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1 MATH 1501 QUICK REVIEW FOR FINAL EXAM FALL 2001 C. Heil Below is quick list of some of the highlights from the sections of the text tht we hve covere. You shoul be unerstn n be ble to use or pply ech item s pproprite. THIS LIST MAY NOT BE COMPLETE! A itionl items to it s you stuy for the finl exm. If there s nything you re not sure of, sk! It s very importnt to be ble to o clcultions, but you lso nee to know WHAT things re n WHY they re wht they re. Specil notes Our finl exm is Tuesy, December 11, 11:30.m. 2:20 p.m., in the lecture room (Weber SST Room 2). Arrive erly so tht we ll be ble to strt on time. The finl is worth 75 points. No books or clcultors llowe on the finl exm. SHOW YOUR WORK on the exm! To get prtil creit, we hve to be ble to figure out wht you re oing! The nswer t the en is the LEAST importnt prt of the problem, it s the WORK tht is importnt! WRITE CLEARLY! The finl is comprehensive, but t lest 50% of the points will be rwn from the mteril covere since Exm 3. Tht is, the mjority of points will be rwn from Sections 5.4 n lter but there will still be substntil number of points rwn from erlier mteril! I will be wy the lst week of clss (December 3-7). During tht week, the TA s will hve their usul office hours. Aitionlly, the following TA s will hve office hours open to stuents from ll three of our sections, plese feel free to go to ny of these office hours: Kevin Khn Skiles p.m., We, Dec 5 Meeti Shh Skiles p.m., Tues, Dec 4 n Thurs, Dec 6 I will be vilble most of Mony of finls week. The homework ssignments for the semester, solutions to the ol quizzes n exms for the semester, n prctice finl exm re vilble in the following irectory: ftp://ftp.mth.gtech.eu/pub/users/heil/1501 Chpter 1 The mteril in this chpter is bckgroun tht you shoul lrey be fmilir with. The following things re especilly importnt.. Know how to work with inequlities, especilly ones involving bsolute vlue. b. Wht is function? Know the menings of omin n rnge n how to fin them. Know how to work with compositions f g, incluing fining the formul n the omin. c. The six trig functions re extremely importnt. Know wht ll of them re. Review pges in the text crefully.. Wht oes the grph of ech of sin x, cosx,tnx,cotx,secx,ncscxlook like? e. Wht re the vlues of ech of those functions t x =0,π/6,π/4,π/3,π/2? (If you know the vlues of sin x n cos x t those points, you cn get ll the others.) f. Cn you use bsic fcts bout chnging the grph of function to plot the grphs of functions like sin 2x, 3cosx, cos(x 3π/2), sin πx, etc.? g. Know bsic trig ientities, especilly sin 2 x +cos 2 x=1 tn 2 x +1=sec 2 x sin 2x = 2 sin x cos x 1

2 Chpter 1 Homework 1.2 # 3, 5, 9, 21, 23, 27, 31, 37, 45, 49, # 5, 7, 9, 11, 15, 19, 23, 31, 35, 39, 41, 43, 47, 53, 55, # 5, 9, 13, 21, 27, 29, 31, 35, 41, 45, 51, # 3, 9, 11, 13, 19, 27, 37, 43, 45, 47, 59, # 12, 16, 17, 26, 33, 53, 61, # 11, 14, 27, 36, 45, 51, 54, Project 1.7 Chpter The ie of limit. Min points:. Wht is the intuitive efinition of limit? (See p. 60.) b. Nottion for limits. c. How to clculte limits.. One-sie limits. 2.2 The efinition of limit. This is the section contining the ε-δ efinition of limit. I will not test this mteril on the finl exm. 2.3 Some limit theorems. Min points:. Limits re unique. b. IF lim f(x) n lim g(x) exist then x x lim [cf(x)] = c lim f(x) x x lim [f(x)+g(x)] = lim f(x) + lim g(x) x x x ( )( ) lim [f(x)g(x)] = lim f(x) lim g(x) x x x f(x) lim x g(x) = lim f(x) x lim g(x) x if lim x g(x) 0 But the limit of f(x)+g(x)orf(x)g(x)orf(x)/g(x) CAN exist even though the limit of f(x) or the limit of g(x) oesn t exist! 