2.7: Toepassing 3: Saamgestelde rente. 2.7: Application 3: Compound interest
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1 Saamgestelde Rente/ Compound Interest SLIDE 1/6 2.7: Toepassing 3: Saamgestelde rente 2.7: Application 3: Compound interest
2 Saamgestelde Rente/ Compound Interest SLIDE 1/6 2.7: Toepassing 3: Saamgestelde rente Probleem: nbedragvanp 0 wordteen nrentekoersvanx% per jaar belê. Hoeveelgeldisindierekeningnatjaar? Rentekoers: r=x%=x/100 Los die probleem op vir die scenarios waar die rentekoers (i) jaarliks, (ii) half-jaarliks,(iii) kwartaliks,(iv) daagliks en(v) kontinu saamgestel word 2.7: Application 3: Compound interest Problem: AnamountofP 0 isinvestedatanannualinterestrate ofx%. Howmuchmoneyisintheaccountaftertyears? Interestrate: r=x%=x/100 Solve the problem for the scenarios where interest is compounded: (i) annually,(ii) biannually,(iii) quarterly,(iv) daily, and(v) continuously
3 Saamgestelde Rente/ Compound Interest SLIDE 2/6 Met jaarlikse samestelling van rente: With interest compounded annually: Aanvanklik/Initially: P(0)=P 0 Na1jaar/After1year: P(1)=P 0 +rp 0 =(1+r)P 0 Na2jaar/After2years: P(2)=(1+r)P(1)=(1+r) 2 P 0 Na3jaar/After3years: P(3)=(1+r) 3 P 0 Natjaar/Aftertyears: P(t)=(1+r) t P 0
4 Saamgestelde Rente/ Compound Interest SLIDE 2/6 Met jaarlikse samestelling van rente: With interest compounded annually: Aanvanklik/Initially: P(0)=P 0 Na1jaar/After1year: P(1)=P 0 +rp 0 =(1+r)P 0 Na2jaar/After2years: P(2)=(1+r)P(1)=(1+r) 2 P 0 Na3jaar/After3years: P(3)=(1+r) 3 P 0 Natjaar/Aftertyears: P(t)=(1+r) t P 0 Met half-jaarlikse samestelling van rente: With interest compounded biannually: Aanvanklik/Initially: P(0)=P 0 Na 1 2 jaar/after 1 2 year: P(1 2 )=P 0+ r 2 P 0= ( 1+ r ) 2 P 0 Na1jaar/After1year: P(1)=(1+ r 2 )P( r 2 2) P0 )=( Natjaar/Aftertyears: P(t)= ( 1+ r 2) 2t P0
5 Saamgestelde Rente/ Compound Interest SLIDE 3/6 Met kwartalikse samestelling van rente: With interest compounded quarterly:: Natjaar/Aftertyears: P(t)= ( 1+ r 4) 4t P0
6 Saamgestelde Rente/ Compound Interest SLIDE 3/6 Met kwartalikse samestelling van rente: With interest compounded quarterly:: Natjaar/Aftertyears: P(t)= ( 1+ r 4) 4t P0 Met daaglikse samestelling van rente: With interest compounded daily: Natjaar/Aftertyears: P(t)= ( 1+ r 365) 365t P0
7 Saamgestelde Rente/ Compound Interest SLIDE 3/6 Met kwartalikse samestelling van rente: With interest compounded quarterly:: Natjaar/Aftertyears: P(t)= ( 1+ r 4) 4t P0 Met daaglikse samestelling van rente: With interest compounded daily: Natjaar/Aftertyears: P(t)= ( 1+ r 365) 365t P0 Met kontinue samestelling van rente: With interest compounded continuously: Natjaar/Aftertyears: P(t) = ( 1+ r n) nt P0,met/withn = lim n (1+ r n) nt P0 Stel/Set:y= ( 1+ r n) nt en/ands=r/n n=r/s y= (1+s) 1/s As/Ifn,dan/thens 0(riskonstant/constant) rt
8 Saamgestelde Rente/ Compound Interest SLIDE 4/6 Laat/Let f =(1+s) 1/s sodat/sothat lnf = ln(1+s) s ln(1+s) limlnf = lim s 0 s 0 s = 1 = lim s s (L Hopital) e lim s 0lnf = lim s 0 e lnf = lim s 0 f =e 1 =e lim 1+ r n n lim s 0 (1+s) 1/s =e nt = lim = e rt s 0 (1+s) 1/s rt
9 Saamgestelde Rente/ Compound Interest SLIDE 5/6 Dus vir kontinue samestelling van rente: Therefore, for interest compounded continuously: P(t)=P 0 e rt
10 Saamgestelde Rente/ Compound Interest SLIDE 5/6 Dus vir kontinue samestelling van rente: Therefore, for interest compounded continuously: P(t)=P 0 e rt Dit bevredig die DV: This satisfies the DE: dp dt =rp met/with P(0)=P 0
11 Saamgestelde Rente/ Compound Interest SLIDE 6/6 Voorbeeld: nbedragvanr1000wordvir3jaarteen nrentekoers van 6% per jaar belê. Hoeveel is dit werd: (a) as rente jaarliks bygevoeg word en(b) as rente kontinu saamgestel word? Antwoorde: (a)r1191(b)r1197 (SELFSTUDIE) Toepassing 4: Newton se wet van afkoeling (pp20&75) Example: An amount of R 1000 is invested for 3 years at an annual interest rate of 6%. How much is it worth: (a) when the interest is compounded annually, and(b) when the interest is compounded continuously? Answers: (a)r1191(b)r1197 (SELF STUDY) Application 4: Newton s law of cooling(pp 20 &75)
12 Saamgestelde Rente/ Compound Interest SLIDE 6/6 Voorbeeld: nbedragvanr1000wordvir3jaarteen nrentekoers van 6% per jaar belê. Hoeveel is dit werd: (a) as rente jaarliks bygevoeg word en(b) as rente kontinu saamgestel word? Antwoorde: (a)r1191(b)r1197 (SELFSTUDIE) Toepassing 4: Newton se wet van afkoeling (pp20&75) Example: An amount of R 1000 is invested for 3 years at an annual interest rate of 6%. How much is it worth: (a) when the interest is compounded annually, and(b) when the interest is compounded continuously? Answers: (a)r1191(b)r1197 (SELF STUDY) Application 4: Newton s law of cooling(pp 20 &75)
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