Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
|
|
- Berniece Farmer
- 5 years ago
- Views:
Transcription
1 . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold th final accuracy mark. (c) dy = (4 ) ( + ) A A 3 Not : (thir u ) + ( + )(thir v ) A: Any on trm corrct. A: Both trms corrct. (d) (4 ) ( ) 0 + = = ( 7)( ) = 0 = 7, A Whn 7 7 y =, = 9, whn, y = = dd A Not dy st : For stting thir found in part (c) qual to 0. nd : Factoris or liminat out corrctly and an attmpt to factoris a 3-trm quadratic or apply th formula to candidat s a + b + c. S ruls for solving a thr trm quadratic quation on pag of this Appndi. 3 rd dd: An attmpt to us at last on -coordinat on y = ( + ). Not that this mthod mark is dpndnt on th award of th two prvious mthod marks in this part. Som candidats writ down corrsponding y-coordinats without any working. It may b ncssary on som occasions to us your calculator to chck that at last on of th two y-coordinats found is corrct to awrt sf. Final A: Both{ = }, y = 7 and { = }, y = 9. cao Not that both act valus of y ar rquird. 7 []
2 . (a) ( + )( ) ( + )( 3) = ( ) ( 3) A af 3 Not : An attmpt to factoris th numrator. : Corrct factorisation of dnominator to giv ( + )( 3). Can b sn anywhr. (b) + 9 ln = = + = 3 = ( ) 3 = 3 d A af cso 4 Not : Uss a corrct law of logarithms to combin at last two trms. This usually is achivd by th subtraction law of logarithms to giv + 9 ln = + Th product law of logarithms can b usd to achiv ln ( + 9 ) = ln (( + )). Th product and quotint law could also b usd to achiv + 9 In = 0 ( ) + d: Rmoving ln s corrctly by th ralisation that th anti-ln of is. Not that this mark is dpndnt on th prvious mthod mark bing awardd. : Collct trms togthr and factoris. Not that this is not a dpndnt mthod mark A: or or. af Not that th answr nds to b in trms of. Th dcimal answr is Not that th solution must b corrct in ordr for you to award this final accuracy mark. [7]
3 3. (i) (a) ln(3 7) = ln(3 7) = Taks of both sids of th quation. This can b implid by 3 7 =. Thn rarrangs to mak th subjct. d 3 7 = + 7 = { = } + 7 Eact answr of. 3 3 A 3 (b) 3 7+ = ln (3 7+ ) = ln Taks ln (or logs) of both sids of th quation. ln 3 + ln 7+ = ln Applis th addition law of logarithms. ln = ln ln = ln A o (ln 3 + 7) = +ln Factorising out at last two trms on on sid and collcting numbr trms on th othr sid. dd + ln = { = } Eact answr of 7 + ln 3 + ln 7 + ln3 A o
4 (ii) (a) f() = +3, y = +3 y 3 = ln (y 3)= Attmpt to mak (or swappd y) th subjct ln y 3 = Maks th subjct and taks ln of both sids Hnc ( ) = ln( 3) ln( 3) or ln ( 3 ) f or ( ) = ln( y 3) f (s appndi) A cao f (): Domain: > 3 or (3, ) Eithr > 3 or (3, ) or Domain > 3. 4 (b) g()=ln( ),, > fg() = ln( ) +3 {=( ) + 3} fg() : Rang: y > 3 or (3, ) An attmpt to put function g into function f. ln( ) +3 or ( ) +3 or + 4. A isw Eithr y > 3 or (3, ) or Rang>3 or fg()>3. 3 [] 4. (a) P = 0 t t = 0 P = 0 o = 0() = 0 0
5 (b) t P = = 0 t 0 t rarrangs quation to mak th subjct. 000 t = ln 0 t = awrt.6 or 3 yars A Not t = or t = awrt.6 t = will scor A0 t (c) dp = 6 dt k and k 0. l 6 t t A (d) t 0 = 6 t 0 = ln 6 { } Using 0 = d P and d t an attmpt to solv t to find th valu of t or P = 0 l 0 ln 6 or P = 0 ( ) Substituts thir valu of t back into th quation for P. d 0(0) P = = 0 0 or awrt 0 A 3 6 []
6 . (a) Curv rtains shap whn > k ln Curv rflcts through th -ais whn > k ln (0, k ) and ( ln, 0 3 in th corrct positions. (b) Corrct shap of curv. Th curv should b containd in quadrants, and 3 (Ignor asymptot) ( k, 0) and ( 0, ln k ) (c) Rang of f: f() > k or y > k or ( k, ) Eithr f() > k or y> k or ( k, )or f > k or Rang > k.
