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2.2 exponential function and compound interest
 

2.2 exponential function and compound interest

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    2.2 exponential function and compound interest 2.2 exponential function and compound interest Presentation Transcript

    • The Exponential Functions
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K =Fractional Powers Rule: A1/k =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K =Fractional Powers Rule: A1/k =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K AFractional Powers Rule: A1/k =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A kFractional Powers Rule: A1/k = A
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A kFractional Powers Rule: A1/k = A AN/k =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)N
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN =Division Rule: AKPower Rule: (AN)K = (AB)N =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN =Division Rule: AKPower Rule: (AN)K = (AB)N =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN = AN – KDivision Rule: AKPower Rule: (AN)K = (AB)N =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN = AN – KDivision Rule: AKPower Rule: (AN)K = ANK (AB)N =
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k = A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN = AN – KDivision Rule: AKPower Rule: (AN)K = ANK (AB)N = ANBN
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k= A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN = AN – KDivision Rule: AKPower Rule: (AN)K = ANK (AB)N = ANBN Exponential FunctionsExponential functions are functions of the form f(x) = bxwhere b>0 and b  0.
    • The Exponential FunctionsDefinitions and Operational Rules of Exponents0-Power Rule (A = 0): A0 = 1Negative Power Rule: A–K = 1 K A k kFractional Powers Rule: A1/k= A AN/k = (A)NThe irrational exponents Ar are approximated by thefractional expansions of r.Multiplication Rule: ANAK = AN+K AN = AN – KDivision Rule: AKPower Rule: (AN)K = ANK (AB)N = ANBN Exponential FunctionsExponential functions are functions of the form f(x) = bxwhere b>0 and b  0. Exponential functions show up infinance, biology and many other scientific disciplines.
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. ofgems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1gems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2gems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4gems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16gems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgems
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x),
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25".
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential Functions
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table:
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1 2 4 8 16
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/2 1 2 4 8 16
    • The Exponential FunctionsExample A. A germ splits into two germs once every day.How many germs will there be after one day? Two days?Three days? Four days? t days?No. of 0 1 2 3 4 tdaysNo. of 1 2 4 8 16 2tgemsHence P(t) = 2t gives the number of germs after t days.The exponential function bx is also written as expb(x).For example, 52 = exp5(2),10x = exp10(x), and "4x = 25"may be expressed as "exp4(x) = 25". Graph of Exponential FunctionsTo graph f(t) = 2t = y we make a table: t -4 -3 -2 -1 0 1 2 3 4 y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (0,1) Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (1,2) (0,1) Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (3,8) (2,4) (1,2) (0,1) Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x t -4 -3 -2 -1 0 1 2 3 4y=2t 1/16 1/8 1/4 1/2 1 2 4 8 16
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x Graph of y = bx where b>1
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x Graph of y = bx where b>1To graph f(t) = (½)t, we make a table.
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x Graph of y = bx where b>1To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x Graph of y = bx where b>1To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t 1 1/2 1/4 1/8 1/16
    • The Exponential Functions (3,8) y=2t (2,4) (1,2) (0,1)(-2,1/4) (-1,1/2) Graph of y = 2x Graph of y = bx where b>1To graph f(t) = (½)t, we make a table. t -4 -3 -2 -1 0 1 2 3 4 y= (½)t 16 8 4 2 1 1/2 1/4 1/8 1/16
    • The Exponential Functions(-3,8) y= (½)t (-2,4) (-1,2) (0,1) (1,1/2) (2,1/4) Graph of y = (½)x
    • The Exponential Functions(-3,8) y= (½)t (-2,4) (-1,2) (0,1) (1,1/2) (2,1/4) Graph of y = (½)x Graph of y = bx where 0<b<1
    • The Exponential Functions (-3,8) y= (½)t (-2,4) (-1,2) (0,1) (1,1/2) (2,1/4) Graph of y = (½)x Graph of y = bx where 0<b<1 Compound InterestWhen an account offer interest on top of previouslyaccumulated interest, it is called compound interest.
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest Formula
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited)
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited) i = periodic rate (interest rate for a period)
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periods
