Diploma Programme First eamiatios 014
33 Topic 6 Core: Calculus The aim of this topic is to itroduce studets to the basic cocepts ad techiques of differetial ad itegral calculus ad their applicatio. 6.1 Iformal ideas of limit, cotiuity ad covergece. Defiitio of derivative from first priciples f( h) f( ) f ( ) lim. h 0 h The derivative iterpreted as a gradiet fuctio ad as a rate of chage. Fidig equatios of tagets ad ormals. Idetifyig icreasig ad decreasig fuctios. The secod derivative. Higher derivatives. si Iclude result lim 1. 0 Lik to 1.1. Use of this defiitio for polyomials oly. Lik to biomial theorem i 1.3. Both forms of otatio, d y d ad first derivative. f Use of both algebra ad techology., for the d y Both forms of otatio, ad f ( ), for d the secod derivative. Familiarity with the otatio d y ad d ( f ) ( ). Lik with iductio i 1.4. 48 hours TOK: The ature of mathematics. Does the fact that Leibiz ad Newto came across the calculus at similar times support the argumet that mathematics eists prior to its discovery? It: How the Greeks distrust of zero meat that Archimedes work did ot lead to calculus. It: Ivestigate attempts by Idia mathematicias (500 1000 CE) to eplai divisio by zero. TOK: Mathematics ad the kower. What does the dispute betwee Newto ad Leibiz tell us about huma emotio ad mathematical discovery? Appl: Ecoomics HL 1.5 (theory of the firm); Chemistry SL/HL 11.3.4 (graphical techiques); Physics SL/HL.1 (kiematics).
34 6. Derivatives of l., si, cos, ta, e ad Differetiatio of sums ad multiples of fuctios. The product ad quotiet rules. The chai rule for composite fuctios. Related rates of chage. Implicit differetiatio. Derivatives of sec, csc, cot, arcsi, arccos ad arcta. a, log a, Appl: Physics HL.4 (uiform circular motio); Physics 1.1 (iduced electromotive force (emf)). TOK: Mathematics ad kowledge claims. Euler was able to make importat advaces i mathematical aalysis before calculus had bee put o a solid theoretical foudatio by Cauchy ad others. However, some work was ot possible util after Cauchy s work. What does this tell us about the importace of proof ad the ature of mathematics? TOK: Mathematics ad the real world. The seemigly abstract cocept of calculus allows us to create mathematical models that permit huma feats, such as gettig a ma o the Moo. What does this tell us about the liks betwee mathematical models ad physical reality? 6.3 Local maimum ad miimum values. Optimizatio problems. Poits of ifleio with zero ad o-zero gradiets. Graphical behaviour of fuctios, icludig the relatioship betwee the graphs of f, f ad f. Not required: Poits of ifleio, where f ( ) is ot defied, for eample, y 13 at (0,0). Testig for the maimum or miimum usig the chage of sig of the first derivative ad usig the sig of the secod derivative. Use of the terms cocave up for f ( ) 0, cocave dow for f ( ) 0. At a poit of ifleio, f ( ) 0 ad chages sig (cocavity chage).
35 6.4 Idefiite itegratio as ati-differetiatio. Idefiite itegral of, si, cos ad e. Other idefiite itegrals usig the results from 6.. The composites of ay of these with a liear fuctio. 6.5 Ati-differetiatio with a boudary coditio to determie the costat of itegratio. Defiite itegrals. Area of the regio eclosed by a curve ad the -ais or y-ais i a give iterval; areas of regios eclosed by curves. Volumes of revolutio about the -ais or y-ais. Idefiite itegral iterpreted as a family of curves. 1 d l c. Eamples iclude 1 5 1 d, d 3 4 1 ad d 5. The value of some defiite itegrals ca oly be foud usig techology. Appl: Idustrial desig.
36 6.6 Kiematic problems ivolvig displacemet s, velocity v ad acceleratio a. Total distace travelled. ds v, dt d d d a v s v v. dt dt ds t Total distace travelled v dt. t1 Appl: Physics HL.1 (kiematics). It: Does the iclusio of kiematics as core mathematics reflect a particular cultural heritage? Who decides what is mathematics? 6.7 Itegratio by substitutio O eamiatio papers, o-stadard substitutios will be provided. Itegratio by parts. Lik to 6.. Eamples: si d ad l d. Repeated itegratio by parts. Eamples: e d ad e si d.