(a) by substituting u = x + 10 and applying the result on page 869 on the text, (b) integrating by parts with u = ln(x + 10), dv = dx, v = x, and

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1 Supplementry Questions for HP Chpter 5. Derive the formul ln( + 0) d = ( + 0) ln( + 0) + C in three wys: () by substituting u = + 0 nd pplying the result on pge 869 on the tet, (b) integrting by prts with u = ln( + 0), dv = d, v =, nd (c) integrting by prts with u = ln( + 0), dv = d nd v = Find the re of the region bounded by the curves y = e 3, y = 2 3 e2 nd the lines = 0 nd = () Show tht n e d = n e n n e d. (b) Use the bove formul to find 5 e d. 4. If we set u =, dv = d, du = d, v =, then we get 2 d = ( ) b ( ) 2 d = + so d = + d Subtrcting d from both sides gives Wht is wrong with this rgument? 0 =. d 5. We hve seen tht the present vlue of A dollrs t yers from now is given by Ae rt where r is the nnul rte of continuous compounding. Suppose revenue of compny flows t time-dependent rte R(t). R(t) tends to increse when business is good nd tends to decrese when business is bd. The present vlue of n n-yer revenue strem is given by P V = n 0 R(t)e rt, i.e., for n yers the revenue is deposited, the mount given by R(t) t ny instnt t. The bove formul gives the present vlue of ll revenue deposited into n ccount up to yer n. Let R(t) = t. () Wht is the present vlue of the first two yers of revenue t 5%? (b) t 0%? (c) Wht is the present vlue of the first three yers t 5%?

2 6. () Find d 3. Hint: try the substitution u = 6. (b) Find +4 d. Hint: try the substitution u = Show tht for n, ( ) n+ ( + ) n+ ( 2 ) n+ d = [ ( ) n ( + ) n ] n ( 2 ) n + C 8. In the following two problems, mke preliminry substitution before using the method of prtil frctions. () (b) e 4t (e 2t ) 3 +ln t t(3+2 ln t) 2 9. A store hs n inventory of q units of certin product t time t = 0. The store sells the product t stedy rte of q w units per week, ehusting the inventory in w weeks. Find the verge inventory level during the period 0 t w. Does this gree with common sense? 0. The men vlue theorem for definite integrls sttes if f() is continuous on [, b], then there eists t lest one number c between nd b such tht (b ) b f() d = f(c) In other words, there eists t lest one c with < c < b such tht f(c) is the verge vlue of f() over the intervl [, b]. For the following two functions, find ll vlues of c tht stisfy the bove theorem. () f() = ( + ), 0 2 (b) f() = 2, e. Two substnces, A nd B rect to form third substnce C in such wy tht if 30 grms of A nd 20 grms of B re brought together t time t = 0, then the mount (t) of C present in the miture hs rte of chnge with respect to time given by d = k(30 )(20 ), k > 0 is constnt nd < 20 Solve for s function of t, ssuming (0) = 0. 2

3 2. For the following differentil equtions, find the eqution of solution which psses through the given point. () dy d = e y, y(0) =. (b) dy (c) dy d d = 0.2y(8+0.) (00+0.5y) = ( + ln )y, y() =., y(0) = 0 (Don t solve for y in this cse.) 3. Let u() be utility function for welth. This mens u() is mesure of the stisfction of owning dollrs in welth. A utility function of constnt reltive risk version stisfies the differentil eqution u () = u ()b (b is constnt) Beginning with the substitution v() = u (), solve for u(). > 0.) (Assume u () > 0 nd 4. () Continuous compounding in bnk ccount t n interest rte of r per yer cn be modeled by the differentil eqution db = rb where B is the blnce in the ccount. Solve this differentil eqution if P is the principl in the ccount t time t = 0. (Write B db = r.) (b) If pyments re mde out of the ccount t continuous rte of N dollrs, then the differentil eqution becomes db/over = rb N. Solve this eqution if P is the principl t time t = 0. (Write db = r.) (c) Let r = 0.05, N = 200. P = 3000, 4000, B N r Sketch the solution to (b) on the sme set of es for 5. In prticulr country, beginning from time t = 0 (in yers), interest rtes incresed ccording to the function r = 0.25t +.50, where r is the interest rte t time t. The rte of chnge of the blnce B in bnk ccount ws described by db = (0.25t )B () Assuming B = when t = 0, find B s function of t. (Assume B > 0.) (b) How much money ws in the ccount when t ws five yers? 6. A certin commodity is being sold t price of $p per unit. Over period of time, mrket forces will mke this price tend towrds the equilibrium price $p 0, t which supply 3