2.4 Continuity. Min points:. Wht is the efinition of continuity? b. If f n g re both continuous t x, thencf, f + g, fg n f/g re ll continuous t x (for f/g you lso require tht g(x) 0). c. Continuity from the right n from the left.. Wht is the ifference between f being continuous t x = c n f hving limit s x c? 2.5 Pinching Theorem n trigonometric limits. Min points:. Wht is the pinching theorem n how o you use it? b. Trigonometric limits. Know the fcts sin x cos x 1 lim = 1 n lim =0 x 0 x x 0 x n be ble to use these to compute other limits. 2

3 2.6 Properties of continuous functions. Min points:. Intermeite Vlue Theorem: wht it is n how to use it. b. Bouneness Theorem: wht it is n how to use it. Chpter 2 Homework 2.1 # 1, 5, 9, 11, 13, 17, 21, 25, 27, 31, 35, 37, 41, 43, # 1, 3, 5, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 41-45, 51 (won t test on this section) 2.3 # 1, 3, 5-37 o, 39, 41, 43, 45, 47, 49, 51, # 1, 5, 7, 11, 15, 17, 19, 23, 27, 29, 33, 35, 41, 43, # 1-15 o, o, # 1, 3, 5, 9, 15, 19, 21, 23, 25, 27, 29, 31, 33 Chpter Derivtives. Min points:. Definition of the erivtive. Mening of secnt lines n tngent lines. b. How to use the efinition of the erivtive to compute the erivtive of functions. c. If f is ifferentible t x = c, thenfis continuous t x = c. But f coul be continuous t x = c WITHOUT being ifferentible t x = c! Give n exmple!. How to tell from sketch of the grph of function where it is ifferentible. e. How to rw n pproximte sketch of the erivtive of function. Importnt! 3.2 Differentition formuls. Min points:. Formuls for the sum, prouct, n quotient of erivtive. KNOW THE PRODUCT RULE AND THE QUOTIENT RULE!! 3.3 nottion, higher erivtives. Min points: x. How to use x nottion, especilly with the chin rule. b. Nottions for higher erivtives. c. Derivtive of x n for integer n. 3.4 Derivtive s rte of chnge. Min points:. Why is rte of chnge erivtive? b. Motion in stright line, velocity, ccelertion. When is it moving right or left, speeing up or slowing own? c. Motion uner grvity. 3.5 Chin Rule. Min points:. Wht is the chin rule in both Newton n Leibniz nottion. b. How to use the chin rule. EXTREMELY IMPORTANT!! 3.6 Trig functions. Min points:. Wht re the erivtives of ALL SIX trig functions? b. Combine prouct rule, quotient rule, n chin rule with trig functions. 3.7 Implicit ifferentition. Min points:. Wht oes it men for function to be efine implicitly? b. How o you fin the erivtive of function if it is only given implicitly? Answer: ifferentite both sies of the eqution tht gives the function implicitly, then solve for y/x. 3

4 3.8 Rtes of chnge per unit time. Wor problems. Min points:. If two or more quntities re relte, then their erivtives will be relte too. b. To solve relte rtes problem: rw picture, fin out wht wht re the functions n wht re the vribles, fin out wht quntities you KNOW (especilly reltionships between the functions), ifferentite both sies of the reltionships, solve for the erivtive you wnt, n finlly figure out wht tht erivtive is AT the time or plce tht you wnt it. Chpter 3 Homework 3.1 # 1, 5, 7, 9, 11, 13, 17, 25, 26, 27, 31, 33, 37, 39, 41, 43, 47, 49, 51, 55, # 3, 5, 9, 13, 17, 19, 23, 25, 29, 37, 43, 47, 55, 61, 67, # 