Exponential & Logarithmic Models

MATH 109 - PrecalculusS. Rook

Overview

• Section 3.5 in the textbook:– Compound interest– Exponential growth & decay

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Compound Interest

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Compound Interest

• We put a principal P (starting amount) into an account with a constant growth rate r (expressed as a %) for t years– The amount gained while in the account is known as

interest

• Each year, interest is added to the account n periods per year– Some common values for n include annually (n =

1), quarterly (n = 4), monthly (n = 12), and daily (n = 365)

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Compound Interest (Continued)• Compound Interest Formula: where A is

the amount when principal P has been invested at a rate r for t years with n compounding periods each year

• The product nt is the total number of compounding periods– e.g. If the amount were to be invested for 5 years with

interest compounded biannually, there would be 5 ∙ 2 = 10 occasions where interest would be added to the account

nt

n

rPA

1

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Compounding Continuously

• Compounding Continuously: when the number of compounding periods increases without bound where A is the amount when investing

principal P at a rate for t years and e is Euler’s constant– Only need to know the formula, not how to derive it

rtPeA

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Compound Interest (Example)

Ex 1: $7500 is deposited into an account with a 6.5 % interest rate for 4 years. Find the amount that results when the interest is compounded:

a) Monthlyb) Three times a year

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Compound Interest (Example)

Ex 2: $10,000 is deposited into an account with an interest rate of 4.05 % that is compounded biannually. Approximately how long would it take for the $10,000 to double?

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Compound Interest & Continuous Compounding (Example)

Ex 3: What interest rate would be required for $1500 to accrue to $2500 in 8 years if the account is compounded:

a) continuouslyb) quarterly

Exponential Growth & Decay

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Exponential Growth & Decay

• Both exponential growth and decay follow the formula N(t) = N0ekt

– If k > 0, the formula models exponential growth– If k < 0, the formula models exponential decay

• Given the points (0, N0) and (t, Nt), we can solve for the constant k– i.e. If we know the initial quantity and the quantity

after some time t

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Half-life & Carbon Dating

• Half-life: the amount of time it takes for half of a substance to dissolve– Don’t need to memorize half-lives for substances

• Carbon Dating: the process of using the amount of carbon-14 left in an organism to determine its age

Model is P(t) = 0.5t/5730 where P(t) is the percent of carbon-14 remaining after t yearsAgain, don’t need to memorize

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Exponential Growth (Example)

Ex 4: Suppose a population has 25,000 members in 2000 and 40,000 members in 2005

a) Find the exponential growth equation that models this situationb) Find the population that the equation predicts in 2010 to the nearest thousand

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Half-life (Example)

Ex 5: Polonium has a half-life of 138 days. Use this information in order to:

a) Construct the exponential decay equation for the amount of Polonium that remains after t daysb) Use the equation to determine the fraction of Polonium that remains after 500 days

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Carbon Dating (Example)

Ex 6: Approximately how old is a bone that contains 70% Carbon-14?

Summary

• After studying these slides, you should be able to:– Solve problems involving:• Compound and continuously compounded interest• Exponential growth and decay

• Additional Practice– See the list of suggested problems for 3.5

• Next lesson– Radian & Degree Measure (Section 4.1)

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