4 ectly blnces demnd. The rte t which the price chnges is described by the Evns Price Adjustment model, which sys tht dp is proportionl to the difference between the mrket price nd the equilibrium price, tht dp = k(p p 0), k < 0 is constnt () Solve this eqution for p s function of t. (Assume p p 0 for ll t.) (b) Wht hppens to p s t? (c) Wht is the price when t = 0? 7. The gmm function is defined for ll > 0 by () Find Γ(). Γ() = 0 t e t. (b) Integrte by prts with respect to t to show tht, for positive n, Γ(n + ) = nγ(n). (Assume lim n e = 0 for ll n.) (c) Find simple epression for Γ(n) is n is positive integer. 8. It is possible to determine whether n improper integrl converges or diverges by compring it to known convergent or divergent integrl. If f() d is convergent nd f() g() for ll, then g() d is convergent. If f() d is convergent nd f() g() for ll, then g() d is convergent. If f() d is divergent nd f() g() for ll, then g() d is divergent. If f() d is divergent nd f() g() for ll, then g() d is divergent. Compre ech of the following integrls to nother integrl in order to determine whether or not they converge. (Do not ttempt to determine their vlue.) () 2 (b) d ( 2 +5) d 2 9. For wht vlues of p is the integrl p d convergent? If it is convergent, wht is its vlue? 4

5 20. The rte, r, t which people get sick during n epidemic of the flu cn be pproimted by r = 000te 0.5t where r is mesured in people per dy nd t is mesured in dys since the epidemic begn. () Sketch grph of r s function of t 0. Assume lim t r(t) = 0. (b) When re people getting sick fstest? (c) How mny people get sick ltogether? 2. On Purchse Timing for Rpidly Improving Consumer s Good In this question it is ssumed tht infltion is t n nnul rte of r compounded continuously. In other words, the present vlue of M dollrs t yers in the future is P = Me rt. Now, rpidly improving consumers goods (the most obvious emple being computers) hve, for the purposes of this question, two min fetures: () Becuse of technicl progress, version of product becomes incresingly obsolete s time psses. Hence, the version s price decreses s time psses. It is ssumed tht the price of ny version will be Ce wt, where C is constnt nd w is referred to s the rte of cost reduction through technicl progress. For emple, if w = for version A, then if A used to cost $5, 000 t time t = 0 (i.e, C = 5000), then A will only cost 5000e w() = 5000e = 5000 e t time t =. (2) Lter versions of product cn generte more revenue per unit of time thn erlier versions of the product. We ssume in this question tht ny fied version of product genertes constnt mount of revenue per unit of time. () Suppose tht consumer with side business wishes to upgrde product. The erlier version genertes constnt $R per unit of time nd the newer version genertes constnt $S per unit of time, where S > R. Also, the price of the newer version is Ce wt t time t, for constnts C nd w. The consumer would like to choose the time T of purchse so s to optimize profits. BY finding, for ny T, the present vlue (i.e, in terms of time t = 0) of profits if the consumer buys the new version t time T, find the time T tht optimizes the present vlue of profits. Note: ssume tht ) the consumer will never upgrde gin, nd 2) the business will lst indefinitely (i.e., her business, for the purposes of simplicity, will lst forever). 5

6 You my ssume tht ny criticl vlue is indeed mimum. Hint: see Integrtion s Applied to Annuities in 7.3 of HP (p. 883). Think of the constnt revenue strems R nd S s continuous nnuities tht re constnt. (b) A prticulr consumer with her own side photocopying business hs decided to upgrde from her current SX-35A Copout to the new improved SX-35B Copyct. The Copout genertes constnt $600/month, wheres the Copyct genertes constnt $700/month. Infltion is t 2% nnully, (but compounded continuously). If, t the beginning of 996, the Copyct ws $5, 000 but t the beginning of Februry, 996 it ws $4, 90, when should she stop Copping out nd buy the Ct? 6