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 33, 35, 39, 43, # 1, 5, 9, 13, 15, 17, 21, 23, 25, 27-36, 37, 41, 45, 47, 51, 55, # 1, 5, 7, 9, 11, 17, 19, 23, 25, 27, 29, 35, 37, 41, 45, 47, 49, 55, 61, 65, 67, # 1, 3, 5, 7, 9, 11, 15, 17, 19, 25, 27, 29, 35, 39, 41, 45, 49, 51, 53, 61, 63, 65, 69, # 3, 5, 7, 9, 11, 15, 17, 19, 25, 27, 31, 39, 41, 51, # 1-39 o 4.1 The Men-Vlue Theorem. Min points:. Wht oes the Men-Vlue Theorem sy? b. How o you use it in problems? Chpter Incresing n ecresing functions. Min points:. Definition of incresing n ecresing. b. IF f is ifferentible AND f (x) > 0forx (, b), then f is incresing on (, b). c. IF f is ifferentible AND f (x) < 0forx (, b), then f is ecresing on (, b).. IF f is ifferentible AND f (x) =0forx (, b), then f is constnt on (, b). e. Be ble to etermine where function is incresing, ecresing, or constnt. f. Given f, how to rw n pproximte sketch of f. g. Given sketch of f, how to get informtion bout f. 4.3 Locl Extreme Vlues. Min points:. Definitions of criticl points, extreme points, locl mx, locl min. b. The First Derivtive test n how to use it. A function must CHANGE from incresing to ecresing to hve locl mx, n must CHANGE from ecresing to incresing to hve locl min. c. The Secon Derivtive test: wht is it n how o you use it?. Be ble to etermine criticl points n tell which of those re extreme points n wht kin of extreme points they re (mx or min). 4.4 Enpoint n bsolute extreme vlues. Min points:. Definition of enpoint extreme vlues. b. Definition of bsolute extreme vlues. c. Be ble to fin ALL the extreme vlues of f on close intervl [, b] look for criticl points within (, b) n test them, AND test the enpoints. YOU MUST CHECK THE ENDPOINTS! 4.5 Mx/min wor problems. Min points:. Determine the function tht you wnt to fin the mx or min of. b. Eliminte vribles until you hve function of one vrible lone. 4

5 c. Fin ALL the extreme vlues of tht function you MUST test the enpoints s well (unless the omin of the function in question hs no enpoints). CHECK THE ENDPOINTS!. Answer the question ske give the quntity ske for, whtever it is. 4.6 Concvity n inflection points. Min points:. Definitions of concve up, concve own, inflection points. b. The concvity MUST CHANGE in orer for point to be n inflection point. An inflection point is NOT just where f (x) = 0!! The concvity MUST CHANGE!! c. f is concve up where f IS INCREASING, n this hppens when f (x) > 0.. Given f, how to rw n pproximte sketch of f n f. e. Given sketch of f, how to get informtion bout f n f. f. Given sketch of f, how to get informtion bout f n f. 4.7 Verticl n horizontl symptotes. Min points:. Be ble to fin the verticl n horizontl symptotes. b. For horizontl symptotes, lso be ble to etermine whether the function is ABOVE or BE- LOW the symptote. c. For verticl symptotes, lso be ble to etermine whether the function is heing to + or to s x pproches the symptote line.. Use this informtion to sketch the function. 4.8 Curve sketching. Min points:. Be ble to fin where f is incresing, ecresing, fin the criticl points n extreme points, etermine the concvity n inflection points, n fin the symptotes. b. Use this informtion to sketch the grph of f. Chpter 4 Homework 4.1 # 3, 7, 9, 11, 13, 15, 19, 21, 23, 29, 33, 37, 43, 44, 45, # 1-13 o, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41-46, # 3, 5, 9, 13, 15, 17, 19, 21, 25, 27, 29, 33, 39, # 1, 5, 9, 11, 17, 19, 21, 25, 29, 31, 33, 39, # 3, 5, 7, 9, 13, 15, 19, 21, 23, 27, 29, 33, 35, 39, 43, 49, # 1, 2, 5-27 o, 31, # 1, 2, 3, 5, 9, 13, 17, 19, 21, 23, 31, 35, 37, # 5, 9, 11, 13, 23, 25, 29, 33, 41, 49, Definite integrls. Min points: Chpter 5. Know wht ech of the following terms mens: prtition P, subintervls [x i 1,x i ], length of the subintervl i, size (norm) P of P, min m i, mx M i, lower sum L f (P ), upper sum U f (P ), points x i, Riemnn sum S (P ). 5