7 (d) y = k y + k = Attmpt to mak ln(y + k) = (or swappd y) th subjct ln( y + k) = Maks th subjct and taks ln of both sids Hnc ( ) = ln( + k) ln( + k) or ln ( + k) A cao 3 f () f (): Domain: > k or ( k, ) Eithr > k or ( k, ) or Domain > k or ft on sidd ft inquality thir part (c) RANGE answr [0] 6. (a) f() = + ( + 4) ( )( + 4) R, 4,. ( )( + 4) ( ) + f() = ( ) ( + 4) An attmpt to combin to on fraction Corrct rsult of combining all thr fractions A = ( )( + 4) + = Simplifis to giv th corrct [( + 4)( )] numrator. Ignor omission of dnominator ( + 4)( 3) = [( + 4)( )] An attmpt to factoris th d numrator. ( 3) = ( ) Corrct rsult A cso AG A
8 (b) g() = 3 R, ln. Apply quotint rul: u = 3 du = v = dv = g () = ( ) ( ( ) 3) vu' uv' Applying v Corrct diffrntiation A = ( ) + 3 = Corrct rsult A AG ( ) cso 3 (c) g () = = = ( ) =( ) Puts thir diffrntiatd numrator qual to thir dnominator. = = A ( 4)( ) = 0 Attmpt to factoris or solv quadratic in = 4 or = = ln 4 or = 0 both = 0, ln 4 A 4 [] 7. (a) g() (b) fg() = f ( ) =3 + ln = + 3 A (fg : + 3 ) (c) fg () 3
9 (d) d ( + 3 ) = + 6 A + 6 = + (6 ) = 0 0, 6 = 0 A = 0, 6 A A 6 [0]. (a) f () = A = 3 (+) = 0 = A f( ) = 3 (b) = 0.96 = = (c) Choosing (0.7, 0.7 6) or an appropriat tightr intrval. f(0.7 ) = f(0.7 6)= A Chang of sign (and continuity) root (0.7, 0.7 6) cso A 3 ( = 0.76, is corrct to 4 dcimal placs) Not: = is accurat [] 9. (a) + = + = ln = (ln ) A
10 (b) dy = + dy = (ln ) = 6 y = 6 (ln ) y = ln A 4 [6] 0. (a) 6 3 ln 3 = ln 6 or ln = ln or ln 3 6 = 0 = (only this answr) Acso Answr = with no working or no incorrct working sn: A ln 6 Not: = from ln = ln 3 = ln M0A0 ln = ln 6 ln 3 = (ln 6 ln 3) allow, = (no wrong working) A (b) ( ) = 0 (any 3 trm form) ( 3)( ) = 0 = 3 or = Solving quadratic dp = ln 3, = 0 (or ln ) A 4 st for attmpting to multiply through by : Allow y, X, vn, for nd is for solving quadratic as far as gtting two valus for or y or X tc 3 rd is for convrting thir answr(s) of th form = k to = lnk (must b act) A is for ln3 and ln or 0 (Both rquird and no furthr solutions) [6]. (a) D = 0, t =, = 0 =.33 awrt A (b) D = 0 + 0, t =, = = 3.49 (*) Acso
11 Alt. (b) = = 3.49 (*) Acso (Main schm is for (0 + 0 ), or {0 + thir(a)} N.B. Th answr is givn. Thr ar many corrct answrs sn which dsrv M0A0 or A0 T (c) = 3 3 T = = T = ln T = or 3. or 3 A 3 T st M is for ( ) = 3 o.. nd T M is for convrting = k (k > 0) to T k = ln indpndnt of st M.. This is Trial and improvmnt: as schm, corrct procss for thir quation (two qual to 3 s.f.) A as schm [7]. (a) 4 C (b) 300 = t t = 7 sub. T = 300 and attmpt to rarrang to 0.0t = a, whr a Q 0.0t 7 = 400 A corrct application of logs t = 7.49 A 4 dt (c) = 0 0.0t ( for k 0.0t ) A dt At t = 0, rat of dcras = (±).64 C / min A 3
12 (d) T >, (sinc 0.0t 0 as t ) [9] 3. dy = A At = 3, gradint of normal = = 3 3 y ln l = 3( 3) y = A [] 00a 4. (a) Stting p 300 at t = = + a (300 = 00a); a = 0. (c.s.o.) (*) d A 3 0.t (b) 00(0.) 0 = 0.t + 0. ; 0.t = 6. A Corrctly taking logs to 0. t = ln k t = 4 (3.9..) A 4 (c) Corrct drivation: (Showing division of num. and dn. by 0.t ; using a) (d) Using t, 0.t 0, p 336 = 00 A 0. [0]. (a) log3 = log taking logs = log log 3 or log3 = log =.46 cao A 3 A
