Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual

Transcription

1 Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, ,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A is the accumulated amout, P is the picipal, is the iteest ate, ad t is the time i yeas. Discouts The discout, D o a discouted loa of M dollas at a sample aual iteest ate of fo t yeas is D = Mt Whee D = discout (iteest paid at time of loa) M = matuity value (amout boowed) = discout ate (aual simple iteest ate) t = legth of loa. The poceeds P of the loa is the actual amout the boowe eceives whe the loa is made ad is give by P = M D. Fo discouted loas, the effective iteest ate is umbe of yeas. Compoud Iteest Fo egula loas, ef = + iteest ate, ad is the umbe of compoudig peiods pe yea. t eff =, whee is the iteest ate ad t is the t, whee e is the effective iteest ate, is the aual P +, whee A is the accumulated amout, P is the picipal, is the aual iteest ate, is the umbe of compoudig peiods pe yea, ad t is the time i yeas fo compoud iteest. Use TVM solve i the calculato. t A= Pe, whee A is the accumulated amout, P is the picipal, is the aual iteest ate compouded cotiuously, ad t is the time i yeas. Peset Value ad Futue Value of Loas, Auities, ad Sikig Fuds Use TVM solve, ete kow ifomatio ad pess ALPHA Ete with cuso o PV (o FV) to solve fo peset value (o futue value). Note: PMT = if thee is a oe-time deposit/loa; PV o FV = if thee ae egula paymets with othig i the begiig. A good estimate fo doublig time is 7 divided by the iteest ate. The Gauss elimiatio method ivolves a sequece of opeatios o a system of liea equatios to obtai equivalet systems util a solutio is foud. Row-Reduced Fom of a Matix. Each ow cosistig etiely of zeos lies below ay othe ow havig ozeo eties.. The fist ozeo ety i each ow is (called a leadig ).. I ay two successive (ozeo) ows, the leadig i the lowe ow lies to the ight of the leadig i the uppe ow. 4. If a colum cotais a leadig, the the othe eties i that colum ae zeos. A colum i a coefficiet matix is i uit fom if oe of the eties i the colum is a ad the othe eties ae zeos. The Gauss-Joda Elimiatio Method. Wite the augmeted matix coespodig to the liea system.. Itechage ows (Opeatio ), if ecessay, to obtai a augmeted matix i which the fist ety i the fist ow is ozeo. The pivot the matix about this ety.

2 Math 66 Week-i-Review - S. Nite // Page of. Itechage the secod ow with ay ow below it, if ecessay, to obtai a augmeted matix i which the secod ety i the secod ow is ozeo. Pivot the matix about this ety. 4. Cotiue util the fial matix is i ow-educed fom. If thee is a ow i the augmeted matix cotaiig all zeos to the left of the vetical lie ad a ozeo ety to the ight of the lie, the the system of equatios has o solutio. Theoem a. If the umbe of equatios is geate tha o equal to the umbe of vaiables i a liea system, the oe of the followig is tue: i. The system has o solutio. ii. The system has exactly oe solutio. iii. The system has ifiitely may solutios. b. If thee ae fewe equatios tha vaiables i a liea system, the the system eithe has o solutio o it has ifiitely may solutios. Usig Matices to Repeset Data A matix is a odeed ectagula aay of umbes. A matix with m ows ad colums has size m x. The ety i the ith ow ad jth colum is deoted by a ij. If A is a m x matix with elemets a ij, the the taspose of A is the x m matix T A with elemets a ji. Scala Multiplicatio If A is a matix ad c is a eal umbe, the the scala poduct ca is the matix obtaied by multiplyig each ety of A by c. Fid the matix X satisfyig the matix equatio X + B = A. Matix Multiplicatio i Geeal If A is a matix of size m x ad B is a matix of size x p, the the matix poduct of A ad B, AB is defied ad is a matix of size m x p. The idetity matix of size ( ows ad colums) is give by The idetity matix has the popety that I A= A fo ay x matix A, ad BI = B fo ay s x matix B. I paticula, if A is a squae matix of size, the The Ivese of a Squae Matix Let A be a squae matix of size. A squae matix I A= AI A. = A of size such that A A= A I the ivese of A. Solvig Systems of Equatios with Ivese Matices If AX = B is a liea system of equatios i ukows ad if A - exists, the X = A - B is the uique solutio of the system. is called

3 Math 66 Week-i-Review - S. Nite // Page of. Tommy eeds $, to pay off a bill. He has take out a discout loa that has a mothly discout ate of.6%. The loa must be paid back i moths. How much will Tommy pay back (i.e., what is the matuity value)?. Fid the effective yield o a discout loa with a discout ate of.% fo moths. Roud you aswe to two decimal places.. A picipal of $4, eas.6% pe yea simple iteest. How log will it take fo the futue value to become $6,7? Roud the aswe to two decimal places. 4. Fid the peset value of $6, due i yeas at at ate of 7% pe yea, compouded quately.. Fid the accumulated amout if $, is ivested at.% pe yea, compouded mothly fo yeas. 6. Fid the amout eeded to deposit ito a accout today to yield semi-aual pesio paymets of $, fo the ext yeas if the accout eas.% pe yea, compouded semiaually. 7. A couple is savig fo thei daughte s educatio at Texas A&M Uivesity. They wat to have $8, at the ed of 6 yeas. (a) How much should they put each moth ito a savigs accout that eas a aual ate of % compouded mothly? (b) How much iteest would they ea ove the life of the accout? (c) Detemie the fud balace afte yeas. 8. A busiess ceates a sikig fud i ode to have $7, to eplace machiey i yeas. (a) How much should be deposited ito the accout at the ed of each quate if the aual iteest ate is % compouded quately? (b) How much iteest would they ea ove the life of the accout? (c) Detemie the balace afte yeas ad 7 yeas. 9. The Juaez s have $6, fo a dow paymet o a house. They pla to sped $ i mothly paymets o the house. If the motgage ate is 6.% pe yea compouded mothly fo the -yea motgage, what pice house ca they affod?. A $6, loa is to be amotized i equal mothly paymets ove yeas with iteest ate of 4% pe yea compouded mothly. Complete the fist thee ows of a amotizatio table.. Six yeas ago, Jey fiaced $8, to puchase a house. The tem of the motgage was yeas with a 6.%/yea iteest ate, compouded mothly o the upaid balace. The iteest ate has dopped to 4.%/yea compouded mothly, ad Jey plas to efiace he house. (a) What is Jey s cuet mothly motgage paymet? (b) What is Jey s cuet outstadig picipal? (c) If Jey decides to efiace by secuig a -yea motgage loa fo the outstadig picipal at 4.%/yea compouded mothly, what will be he mothly motgage paymet? (d) How much iteest will Jey save by efiacig?

4 Math 66 Week-i-Review - S. Nite // Page 4 of. Jig plas to etie i yeas ad wats to stat a etiemet accout to povide $, pe moth fo yeas whe she eties. She ca get at 4% aual iteest ate, compouded mothly. If she stats with $ ad makes mothly paymets fom ow util etiemet, how much must she deposit each moth.. Detemie which of the followig matices ae i ow-educed fom. If ot, why ot? a) b) c) 7 d) e) 4. Pivot the matix about the cicled elemet. 4. Solve the followig systems ad itepet the solutio. (a) -x + y 4z = 7 x - y + = 4z -y + 4z 6 = x (b) x y + z = x + y z = - -x + y - = z (c) x + 4y 6 = z -x y + z = - y + 4z x = -4 (d) -x + y + = z x y z = 4 4x + z = 6y 6. Set up, solve, ad itepet the solutio: Julio has $, to ivest i thee stocks A, B, ad C. He wats to ivest thee times as much i stock C as i stocks A ad B combied. He estimates a etu of % o stock A, 4% o stock B, ad 6% o stock C. How much should he ivest i each stock to have a etu of $7 pe yea o the total ivestmet?

5 Math 66 Week-i-Review - S. Nite // Page of 7. Set up, solve, ad itepet the solutio: A shop makes ecklaces, eaigs, ad bacelets. The table shows the umbe of each item speds i each aea. If thee ae hous available i the stigig aea, 4 hous available i the paitig aea, ad 8 hous available i the fiishig aea, how may of each item ca be made if all the available hous ae to be used? Stigig Paitig Fiishig Necklace 8 Eaig pai 8 6 Bacelet 8 8. Give c ad B =, fid AB. a 9. Give c ad B =, fid BA. a. Give 4 ad B =, fid A T - B.. Give, B =, ad C =, fid matix K such that KA + B = C.. Set up the matix equatio ad solve the system by ivese matices. x y + = z -z + x + y = -7 y x = -4