Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the following concepts, give emples fo ech nd use them in poblems. Wht is/e: 1 logithmic function? How does it elte to eponentil functions? The popeties of ln? eponentil gowth nd eponentil decy? How does the initil size nd the te of gowth/decy of the popultion ffect the fomul fo the popultion size? 3 the hlf life of dioctive element? How do we clculte it given the decy constnt λ? Convesely, how do we use to to clculte λ? 4 cbon dting? How do we use it to detemine the ge of n tifct? 5 continuously compounded inteest? 6 the pesent vlue of money to be eceived in t yes? 7 the logithmic deivtive of ft? How do we use it to find the eltive te of chnge of ft? Wht is the pecentge te of chnge of ft? Which functions hve constnt eltive te of chnge? 8 n ntideivtive? How mny ntideivtives does function f hve? Wht is the connection between ntideivtives of f nd the indefinite integl f d? 9 the bsic emples of functions nd thei ntideivtes? 10 the lineity popeties of indefinite integls? How do we use them to compute indefinite integls? 11 diffeentil eqution DE? Wht do we solve fo in diffeentil eqution? Wht is n initil condition of DE? 1 Riemnn sum of function f on the intevl [, b]? Wht is? the numbe n of subintevls? left hnd, ight hnd, o midpoint Riemnn sum? How do we choose the smple points i in ech subintevl? To wht does the Riemnn sum ppoch when n? 13 the definite integl f d? Wht is its connection with the indefinite integl of f nd with the net e unde f? 14 the lineity popeties of definite integls? How do we use them to compute definite integls? 15 the e between two cuves? Wht is the cuves intesect, o wht if some of the e ppes unde the -is: do we still tke with positive sign? 16 the vege of finitely mny numbes? the vege vlue of function f on intevl [, b]? 17 consumes suplus? How do we clculte it?. Theoems nd Poblem Solving Techniques Undestnd ech of the following theoems nd be ble to pply ech theoem ppopitely in poblems. 1 Popeties of ln. The ntul logithmic function ln is the invese of the ntul eponentil function e, i.e. lne = fo ll, nd e ln = fo ll > 0. Hence, the following two equtions e equivlent: ln = b nd = e b. ln is defined only fo > 0.
b ln 1 = 0. c lne = fo ll. d ln fo ll > 0. e ln > 0 fo ll > 1, nd ln < 0 fo ll < 1. f ln is concve up fo ll > 0. g ln = 1 fo ll > 0. Moe genelly, ln = 1 fo ll 0. h lny = ln + ln y fo ll, y > 0 ln tuns poducts into sums. i ln y = y ln fo ll y nd fo ll > 0. j ln = ln ln y fo ll, y > 0 ln tuns quotients into diffeences. y k ln 1 = ln fo ll > 0. Genel eponentil functions. b = ln b b fo ny b > 0, fo ll. 3 Eponentil Gowth. If P t gows t te popotionl to its size, i.e. if P t = kp t fo some gowth constnt k > 0, then P t is the function given by: P t = P 0 e kt, whee P 0 = P 0 is the initil size of the popultion i.e. t time t = 0. 4 Eponentil Decy. If P t decys t te popotionl to its size, i.e. if P t = kp t fo some decy constnt k < 0, then P t is the function given by: P t = P 0 e kt = P 0 e λt, whee P 0 = P 0 is the initil size of the popultion, nd λ = k > 0 is positive constnt. 5 Hlf life. The hlf life t of dioctive element is clculted by setting P t = 1 P 0, i.e. P 0 e λt = P 0 ln, nd solving fo λ: λt = ln1/ = ln, i.e. t = λ. 6 Cbon Dting. Given tht n tifct contins % of the 14 C level in living mtte, we detemine the ge of the tifct by setting the following eqution nd solving it fo t: P t = P 0 e 0.0001t = 100 P 0, 0.0001t = ln 100 = t = ln 100 0.0001. Hee we used tht λ = 0.0001 is the decy constnt fo 14 C. 7 Inteest compounded sevel times. If P is the pincipl mount, is the yely inteest, nd m is the numbe of times the inteest in compounded yely, the mount of money in t yes is given by At = P 1 + m mt. Note tht hee 5% inteest te tnsltes into = 0.05 in the bove fomul. 8 Continuously compounded inteest. If P is the pincipl mount, is the yely inteest, nd the inteest is compounded continuously, the mount of money in t yes is given by At = P e t, i.e. the money gows eponentilly with gowth constnt. Note tht hee 5% inteest te tnsltes into = 0.05 in the bove fomul. 9 Pesent Vlue. If A mount of money is to be eceived in t yes t inteest te, the pesent mount P of A cn be clculted by setting A = At = P e t, nd solving fo P : P = Ae t. 10 Reltive Rte of Chnge. The eltive te of chnge of ft is the logithmic deivtive of ft: eltive te of chnge = d dt ln ft = f t ft.
