .(a)cos (b)cos 4 (c)sin (d)sin2 3(e)tan (f)tan (g)sin( 8 )(h)tan 4(i)cos 3 (j)cos 5 2!(k)sin the following derivatives.(a)ddx(ex)(b)ddy(e2y)(c)ddt( e 7t)(d)ddx(e x/3)(e)ddz(e2z/ )(f)ddx(e )(g)ddx(3x) the following derivatives.(a)ddx(lnx)(b)ddz(ln 5z)(c)ddx ln2x3 62. Linearity in differentiationThelinearity rulesenable us to differentiate sums and differences of functions,andconstant multiples of functions. Specificallyddx(f(x) g(x)) =ddx(f(x)) ddx(g(x)),ddx(kf(x)) =kddx(f(x)). each of the following with respect tox.(a)3x+ 2(b)2x x2(c) cosx sinx(d)3x 3+ 4 sin 4x(e)2ex+e 2x(f)1x 4 3 lnx(g)4x5 3 tan 8x the following derivatives.

5 (a)ddt 5t1/5+t88!(b)dd 2 cos 4 3e /4!(c)ddx 3e3x/55!(d)ddx 29tan3x2 34cos 8x (e)ddz 14z4/3 13e 4z/3 7In Questions 3-5 you don t need the product rule, quotient rule or chain rule todifferentiate any of these if you do the algebra first! the powers or roots and hence find the following derivatives.(a)ddy q2y (b)ddx (2x)3 1(2x)3!(c)ddy 12ey 4!(d)ddt 3 5e 2t or expand each of the following expressions, and then differentiate withrespect tox.(a)x x2x3(b)x( x x2)(c) 2x 2x 3x2+x (d)(e2x 1)(3 e3x)(e)1 e 2xe the laws of logarithms to find the following derivatives.(a)ddx lnx9/2 (b)ddx ln 1 6x!!(c)ddt ln t3e3t!!(d)ddt ln te 2t 1/3 83. Higher the following second derivatives.(a)d2dx2(x5)(b)d2dx2(cos 3x)(c)d2dz2(e2z e 2z)(d)d2dy2(8 13y)(e)d2dx2 1x 3x 3x3 (f)d2dt2(ln 2t 6t)(g)d2dx2 x3/2 1x3/2 (h)d2dx2(ex+e x+ sinx+ cosx)(i)d2dt2 12sin 2t 14ln 4t 94. The product rule for differentiationThe rule for differentiating the product of two functionsf(x)andg(x)isddx(f(x)g(x)) =f (x)g(x) +f(x)g (x).

6 Each of the following with respect tox.(a)xsinx(b)x3cos 2x(c)x 1/3e 3x(d) xln 4x(e)(x2 x) sin 6x(f)1x tanx3 cosx3 the following derivatives.(a)dd (sin cos )(b)ddt(sin 2ttan 5t)(c)ddz(sinzln 4z)(d)ddx e x/2cosx2 (e)ddx(e6xln 6x)(f)dd (cos cos 3 )(g)ddt(lntln 2t)105. The quotient rule for differentiationThe rule for differentiating the quotient of two functionsf(x)andg(x)isddx f(x)g(x)!=f (x)g(x) f(x)g (x)(g(x)) each of the following with respect tox.(a)x1 x2(b)x41 x(c)2 x1 + 2x(d)3x2 2x32x3+ 3(e)1 + x x the following derivatives.(a)ddx sinxx (b)ddx lnxx4/3!(c)dd 2tan 2 !(d)ddz ez z!(e)ddx x2ln 2x! the following derivatives.(a)ddt sin 2tsin 5t (b)ddx e 2xtanx!(c)ddx lnxcos 3x!(d)ddx ln 3xln 4x!126. The chain rule for differentiationThe chain rule is used to differentiate a function of a function :ddx(f(g(x))) =f (g(x)).g (x). each of the following with respect tox.(a)(4 + 3x)2(b)(1 x4)3(c)1(1 2x)2(d) 1 +x2(e) x 1x 1/3(f)(2x2 3x+ 5)5/2(g)qx 2 x(h)1 4x2 the following derivatives.

7 (Remember the notation forpowers of trigono-metric functions: sin2x means(sinx)2, etc.)(a)dd (sin2 )(b)dd (sin 2)(c)dd (sin(sin ))(d)ddx(tan(3 4x))(e)ddz(cos55z)(f)ddx 1cos3x (g)ddt(sin(2 t 3t2)) the following derivatives. (The notation expx is used rather than ex where it is clearer.)(a)ddy exp( y2) (b)ddx(exp(cos 3x))(c)ddx(cos(e3x))(d)ddx(ln(sin 4x))(e)ddx(sin(ln 4x))(f)ddx(ln(ex e x))(g)ddt e3t 3 cos 3t 147. Differentiation of functions defined terms ofywhenxandyare related by the following equations. Youwill need the formuladydx= 1/dxdy.(a)x=y y3(b)x=y2+1y(c)x=ey+e2y(d)x= ln(y e y) terms ofxand/orywhenxandyare related by the followingequations.(a)cos 2x= tany(b)x+y2=y x2(c)y siny= cosx(d)ex x=e2y+ 2y(e)x+ey= lnx+ lny(f)y= (x y)3158. Differentiation of functions defined parametricallyIfxandyare both functions of a parametert, thendydx=dydt terms oftwhenxandyare related by the following pairs of parametricequations.(a)x= sint, y= cost(b)x=t 1t, y= 1 t2(c)x=e2t+t, y=et+t2(d)x= lnt+t, y=t terms oftwhenxandyare related by the following pairs of parametricequations.