13 + (b) = log + = 4 + = 4 = or quivalnt; 4 multiplying by to gt a linar quation A 4 (c) sc = / cos sin = cos tan = = 4, A 3 us of tan [0] 6. (a) I = 3 + Using limits corrctly to giv +. (c.a.o.) A 3 must subst 0 and and subtract (b) A = (0, ); y = dy = Equation of tangnt: y = +; c =. ; A 4 attmpting to find q. of tangnt and subst in y = 0, must b linar quation (c) + y = y + 4y = + 4 y = y + 4 ; A putting y = and att. to rarrang to find. g 4 ( ) = or quivalnt A 3 must b in trms of (d) gf(0) = g(); =3 ; A att to put 0 into f and thn thir answr into g []
14 7. (a) (i) = a y (ii) In both sids of (i) i. ln = ln a y ln or ( y =) log a = ln a = y ln a * y ln a = ln cso = ylna is BO Must s ln a y or us of chang of bas formula. (b) y = ln a ln dy,, = * ln a, A cso dy dy ALT. or = ln a, = ln a, A cso nds som corrct attmpt at diffrntiating. (c) log 0 0 = A is (0, ) y A = from(b) m = or or (or bttr) 0 ln a 0 ln0 qu of targt y = m ( 0) i. y = ( 0) or y = + (o.) 0ln0 0 ln0 ln0 A 4 Allow ithr ft thir y A and m (d) y = 0 in (c) 0 = +, = 0 ln 0 0ln0 ln0 ln0 = 0 0 ln 0 or 0( ln0) or 0 ln0( ) ln0 A Attmpt to solv corrct quation. Allow if a not = 0. [0] y. (a) O Shap 3 p = 3 or { 3, 0} sn
15 (b) Gradint of tangnt at Q = q Gradint of normal = q Attmpt at quation of OQ [y = q] and substituting = q, y = ln 3q or attmpt at quation of tangnt [y 3 ln q = q( q)] with = 0, y = 0 or quating gradint of normal to (ln 3q)/q q + ln 3q = 0 (*) A 4 (c) ln 3 = 3 = ; = 3 ; A (d) = 0.90; = , 3 = , 4 = ; A Root = (3 dcimal placs) A 3 [] 9. dy = 6 A At =, dy = ; y = 4 ln A; Tangnt is y 4 + ln = ( ) At y = 0, = + ln = ln + ln = ln A [7] 0. (a) A is (, 0); B is (0, ) ; (b) y = Chang ovr and y, = y y = ln ( + ) y = + ln ( + ) A f : / + ln ( + ), > A A
16 (c) f() = 0 is quivalnt to = 0 Lt g() = g(3) =. g(4) =.3 Sign chang root α A (d) n + = + ln( n + ), = 3. = A 3 = A 4 = 3.04 = 3.0 Nds convincing argumnt on 3 d.p. accuracy Tak 3.03 and nt itration is rducing 3.0 Answr: 3.0 (3 d.p.) A [4]. (i) + 3 = = ln 6 = (ln 6 3) A 3 (ii) ln (3 + ) = = 4 = ( 4 ) 3 A 3 [6]. (a) y y = ln O shap -intrcpt lablld
17 (b) d y = so tangnt lin to (, ) is y = + C th lin passs through (, ) so = + C and C = 0 Th lin passs through th origin. A 3 y y = y= ln O (c) All lins y = m passing through th origin and having a gradint > 0 li abov th -ais. Thos having a gradint < will li blow th lin. y = so it cuts y = ln btwn = and =. (d) 0 =.6 n = 3 =.9 =. A 3 =. 4 =. =.7 A 3 () Whn =.7, ln = > 0 3 Whn =.6, ln = < 0 A Chang of sign implis thr is a root btwn. A 3 [3]
4037 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,
More informationCondensed. Mathematics. General Certificate of Education Advanced Level Examination January Unit Pure Core 3. Time allowed * 1 hour 30 minutes
Gnral Crtificat of Education Advancd Lvl Eamination January 0 Mathmatics MPC Unit Pur Cor Friday 0 January 0.0 pm to.00 pm For this papr you must hav: th blu AQA booklt of formula and statistical tabls.