The pecentge te of chnge of ft is the eltive te of chnge of ft epessed s pecentge: pecentge te of chnge = f t 100 ft %. 11 Antideivtives. If f is continuous function on, b, then ny two ntideivtives of g diffe by constnt, i.e. if F nd G e two ntideivtives of g, then G = F fo some constnt C. Consequently, to find ll ntideivtives of f, it suffices to find one such ntideivtive F, nd then dd C: f d = F, fo C R. 1 Eveywhee zeo deivtive. If F = 0 fo ll.b, then F is constnt function on, b, i.e. F = C fo some constnt C. 13 Bsic emples of ntideivtives. f f Check k k k = k n n+1 n+1, n 1 n + 1 n + 1 = n 1 = 1 ln ln = 1 e e e = e e e e = e, > 0 ln ln = + b n + b n+1 + b n+1, n 1 n + 1 n + 1 1 ln + b ln + b + b = + b n = 1 + b 14 Lineity Popeties of Indefinite Integls IL s. The integl of sum is the sum of the integls: f + gd = f d + g d. b The integl of diffeence is the diffeence of the integls: f gd = f d g d. c Constnts jump in font of integls: c fd = c f d. Wning: Integls of poducts e not equl to poducts of integls: f gd f d g d. 15 Diffeentil Equtions DEs. Given diffeentil eqution f = g with initil condition f = b, we solve fo f. Fist, find n ntideivtive G fo g nd set f = G. Net, plug in the initil condition: f = b = G nd solve fo C. Finlly, list the function f = G fo the newly found G nd C. 3
16 Velocity distnce poblems. Given the velocity vt of n object, to find the distnce st tvelled between times t = nd t = b, set up the DE: s t = vt nd hence st = vt dt. Find n ntideivtive Gt of vt nd set st = Gt. Finlly, subtct: sb s = Gb G. 17 Riemnn sums. Let f be continuous function on [, b]. To set up the nth Riemnn sum of f, we divide the intevl [, b] into n subintevls of length = b /n. We choose some i in ech subintevl: if sked fo left endpoint ight endpoint, midpoint ppoimtion, choose i to be the left endpoint ight endpoint, midpoint of the ith subintevl. Add up the es of the esulting ectngles to fom the desied Riemnn sum: S n = f 1 + f + + f n. When n, the Riemnn sum ppoches the net e unde the gph of f: lim S n = lim f 1 + f + + f n = net e unde the gph of f = n n Wning: Aes e lwys positive. positive o negtive! f d. Net es, definite integls nd Riemnn sums my be 18 Fundmentl Theoem of Clculus. Let f be continuous function on [, b]. Then f d = F b = F b F, whee F is one ntideivtive of f, i.e. F = f. 19 Lineity Popeties of Definite Integls. The integl of sum is the sum of the integls: f + gd = f d + b The integl of diffeence is the diffeence of the integls: f gd = c Constnts jump in font of integls: c fd = c f d f d. g d. g d. 0 Ae between cuves. Let f nd g be two continuous functions on [, b] such tht f g, i.e. f is bove g on [, b]. Then the e between the two cuves is A = f g d. If the two cuves intesect t sevel points, we ptition the intevl [, b] into sevel subintevls [ 1, ], [, 3 ], [ 3, 4 ],... so tht on ech intevl one of the two functions is entiely bove the othe function, nd then we dd up the coesponding es: A = 1 f g d + 3 g f d + 4 3 f g d +, whee f g on [ 1, ], g f on [, 3 ], f g on [ 3, 4 ], etc. Thus, ech piece of e between f nd g ends up being tken with positive sign. 1 Intesection points. To find the intesection points of two cuves f nd g, set f = g nd solve fo. Finlly, plug in the found vlues fo into f o g to ive t the ctul points of intesection, f. 4
Avege Vlues. Given sevel numbes 1,,..., n, thei ithmetic vege is clculted by 1 + + + n /n. Given function f on intevl [, b], the vege vlue of f on [, b] is clculted by f d vege vlue of f = b 3 Ae of Cicle. The e of cicle of dius is π. 4 Volume of Solid of Revolution. The volume of solid obtined by otting the gph of f bout the is ove [, b] is give by: πf d. 5 Volume of Sphee. The volume of sphee of dius R is 4 3 πr3. 6 Volume of Pymid. The volume of pymid is given by 1 bse e height. The volume 3 of ight cicul cone of dius nd height h is given by 1 3 π h. 7 Consumes Suplus. Given demnd cuve p = f, the consumes suplus is clculted by A 0 f fa d, whee the quntity demnded is A, nd the pice sked is fa = B. 8 Fomuls fo Sums. 1 + + 3 + + n = 1 + + 3 + + n = 1 3 + 3 + 3 3 + + n 3 = nn + 1 nn + 1n + 1 6 nn + 1 + + + 3 + + n = 1 n+1, 1. 1 3. Eecises to Review Review ll clss, homewok nd quiz poblems nd solutions. These should be sufficient to do well on the em. 4. Chet Sheet Fo the em, you e llowed to hve chet sheet - one pge of egul 8 11 sheet. You cn wite whteve you wish thee, unde the following conditions: The whole chet sheet must be hndwitten by you own hnd! No eoing, no copying, nd fo tht mtte, no teing pges fom the tetbook nd psting them onto you chet sheet. Any violtion of these ules will disqulify you chet sheet nd my end in disqulifying you em. I my decide to ndomly check you chet sheets, so let s ply it fi nd sque. : Don t be feksuus! Stt studying fo the em sevel dys in dvnce, nd pepe you chet sheet t lest dys in dvnce. This will give you enough time to become fmili with you chet sheet nd be ble to use it moe efficiently on the em. 5