6 b. Definition of the efinite integrl f(x) x: it is the unique number (if one exists) such tht L f (P ) f(x) x U f(p ) for EVERY prtition P. Ech lower n upper sum therefore gives n estimte of the vlue of the integrl. c. How o the Riemnn sums S (P ) compre to the lower n upper sums n to the integrl f(x) x? Wht hppens to the lower, upper, n Riemnn sums s the size P of the prtition ecreses to zero?. Given function f n specifie prtition P, be ble to compute L f (P ), U f (P ), S (P )nto sketch the boxes involve in computing those sums. 5.2 F (x) = x f(t)t. Min points:. If f is continuous on [, b], then F is one ntierivtive of f. This mens tht F (x) =f(x), or: x f(t) t = f(x). x b. Any other ntierivtive of f hs the form G(x) =F(x)+C where C is constnt. c. KnowbsicpropertiesofF : itisifferentibleon (, b) (eveniff isn t ifferentible!), F () = 0, if f(x) 0 everywhere then F (x) is incresing, etc.. Know integrtion formuls such s f(x) x = c f(x) x + c f(x) x, f(x) x = b f(x) x. e. Do computtions with vritions on F (x) suchs x 3 f(t)t. Nee to pply the CHAIN RULE here! 5.3 Funmentl Theorem of Clculus (FTC). Min points:. Wht oes the FTC sy? b. Use the FTC to compute efinite integrls involving simple functions such s x r,sinx,cosx, sec 2 x,secxtn x, n more complicte ones such s 1 0 (x3 +1)3x 2 x. These re esy cses of the more complicte u-substitutions tht we o lter. 5.4 Are problems. Min points:. Given some curves, be ble to sketch the region enclose by them. b. Then be ble to set up n work out the integrl the gives the re enclose by the curves. c. Be especilly creful if the curves cross: then you hve to brek up into subregions, n compute seprte integrl for ech subregion. 5.5 Inefinite integrls. Min points:. f(x) x enotes generic ntierivtive of f(x). b. Therefore, if F (x) is ny ONE ntierivtive of f(x), then f(x) x = F (x) +C gives the generic form of ll the possible ntierivtives of f(x). 5.6 u-substitutions. Min points:. You wnt to fin u so tht your integrl cn be written s f(u) u. Theniffis nice enough, you cn work out the integrl in terms of u n substitute bck for u t the en. b. PRACTICE, PRACTICE, PRACTICE until you cn see how to choose u! c. The tble on p. 302 is goo summry of some bsic formuls. 5.7 Aitionl properties. Min points:. If f(x) 0 everywhere then f(x) x 0. 6

7 b. f(x) x f(x) x. Chpter 5 Homework 5.1 # 1-9 o, 11-17, 21, 23, 25, 35 Typos: #13, 14 shoul sy Repet exercise 12, #35c shoul sy Use exercise # 1-29 o, # 1-35 o, 37, 41, 45, 49, 51, 53, 55, # 3, 5, 9, 11, 15, 19, 23, 25, 27, 28, 29, 30, 33, # 1, 5, 9, 13, 15, 17, 23, 27, 31, 37, # 1, 7, 9, 11, 17, 19, 23, 25, 29, 33, 37, 39, 41, 45, 47, 49, 53, 61, 65, 69, 71, 73, # 1, 3, 5, 7, 9, 11, 13, 15, 17, 23 Chpter More on re. Min points:. Relly unerstn the connection between n integrl n Riemnn sum. b. Given region boune by some curves, be ble to prtition either the x-xis or the y-xis n fin the corresponing Riemnn sum tht pproximtes tht re. Then by looking t the formul for the Riemnn sum, be ble to etermine the integrl tht represents