7.4 B – Applying calculus to

Exponentials

Big Idea

• This section does not actually require calculus. You will learn a couple of formulas to model exponential growth and decay that were found using calculus. You do not have to derive the formulas, you can simply use them to solve the problems.

Exponentials.

• We know enough about exponentials to know the amount if increase depends on the amount present.

• In other words

• If when t=0 then (Note: This was derived using calculus)

• Growth If • Decay if

Money Application

• Interest can be compounded in intervals (weekly, monthly, annually, etc).

• If money is invested (p into an account where the interest rate (r) is added to the account n times per year, the amount present after t years is:

• Or it can be compounded continuously.

A = Pert

Example• Suppose you deposit $800 into an account that oats 6.3% annual interest. How much will you have in 8 years if

a) It is compounded continuously?

b) It is compounded quarterly?

These are surprisingly close…interesting.

Population Problems

Equation: Population = Initial (base) (1/time to increase)time

We can choose a convenient base to help figure out a growth pattern.

Example

A population of ants started at 10 on June 1, and reached 40 in 30 days. If the growth continues to follow this model, how many days after June 1 will the population reach 100?

Continued

Approximately 50 days will pass after June 1 before the population reaches 100.

The equation for the amount of a radioactive element left after time t is:

If k<0, it is decay, if k>0 it is growth.

The half-life is the time required for half the material to decay.

Radioactivity Application

ktOeyy

60 mg of radon has a half-life of 1690 years. How much is left after 100 years?

Example

ke16906030

ke16902

1

k16902

1ln

00041.k)100)(00041.(60 ey

mgy 58

ktOeyy

100 bacteria are present initially and double every 12 minutes. How long before there are 1,000,000

You try!

ke12100200

ke122

k122ln

0577.k))(0577(.100000,000,1 te

te 0577.10000

ktOeyy

minutes159t

Half-life

0 0

1

2kty y e

1ln ln2

kte

ln1 ln 2 kt 0

ln 2 kt ln 2t

k

Half-life:

ln 2half-life

k

Equation for Half-Life (you do not need to write this work)

Warm Up

• The number of radioactive atoms remaining after t days in a sample is given by the equation, y = y0e-0.137t Find the element’s half-life.

ln 2half-life

k

Coffee left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air.This observation is Newton’s Law of Cooling, although it applies to warming as well, and there is an equation for it.

Newton’s Law of Cooling

Newton’s Law of Cooling

0kt

s sT T T T e

If T is the temperature of the object at time t, is the original temperature and is the surrounding temperature.

sT

Formula

Example• A hard-boiled egg at 98o C is put in a pan under running 18o C

water to cool. After 5 minutes, the egg’s temperature is found to be 38o C. How much longer will it take the egg to reach 20o C?

0kt

s sT T T T e

Continued

• To find k, we use the information that T=38 when t=5

Continued

The egg’s temperature T at time t is

Could solve for t by taking the natural log, OR we could use our calculators.

Now we will use our calculators to find the time when the egg’s temperature is 20 degrees.

Continued

Find intersection. Intersect at about t=13.3.The egg’s temperature will reach 20 in about 13.3 minutes after it is put under the cool water.

Homework

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