8 (a)x= 3t+t3, y= 2t2+t4(b)x= cos 2t, y= tan 2t169. Miscellaneous differentiation the following derivatives, each of which requires one of the techniques cov-ered in previous sections. You have to decide which technique is required for eachderivative!(a)ddx(x3tan 4x)(b)ddt(tan34t)(c)ddx(exp(3 tan 4x))(d)dd 3 tan 4 !(e)ddx(exp(x ex))(f)ddy y4+y 4y+y 1!(g)ddx(2xx2)(h)ddx 1lnx x (i)ddx 5 3x (j)ddt(ln(lnt))(k)ddz ln 1 z1 +z 2! the following derivatives, which require both the product and quotient rules.(a)ddx xcosx1 cosx (b)ddz ezzlnz (c)dd sin 3 cos 2 tan 4 ! the following derivatives, which require the chain rule as well as either theproduct rule or the quotient rule.(a)ddt(e tln(et+ 1))(b)dd (sin23 cos43 )(c)ddx 1 x21 +x2!3/2 (d)dd (exp( cos ))(e)ddx (xlnx)3 (f)ddx exp 1 x1 +x (g)ddy 1y2 y2 1! the following derivatives, which require use of the chain rule more than once.(a)ddx( 1 cos3x)(b)ddx exp (x x2)1/4 (c)dd ln tan1 the following second derivatives.(a)d2dx2( 1 +x2)(b)d2dz2(exp(z2))(c)d2d 2(sin3 )(d)d2dx2 1(1 x4)4!

9 That cosecx=1sinx,secx=1cosxandcotx=cosxsinx, find thefollowing derivatives.(a)ddx(cosec2x)(b)dd (sec2 )(c)ddz( 1 + cotz)(d)dd (cosec2 cot3 )(e)ddx(ln(secx+ tanx))(f)dd (tan(sec ))1910. Integrals of basic functionsTry to find all the integrals in this section without referring to a table of integrals of these functions occur so frequently that you should try to memorisethe appropriate rules. If you are really stuck, consult the Tables on page each of the following with respect tox.(a)x4(b)x7(c)x1/2(d)x1/3(e) x(f)x 1/2(g)4 x(h)1x3(i) (j) (k)1 x(l)1 x3(m)x 2(n)x4 each of the following with respect tox.(a)cos 5x(b)sin 2x(c)sin12x(d)cosx2(e)1x(f)e2x(g)e 2x(h)ex/3(i) (j)1ex(k)1e2x(l)cos( 7x)2011. Linearity in integrationThelinearity rulesenable us to integrate sums (and differences) of functions, andconstant multiples of functions. SpecificallyZ(f(x) g(x))dx=Zf(x) dx Zg(x)dx,Zk f(x)dx=kZf(x) each of the following with respect tox.(a)7x4(b) 4x7(c)x1/2+x1/3(d)17x1/3(e) x 1 x(f)x2+1x(g)x3+1x2(h)17x3(i)11(j) (k)2x 2x(l)7x each of the following with respect tox.

10 (a)3x+ cos 4x(b)4 + sin 3x(c)x2+ sinx2(d)4ex+ cosx2(e)e 2x+ e2x(f)3 sin 2x+ 2 sin 3x(g)1kx,kconstant (h) 1 4x(i)1 +x+x2(j)13x 7(k)12cos12x(l)12x2 3x 1 each of the following expressions first and then integrate them withrespect tox.(a)6x(x+ 1)(b)(x+ 1)(x 2)(c)x3+ 2x2 x(d)( x+ 2)( x 3)(e)e2x(ex e x)(f)e3x e2xex(g)x+ 4x(h)x2+ 3x+ 2x+ 22212. Evaluating definite each of the following definite integrals.(a)Z107x4dx(b)Z3 2 4t7dt(c)Z21(x1/2+x1/3)dx(d)Z 1 217t1/3dt(e)Z31(2s+ 8s3)ds(f)Z511x2dx(g)Z30(t2+ 2t)dt(h)Z /40cos 2xdx(i)Z1/20e3xdx(j)Z421 exdx(k)Z /40(2 + sin )d (l)Z10(ex+ e x) the functionf(x) =x2+ 3x 2verify thatZ20f(x)dx+Z32f(x)dx=Z30f(x) the functionf(x) = 4x2 7xverify thatZ1 1f(x)dx= Z 11f(x) Integration by partsIntegration by parts is a technique which can often be used tointegrate products offunctions. Ifuandvare both functions ofxthenZudvdxdx=uv ZvdudxdxWhen dealing with definite integrals the relevant formula isZbaudvdxdx= [uv]ba each of the following with respect tox.