the true re. 6.2 Volumes by iscs n wshers. Min points:. Given soli, be ble to prtition n xis n then fin the volume of one slice of the soli. b. For solis of revolution, such slice will be isc or wsher (this section) or shell (Section 6.3). c. For something tht isn t soli of revolution, you nee to think crefully bout wht the volume of the slice will be. See especilly problems in this section. Importnt!. Get the Riemnn sum pproximtion to the volume of the soli by ing up the volumes of ll the slices. e. From the Riemnn sum pproximtion, be ble to etermine the integrl tht represents the true volume of the soli. f. The MAIN POINT is to unerstn how to o these problems by Riemnn sums. Formuls such s eqution (6.2.3) on pge 331 follow from this you nee to unerstn Riemnn sums, not memorize those formuls. Besies, those formuls ONLY pply to solis of revolution. 6.3 Volumes by shells. Min points:. Given soli of revolution, be ble to prtition the xis n fin the volume of one slice of the soli. b. If the slice is shell, be ble to fin the volume of tht shell, n then by choosing x i (or yi if prtitioning the y-xis) to be the mipoint of the ith subintervl, turn tht volume into something tht involves only x i n x i (or yi n y i if prtitioning the y-xis). c. Then fin the Riemnn sum n then fin the integrl tht gives the true volume of the soli. 6.5 Work. Min points:. Work is force times istnce. If the force is from grvity (s in ll the problems tht we will o), then force = weight. b. In English units, weight = pouns (lbs). In Metric units, weight = mss times ccelertion ue to grvity, with mss given in kilogrms (kg) n the ccelertion ue to grvity being 9.8 m/sec 2. c. Weight = volume times weight per unit volume. For exmple, wter weighs 62.5 lbs per cubic foot, so x ft 3 of wter weighs 62.5x lbs. 7

8 . Much s in the volume problems, prtition your object verticlly, then fin the work to move ech slice of the object. Tht work is the weight of tht slice times the istnce tht slice moves. A up the work for ech slice to get the Riemnn sum pproximtion to the totl work. Tht les you to the integrl tht represents the true work performe on the entire object. The big ifference between these problems n volume problems is tht you lso hve to tke into ccount the istnce tht the slice moves. Chpter 6 Homework 6.1 # 3, 7, 11, 17, 21, 23, 25, 29, 35, 37, 44, # 5, 11, 13, 19, 25, 27, 29, 31, 33, 39, 43, # 3, 5, 9, 11, 15, 17, 25-30, # 1, 3, 13, 15, 17, 19, 21, 25, 27 Chpter One-to-one functions n inverse functions. Min points:. A function f is 1-1 if f(x 1 ) f(x 2 ) whenever x 1 x 2. b. To test for 1-1, use the horizontl lines test. c. A function tht is lso incresing or lwys ecresing on connecte omin is sure to be A 1-1 function f hs n inverse function f 1. The min property of the inverse function is tht f(f 1 (x)) = x n f 1 (f(x)) = x. e. Given f(x), be ble to fin formul for the inverse function. Be ble to fin the omin of the inverse function. f. The grph of y = f 1 (x) is foun by tking the mirror imge of the grph of y = f(x) bout the line y = x. g. The erivtives of f n f 1 re relte: (f 1 ) (b) = 1 f () if b = f(). 7.2 Logrithms. Min points:. log x is the exponent tht must be rise to in orer to get x. Thtis, y=log x y = x. In other wors: x n log x re INVERSE FUNCTIONS. If f(x) = x then f 1 (x) =log x! b. Some importnt properties: log x log x = x = x log = 1 log 1 = 0 log xy = log x+log y log x/y = log x log y log 1/x = log x log x y = y log x omin(log x) = (0, ) 8