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationPrelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationwww.onlineamhlp.com www.onlineamhlp.com UIVERSITY OF CAMBRIDGE ITERATIOAL EXAMIATIOS GCE Advancd Lvl MARK SCHEME for th Octobr/ovmbr qustion papr for th guidanc of tachrs 9 FURTHER MATHEMATICS 9/ Papr,
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationy cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationMATHEMATICS (MEI) 4753/01 Methods for Advanced Mathematics (C3)
ADVANCED GCE MATHEMATICS (MEI) 4753/0 Mthods for Advancd Mathmatics (C3) QUESTION PAPER Candidats answr on th printd answr book. OCR supplid matrials: Printd answr book 4753/0 MEI Eamination Formula and
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More information7' The growth of yeast, a microscopic fungus used to make bread, in a test tube can be
N Sction A: Pur Mathmatics 55 marks] / Th rgion R is boundd by th curv y, th -ais, and th lins = V - +7 and = m, whr m >. Find th volum gnratd whn R is rotatd through right angls about th -ais, laving
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C.
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Tim: 3hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A, B and C. SECTION -A Vry Short Answr Typ Qustions. 0 X = 0. Find th condition
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More information( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationMAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP
EXERCISE - MAXIMA-MINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6.
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationTEMASEK JUNIOR COLLEGE, SINGAPORE. JC 2 Preliminary Examination 2017
TEMASEK JUNIOR COLLEGE, SINGAPORE JC Prliminary Eamination 7 MATHEMATICS 886/ Highr 9 August 7 Additional Matrials: Answr papr hours List of Formula (MF6) READ THESE INSTRUCTIONS FIRST Writ your Civics
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationPhysicsAndMathsTutor.com
. (a) Simplify fully + 9 5 + 5 (3) Given that ln( + 9 5) = + ln( + 5), 5, (b) find in terms of e. (Total 7 marks). (i) Find the eact solutions to the equations (a) ln (3 7) = 5 (3) (b) 3 e 7 + = 5 (5)
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationSUMMER 17 EXAMINATION
(ISO/IEC - 7-5 Crtifid) SUMMER 7 EXAMINATION Modl wr jct Cod: Important Instructions to aminrs: ) Th answrs should b amind by ky words and not as word-to-word as givn in th modl answr schm. ) Th modl answr
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More information2013 Specialist Mathematics GA 3: Written examination 2
0 0 Spcialist Mathmatics GA : Writtn xamination GENERAL COMMENTS Th 0 Spcialist Mathmatics xamination comprisd multipl-choic qustions (worth marks) and fiv xtndd qustions (worth 8 marks). Th papr smd accssibl
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationas a derivative. 7. [3.3] On Earth, you can easily shoot a paper clip straight up into the air with a rubber band. In t sec
MATH6 Fall 8 MIDTERM II PRACTICE QUESTIONS PART I. + if
More informationCHAPTER 24 HYPERBOLIC FUNCTIONS
EXERCISE 00 Pag 5 CHAPTER HYPERBOLIC FUNCTIONS. Evaluat corrct to significant figurs: (a) sh 0.6 (b) sh.8 0.686, corrct to significant figurs (a) sh 0.6 0.6 0.6 ( ) Altrnativly, using a scintific calculator,
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationObjective Mathematics
x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8
More informationHyperbolic Functions Mixed Exercise 6
Hyprbolic Functions Mid Ercis 6 a b c ln ln sinh(ln ) ln ln + cosh(ln ) + ln tanh ln ln + ( 6 ) ( + ) 6 7 ln ln ln,and ln ln ln,and ln ln 6 artanh artanhy + + y ln ln y + y ln + y + y y ln + y y + y y
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationnd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).
Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationAnswers & Solutions. for MHT CET-2018 Paper-I (Mathematics) Instruction for Candidates
DATE : /5/8 Qustion Booklt Vrsion Rgd. Offic : Aakash Towr, 8, Pusa Road, Nw Dlhi-5 Ph.: -75 Fa : -77 Tim : Hour Min. Total Marks : Answrs & Solutions for MHT CET-8 Papr-I (Mathmatics) Instruction for
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &D) AND CALCULUS. TIME : hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.If th portion
More information3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc
More informationLogarithms. Secondary Mathematics 3 Page 164 Jordan School District
Logarithms Sondary Mathmatis Pag 6 Jordan Shool Distrit Unit Clustr 6 (F.LE. and F.BF.): Logarithms Clustr 6: Logarithms.6 For ponntial modls, prss as a arithm th solution to a and d ar numrs and th as
More informationChapter 3 Exponential and Logarithmic Functions. Section a. In the exponential decay model A. Check Point Exercises
Chaptr Eponntial and Logarithmic Functions Sction. Chck Point Erciss. a. A 87. Sinc is yars aftr, whn t, A. b. A A 87 k() k 87 k 87 k 87 87 k.4 Thus, th growth function is A 87 87.4t.4t.4t A 87..4t 87.4t
More informationMATH 1080 Test 2-SOLUTIONS Spring
MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =
More informationMark Scheme (Results) January 2011
Mark (Results) January 0 GCE GCE Core Mathematics C (6664) Paper Edecel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn, London WCV 7BH Edecel is one of the leading
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationSupplemental Appendix: Equations of Lines, Compound Inequalities, and Solving Systems of Linear Equations in Two Variables
0000000707688_t.pdf /9/ : AM - 99 - ( ) Supplmntal Appndi: Equations of Lins, Compound Inqualitis, and Solving Sstms of Linar Equations in Two Variabls 0000000707688_t.pdf /9/ : AM - 90 - ( ) 0000000707688_t.pdf
More informationIntegration by Parts
Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'(
More informationLecture 19: Free Energies in Modern Computational Statistical Thermodynamics: WHAM and Related Methods
Statistical Thrmodynamics Lctur 19: Fr Enrgis in Modrn Computational Statistical Thrmodynamics: WHAM and Rlatd Mthods Dr. Ronald M. Lvy ronlvy@tmpl.du Dfinitions Canonical nsmbl: A N, V,T = k B T ln Q
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationMark Scheme Summer 2009
Mark Summer 009 GCE Core Mathematics C (666) Edecel is one of the leading eamining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic,
More informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationDivision of Mechanics Lund University MULTIBODY DYNAMICS. Examination Name (write in block letters):.
Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions
More informationSystems of Equations
CHAPTER 4 Sstms of Equations 4. Solving Sstms of Linar Equations in Two Variabls 4. Solving Sstms of Linar Equations in Thr Variabls 4. Sstms of Linar Equations and Problm Solving Intgratd Rviw Sstms of
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationSouthern Taiwan University
Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:
More informationEngineering Mathematics I. MCQ for Phase-I
[ASK/EM-I/MCQ] August 30, 0 UNIT: I MATRICES Enginring Mathmatics I MCQ for Phas-I. Th rank of th matri A = 4 0. Th valu of λ for which th matri A = will b of rank on is λ = -3 λ = 3 λ = λ = - 3. For what
More informationCore Mathematics C3 Advanced Level
Paper Reference(s) 666/0 Edecel GCE Core Mathematics C Advanced Level Wednesda 0 Januar 00 Afternoon Time: hour 0 minutes Materials required for eamination Mathematical Formulae (Pink or Green) Items included
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationMath-3. Lesson 5-6 Euler s Number e Logarithmic and Exponential Modeling (Newton s Law of Cooling)
Math-3 Lsson 5-6 Eulr s Numbr Logarithmic and Eponntial Modling (Nwton s Law of Cooling) f ( ) What is th numbr? is th horizontal asymptot of th function: 1 1 ~ 2.718... On my 3rd submarin (USS Springfild,
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationJune Further Pure Mathematics FP1 (new) Mark Scheme
June 009 6667 Further Pure Mathematics FP (new) Mar Q (a) Mars B () (c) (d) z + ( ) 5 (or awrt.4) α arctan or arctan arg z 0.46 or 5.8 (awrt) (answer in degrees is A0 unless followed by correct conversion)
More informationFP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
FP Mark Schemes from old P, P5, P6 and FP, FP, FP papers (back to June 00) Please note that the following pages contain mark schemes for questions from past papers. The standard of the mark schemes is
More information