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periodsCompound Interest Formula: A = P (1 + i )N
    • Compound InterestsExample B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $ Compound Interest FormulaLet P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periodsCompound Interest Formula: A = P (1 + i )NWith this formula, we may compute the return after Nperiods directly.
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )N
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $After 4 months, N = 4,
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $After 4 months, N = 4,A = 1000(1 + 0.01)4
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $After 4 months, N = 4,A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $After 4 months, N = 4,A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $Often for compound interest, the annual (yearly) rate r isgiven, and the number of periods in one year K is given.
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months?Use the formula A = P (1 + i )NP = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $After 4 months, N = 4,A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $Often for compound interest, the annual (yearly) rate r isgiven, and the number of periods in one year K is given.In this case, the periodic rate is i = r K A = P (1 + i )N
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months? Use the formula A = P (1 + i )N P = 1000, i = 0.01, after 3 months, N = 3, A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4, A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $Often for compound interest, the annual (yearly) rate r isgiven, and the number of periods in one year K is given.In this case, the periodic rate is i = r KFor example, if the annual compound interest rate isr = 8% and compounded 4 times a year,
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months? Use the formula A = P (1 + i )N P = 1000, i = 0.01, after 3 months, N = 3, A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4, A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $Often for compound interest, the annual (yearly) rate r isgiven, and the number of periods in one year K is given.In this case, the periodic rate is i = r KFor example, if the annual compound interest rate isr = 8% and compounded 4 times a year, then the perioic rate isi = 0.08 = 0.02. 4
    • Compound InterestsExample C. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is thereafter 3 months? and after 4 months? Use the formula A = P (1 + i )N P = 1000, i = 0.01, after 3 months, N = 3, A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4, A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $Often for compound interest, the annual (yearly) rate r isgiven, and the number of periods in one year K is given.In this case, the periodic rate is i = r KFor example, if the annual compound interest rate isr = 8% and compounded 4 times a year, then the perioic rate isi = 0.08 = 0.02. If its compunded 12 times a year, then i = 0.08 . 4 12
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,N = (20 years)(4 times per years) = 80
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80  4875.44 $
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80  4875.44 $If K is the number of times compounded in a year and t is thenumber of years, then N = Kt is the total number of periods.
    • Compound InterestsExample D: We deposited $1000 in an account with annualcompound interest rate r = 8%, compounded 4 times a year.How much will be there after 20 years?We have 0.08P = 1000, i = 4 = 0.02,N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80  4875.44 $If K is the number of times compounded in a year and t is thenumber of years, then N = Kt is the total number of periods.Together with i = r/k, the formula A = P (1 + i )N is r A = P (1 + K )Ktwhere P = principal, r = annual rate,K = number of periods in one year, t = number of years.
    • Compound Interests Exercise A. 1. Graph y = (½)x 2. Graph y = 3x 3. Graph y = (1/3)x Exercise B. Periodic compound interest formulas. A = P (1 + i )N A = P (1 + r )Kt K Given the periodic rate and the time, find the missing variable. 4. i = 1% per month, t = 2 years, P = $2,000, find A. 5. i = 1% per month, t = 2 years, A = $2,500, find P. 6. i = 1.2% per month, t = 1½ years , A = $10,000, find P. 7. i = 3/4% per month, t = 30 years, P = $2,000, find A. 8. i = ½ % per month, t = 40 years, A = $500,000, find P. 9. i = 1.5% per quarter (3 m), t = 8 year, P = $2,000,000, find A.10. i = 0.75 % per day, t = 15 years, P = $15,000, find A.11. i = 0.033 % per day, t = 3 months, P = $2,000,000, find A.12. i = 0.025 % per day, t = 8 months, P = $25,000,000, find A.
    • Compound InterestsExercise C.Given the rate r, method of compounding, and the time t,find the missing variable.13. r = 12%, compounded yearly, t = 20 years, and P = $1,000, find A.14. r = 12%, compounded monthly, t = 20 years, and P = $1,000, find A.15. r = 8%, compounded monthly, t = 15 years, and A = $15,000, find P.16. r = 6%, compounded quarterly, t = 40 years, and P = $100,000, find A.17. r = 9%, compounded yearly, t = 24 years, and A = $2,000, find P.18. r = 3.5%, compounded monthly, t = 2 years, and P = $2,000, find A.