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{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 206 60 "Mortgage payments, in tererst rate, loan, and term of mortgag" }{TEXT 205 1 "e" }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 54 "Assuming compounding interest takes place continuosly." }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 8 "restart;" }{MPLTEXT 1 207 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 110 "Assuming annual interest rate of 9%, and monthly payments of $800 a month. The equation will be the following:" }{TEXT 205 0 "" }} {PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 38 "eq1:= diff(y(t),t) = 0.09*y(t) - 9600;" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 17 "dsolve(eq1 ,y(t));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 46 "Here is the solution assuming a \+ $100000 loan." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 31 "dsolve(\{eq1,y(0)=100000 \},y(t));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 47 "Let's define a function out of \+ this expression." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 43 "tempv:=rhs(dsolve(\{eq 1,y(0)=100000\},y(t)));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 21 "y1:=unapply(tempv,t);" }{MPLTEXT 1 207 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 21 "plot(y1(t), t=0..40);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }} {PARA 200 "" 0 "" {TEXT 205 65 "We can ask several questions regarding this problem. For example," }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 55 "1.1 when is the loan going to be completely cancelled?" }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 84 "1.2 What interest \+ rate may allow the cancellation of the house in exactly 30 years?" } {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 3 "1.1" }{TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 25 "Years:=fsolve(y1(t)=0,t);" }{MPLTEXT 1 207 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 29 "months:=round((Years-30) *12);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 1 " \+ " }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 40 "So the answer is 3 0 years and 10 months." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 4 "1.2 " }{TEXT 205 0 "" }} }{EXCHG {PARA 200 "" 0 "" {TEXT 205 103 " We will be working with the \+ same equation but this time will let the interest rate, r as a parame ter." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 36 "eq11:= diff(y(t),t) = r*y(t) - \+ 9600;" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 32 "dsolve(\{eq11,y(0)=100000\},y(t));" }{MPLTEXT 1 207 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 44 "tempv:=rhs(dsolve(\{eq11 ,y(0)=100000\},y(t)));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 24 "y11:=unapply(tempv,t,r);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 34 "100*fsolve(y11(30,r) =0,r,0.01..1);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 116 "That means that a sl ight variation in the interest rate to 8.94% is enough for paying the \+ house in exactly 30 years." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 134 "Here is another situation. If the loan were 110000 instead of 100000, when is it going to be pai d under the same finance restrictions?" }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 43 " tempv:=rhs(dsolve(\{eq1,y(0)=110000\},y(t)));" }{MPLTEXT 1 207 0 "" }} }{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 21 "y2:=unapply(tempv,t);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 21 " plot(y2(t), t=0..30);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 " " {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 150 "So the answer for \+ this question is NEVER. It means $800 a month for the $110000 at 9% in terest rate will grow instead of decreasing as time increases." } {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 85 "Another possibility is to study the following situation \+ (problem 10 in Boyce's book):" }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 92 "1.3 What is the maxim um loan that this buyer can afford under the same finance conditions? \+ " }}{PARA 200 "" 0 "" {TEXT 205 67 " term: 30 years, $800 mont hly payments and 9% interest rate?" }{TEXT 205 0 "" }}{PARA 200 "" 0 " " {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 39 "temp v:=rhs(dsolve(\{eq1,y(0)=yo\},y(t)));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 21 "y3:=unapply(tempv,t);" } {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 20 "f solve(y3(30)=0,yo);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 45 "If the term were 20 y ears instead of 30 years" }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 26 "Loan:=fsolve (y3(20)=0,yo);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 63 "Total interest paid: \+ Total payments - initial amount borrowed." }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 20 "evalf(9600*2 0-Loan);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 46 "Another interesting question is the following:" }{TEXT 205 0 "" }} {PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 100 "1. 3 How much does the mortgage payment, Mp, (Annual payments = Ap = Mp*1 2) need to be for the buyer " }}{PARA 200 "" 0 "" {TEXT 205 69 "to aff ord the $110000 loan with a 30 years term and 9% interest rate?" } {TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 37 "eq13:= diff(y(t),t) = 0.09*y(t) - Ap;" } {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 32 "d solve(\{eq13,y(0)=110000\},y(t));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 44 "tempv:=rhs(dsolve(\{eq13,y(0)=1 10000\},y(t)));" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 25 "y13:=unapply(tempv,t,Ap);" }{MPLTEXT 1 207 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 25 "fsolve(y13(30,Ap),Ap)/12 ;" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 205 0 "" }} {PARA 200 "" 0 "" {TEXT 205 95 "That means that with a quantity sligth ly greater of $885 the buyer can afford the $110000 loan." }{TEXT 205 0 "" }}{PARA 200 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 207 29 "plot(y13(t,885*12), t=0..35);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 208 0 "" }}}{PARA 200 "" 0 "" {TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }