Markscheme May 2017 Calculus Higher level Paper 3

Transcription

1 M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages

2 M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted to any othe peson without the authoization of the IB Global Cente, Cadiff.

3 3 M7/5/MATHL/HP3/ENG/TZ0/SE/M Instuctions to Eamines Abbeviations M (M) A (A) R N AG Maks awaded fo attempting to use a valid Method; woking must be seen. Maks awaded fo Method; may be implied by coect subsequent woking. Maks awaded fo an Answe o fo Accuacy; often dependent on peceding M maks. Maks awaded fo an Answe o fo Accuacy; may be implied by coect subsequent woking. Maks awaded fo clea Reasoning. Maks awaded fo coect answes if no woking shown. Answe given in the question and so no maks ae awaded. Using the makscheme Geneal Mak accoding to RM Assesso instuctions and the document Mathematics HL: Guidance fo e-making May 07. It is essential that you ead this document befoe you stat making. In paticula, please note the following: Maks must be ecoded using the annotation stamps. Please check that you ae enteing maks fo the ight question. If a pat is completely coect, (and gains all the must be seen maks), use the ticks with numbes to stamp full maks. If a pat is completely wong, stamp A0 by the final answe. If a pat gains anything else, it must be ecoded using all the annotations. All the maks will be added and ecoded by RM Assesso. Method and Answe/Accuacy maks Do not automatically awad full maks fo a coect answe; all woking must be checked, and maks awaded accoding to the makscheme. It is not possible to awad M0 followed by, as A mak(s) depend on the peceding M mak(s), if any. Whee M and A maks ae noted on the same line, eg M, this usually means M fo an attempt to use an appopiate method (eg substitution into a fomula) and fo using the coect values. Whee the makscheme specifies (M), N3, etc., do not split the maks.

4 4 M7/5/MATHL/HP3/ENG/TZ0/SE/M Once a coect answe to a question o pat-question is seen, ignoe futhe coect woking. Howeve, if futhe woking indicates a lack of mathematical undestanding do not awad the final. An eception to this may be in numeical answes, whee a coect eact value is followed by an incoect decimal. Howeve, if the incoect decimal is caied though to a subsequent pat, and coect FT woking shown, awad FT maks as appopiate but do not awad the final in that pat. Eamples Coect answe seen Futhe woking seen Action Awad the final 8 (incoect decimal value) (ignoe the futhe woking). sin 4 sin Do not awad the final 4 3. log a log b log ( ab) Do not awad the final 3 N maks Awad N maks fo coect answes whee thee is no woking. Do not awad a mitue of N and othe maks. Thee may be fewe N maks available than the total of M, A and R maks; this is delibeate as it penalizes candidates fo not following the instuction to show thei woking. 4 Implied maks Implied maks appea in backets eg (M), and can only be awaded if coect wok is seen o if implied in subsequent woking. Nomally the coect wok is seen o implied in the net line. Maks without backets can only be awaded fo wok that is seen. 5 Follow though maks Follow though (FT) maks ae awaded whee an incoect answe fom one pat of a question is used coectly in subsequent pat(s). To awad FT maks, thee must be woking pesent and not just a final answe based on an incoect answe to a pevious pat. If the question becomes much simple because of an eo then use discetion to awad fewe FT maks. If the eo leads to an inappopiate value (eg sin.5), do not awad the mak(s) fo the final answe(s). Within a question pat, once an eo is made, no futhe dependent A maks can be awaded, but M maks may be awaded if appopiate. Eceptions to this ule will be eplicitly noted on the makscheme.

5 5 M7/5/MATHL/HP3/ENG/TZ0/SE/M 6 Misead If a candidate incoectly copies infomation fom the question, this is a misead (MR). A candidate should be penalized only once fo a paticula misead. Use the MR stamp to indicate that this has been a misead. Then deduct the fist of the maks to be awaded, even if this is an M mak, but awad all othes so that the candidate only loses [ mak]. If the question becomes much simple because of the MR, then use discetion to awad fewe maks. If the MR leads to an inappopiate value (eg sin.5), do not awad the mak(s) fo the final answe(s). 7 Discetionay maks (d) An eamine uses discetion to awad a mak on the ae occasions when the makscheme does not cove the wok seen. In such cases the annotation DM should be used and a bief note witten net to the mak eplaining this decision. 8 Altenative methods Candidates will sometimes use methods othe than those in the makscheme. Unless the question specifies a method, othe coect methods should be maked in line with the makscheme. If in doubt, contact you team leade fo aice. Altenative methods fo complete questions ae indicated by METHOD, METHOD, etc. Altenative solutions fo pat-questions ae indicated by EITHER... OR. Whee possible, alignment will also be used to assist eamines in identifying whee these altenatives stat and finish. 9 Altenative foms Unless the question specifies othewise, accept equivalent foms. As this is an intenational eamination, accept all altenative foms of notation. In the makscheme, equivalent numeical and algebaic foms will geneally be witten in backets immediately following the answe. In the makscheme, simplified answes, (which candidates often do not wite in eaminations), will geneally appea in backets. Maks should be awaded fo eithe the fom peceding the backet o the fom in backets (if it is seen). Eample: fo diffeentiating f () sin(53), the makscheme gives: ( ) cos(5 3) 5 0cos(5 3) f Awad fo cos(5 3) 5, even if 0cos(5 3) is not seen.