9 c. Nturl logs re logs with bse = e Nottion: ln x =log e x. Write own ll the properties given in prt b bove for the function y =lnx!. Aitionlly, ln x is n ntierivtive of 1/x: x 1 ln x = 1 t t, i.e., ln x is the re uner the curve y =1/t between 1 n x. Consequently, x ln x = 1 x, x > More on nturl logs. Min points:. While ln x is only efine when x > 0, the function ln x is efine for ll x 0 n is ifferentible for ll x 0. b. x ln x = 1 x. Applying the chin rule, we therefore get: x ln u = 1 u u x n 1 u = ln u +C u Be ble to use these! c. The formuls in the boxe eqution (7.3.6) on p. 387 will be supplie to you on the finl exm, but you nee to be fmilir with them n know how to use them.. There will be no problems on logrithmic ifferentition (see pges of the text). 7.4 e x. Min points:. e x n ln x re inverse functions. b. x ex = e x. By the chin rule, we then get: x eu = e u u n x Be ble to use these! e u u = e u + C 7.6 Exponentil growth n ecy. Min points:. The solution to f (x) =kf(x) is f(x)=ce kx where C is constnt. In fct, C = f(0). b. If k>0 then this is exponentil growth, ifk<0thenitisexponentil ecy. c. Be ble to o wor problems bout popultion growth, rioctive ecy, etc. Chpter 7 Homework 7.1 # 5, 9, 13, 17, 21, 28-31, 37, 39, # 1, 11, 13, 15, 17, 19, 23, # 3, 7, 11, 13, 17, 21, 25, 31, 35, 41, 45, 57, # 3, 5, 7, 11, 13, 17, 19, 21, 25, 29, 31, 33, 35, 37, 39, 41, 43, 51, 61, 65, # 5, 7, 9, 11, 15, 17, 21, 25 Chpter Review of integrtion formuls. Min points:. You shoul memorize equtions 1 4, 6 7, n on pge 441 of the text. b. Formuls 8 11 on pge 441 of the text will be supplie to you, but you nee to be fmilir with them. 9

10 8.2 Integrtion by prts. Min points:. Be fmilir with the integrtion by prts formuls in both the inefinite n efinite forms: uv = uv vu n uv = uv b vu = u(b)v(b) u()v() b. PRACTICE, PRACTICE, PRACTICE until you re ble to see how to choose u n v. c. Once u n v re chosen, you fin u n v n use the integrtion by prts formuls bove.. Sometimes you hve to o integrtion by prts twice, then solve for the integrl you wnt. e. Be ble to tell whether n integrl nees to be one by u-substitution or by integrtion by prts. Chpter 8 Homework 8.1 # 1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 27, 29, 31, 35, 37 Hints: ln x 3 =3lnx,tn 2 x+1=sec 2 x,e 1/x = e (x 1 ) 8.2 # 3, 5, 7, 9, 13, 17, 19, 21, 23, 25, 29, 39, 41 Hints: ln x 1/2 =(1/2) ln x, #7 o integrtion by prts twice, #13 tke v = x vu Complex Numbers Complex number hnout. Min points:. The imginry unit is i = 1. So i 2 = 1. b. A complex number hs the form z = + bi where, b re rel numbers. Know wht the complex plne is. c. Know how to fin the rel n imginry prt of complex number: if z = +bi then Re(z) = n Im(z) =b.. The moulus or bsolute vlue of z = + bi is z = 2 + b 2. e. The rgument of z is the ngle from the rel xis to z. Writerg(z) for the rgument of z. It must be in the rnge 0 rg(z) < 2π. f. Given z, be ble to fin z n rg(z). Given z n rg(z), be ble to fin z. g. The polr representtion of z is z =(rcos θ)+(rsin θ)i where r = z n θ =rg(z). h. The complex conjugte of z = + bi is z = bi. i. Be ble to n multiply complex numbers. j. Be ble to multiply complex numbers in the polr representtion form. Remember tht zw = z w n rg(zw) =rg(z)+rg(w). Complex number hnout Homework # 1, 3, 5, 7, 9, 13, 15, 16, 21, 23, 25, 27, 29, 33, 35, 37, 39, 47 10

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