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Math Wizard's Notes

Exponential growth and decay are rates. They represent the change in some quantity through time.

**Exponential Growth**

Exponential growth is any increase in a quantity P.

P(t)= Poe^(kt)

Where Po is the initial quantity, t is time, k is a constant, P(t) is the quantity after time t, and e^x is the exponential function.

**Exponential Decay**

Exponential Decay is an decrease in a quantity P.

P(t)= Poe^(-kt)

Where Po is the initial quantity, t is time, k is a constant, P(t) is the quantity after time t, and e^x is the exponential function.

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Math Wizard's Notes

**Exponential Growth and Decay Models**

Po<k is a graphical interpretation of exponential decay.

Po>k is a graphical interpretation of exponential growth.

Po=k is a graphical interpretation of constant exponential growth/decay.

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Math Wizard's Exercise

1. If a city has a population of 340 people, and if the population grows continuously at an annual rate of 2.3%, what will the population be in 6 years?

We are given Po= 340 people and k= .023.

P(t)= 340e^(.023t)

When t= 6 years

P(t)= 340e^(.023*6)= ~390 people

2. If China has a population of 420 tigers, and if the population doubles every 9 years, what will the population be in 7 years?

We are given Po= 420 and t= 7 and P(9)= 840.

A doubling time of 9 years means that 840= P(9)= Poe^(kt), and follows that 2= e^(9k).

Take the natural log of both sides to get ln2=9k.

So k=(ln2)/9

Hence, P(7)= 420e^(7k)= 420e^((7ln2)/9)= ~720 tigers