6 6 M7/5/MATHL/HP3/ENG/TZ0/SE/M 0 Accuacy of Answes Candidates should NO LONGER be penalized fo an accuacy eo (AP). If the level of accuacy is specified in the question, a mak will be allocated fo giving the answe to the equied accuacy. When this is not specified in the question, all numeical answes should be given eactly o coect to thee significant figues. Please check wok caefully fo FT. Cossed out wok If a candidate has dawn a line though wok on thei eamination scipt, o in some othe way cossed out thei wok, do not awad any maks fo that wok. Calculatos A GDC is equied fo pape 3, but calculatos with symbolic manipulation featues (eg TI-89) ae not allowed. Calculato notation The mathematics HL guide says: Students must always use coect mathematical notation, not calculato notation. Do not accept final answes witten using calculato notation. Howeve, do not penalize the use of calculato notation in the woking. 3 Moe than one solution Whee a candidate offes two o moe diffeent answes to the same question, an eamine should only mak the fist esponse unless the candidate indicates othewise.

7 7 M7/5/MATHL/HP3/ENG/TZ0/SE/M. attempt to use l Hôpital s ule, M sin cos sin limit lim o 0 ln( ) ln( ) Note: Awad fo numeato fo denominato. this gives 0/0 so use the ule again lim cos sin cos o 0 ( ) ( ) (M) Note: Awad fo numeato fo denominato. = Note: This is dependent on all pevious maks being awaded, ecept when the fist application of L'Hopital's does not lead to 0/0, when it should be awaded fo the coect limit of thei deived function. [7 maks]. (a) (i) (sec ) a 3a 5a sec a a a 3 5 (ii) 3 5 a a a M 4 3 Note: Condone the pesence of tems with powes geate than fou. (b) equating constant tems: a equating tems: 3a3 a a3 3 4 equating tems: 5a5 aa 3 a5 3 5 [3 maks] [3 maks] Total [6 maks]

8 8 M7/5/MATHL/HP3/ENG/TZ0/SE/M 3. conside I N d ln M Note: Do not awad if n is used as the vaiable o if lowe limit equal to, but some subsequent A maks can still be awaded. Allow as uppe limit. let y ln d d y, [, N] [ln, ln N] I ln N ln dy y (M) () () Note: Condone absence of limits, o wong limits. ln N y ln Note: is fo the coect integal, iespective of the limits used. Accept coect use of integation by pats. lnn ln (M) Note: M is fo substituting thei limits into thei integal and subtacting. as N Notes: Allow =, limit does not eist, diveges o equivalent. Do not awad if wong limits substituted into the integal but allow N o as an uppe limit in place of ln N. (by the integal test) the seies is divegent (because the integal is divegent) Notes: Do not awad this mak if used as uppe limit thoughout. [9 maks]

9 9 M7/5/MATHL/HP3/ENG/TZ0/SE/M 4. (a) dy y v v d d M the diffeential equation becomes v f() v d d f () v v integating, ln Constant f() v v AG [3 maks] (b) EITHER f () v 3v v ln ( ) C f v v 3v v v ln C v () M Note: is fo coect factoization. ln C v OR v v v d 3 d vv M Note: v d is fo coect factoization. () ln C v continued

10 0 M7/5/MATHL/HP3/ENG/TZ0/SE/M Question 4 continued THEN substitute y o v when theefoe C Note: This can be awaded anywhee in thei solution. substituting fo v, ln y (M) M Note: Awad fo coect substitution of y into thei epession. y ln Note: Awad fo any eaangement of a coect epession that has y in the numeato. () y ln ln ln (o equivalent) [0 maks] Total [3 maks]

11 M7/5/MATHL/HP3/ENG/TZ0/SE/M 5. (a) Note: Cuve, both ectangles and coect - values equied. aea of ectangles and Note: Coect values on the y-ais ae sufficient evidence fo this mak if not othewise indicated. in the above diagam, the aea below the cuve between and + is between the aeas of the lage and smalle ectangle d o d integating, ln ln ln ln (R) AG [4 maks] (b) (i) summing the ight-hand pat of the above inequality fom to n, n n ln M 3 n n ln ln ln ln n n EITHER 3 n n ln n n OR () ln lnln 3 ln ln ( n) ln ( n) ln ( ) n AG continued

12 M7/5/MATHL/HP3/ENG/TZ0/SE/M Question 5 continued (ii) n 3 n... ln ln... ln 3 n n M Note: n n ln M is fo using the coect inequality fom (a), fo both sides beginning with, fo completely coect epession. Note: The might be added afte the sums have been calculated. lnn AG [6 maks] (c) (i) fom (b)(i) U ln( n) ln n 0 (ii) n n n Un Un ln( n ) ln n M n ln n n 0 (using the esult poved in (a)) U n Un AG [4 maks] (d) it follows fom the two esults that { U n } cannot be divegent eithe in the sense of tending to o oscillating theefoe it must be convegent Note: Accept the use of the esult that a bounded (monotonically) deceasing sequence is convegent (allow positive, deceasing sequence ). R [ mak] Total [5 maks]