mat 1235 calculus ii section 6.5 exponential growth and decay

MAT 1235Calculus II

Section 6.5

Exponential Growth and Decay

http://myhome.spu.edu/lauw

Homework and …

WebAssign HW 6.5

Preview

The problems from this section are at most at pre-cal level.

It was moved, in the 6th edition, from section 9 to section 7.

We will look at how to find the formula in additional to verifying the formula.

Two Common Ways…

2 ways to introduce a mathematical fact…

1. Verification

2. Show (Prove)

Two Common Ways…

2 ways to introduce a mathematical fact…

1. Verification

2. Show (Prove)

21 is a solution of 3 2 0.x x x

Two Common Ways…

2 ways to introduce a mathematical fact…

1. Verification

2. Show (Prove)

21 is a solution of 3 2 0.x x x

2 1 3 1 2

Two Common Ways…

2 ways to introduce a mathematical fact…

1. Verification

2. Show (Prove)

21 is a solution of 3 2 0.x x x

2 1 3 1 2

2 3 2 0

1 2 0

1,2

x x

x x

x

Definitions

Differential Equation (D.E.): An equation involves derivatives

Initial Value Problem (IVP): A D.E. with an initial condition

Section 9

Example 1

D.E.

IVP

dyky

dt

; (0) 2dy

ky ydt

Theorem

The solution of

is

where c is some constant.

dyky

dt

kty ce

Solutions

In addition to verification as done in the book, we are going to look at how to actually show that there are no more solutions.

Verificationkty ce

dy

dt

dyky

dt

Separable Equations (10.3)

dyky

dt

kty ce

Application Examples

Elementary, at pre-cal level.

Population Model: Unlimited Growth

Size of Population = Assumption: Rate of change of

population proportion to its size

= relative growth rate

dPkP

dt

Population Model: Unlimited Growth

Suppose , or Solution:

0( ) ktP t P e

kt

dyky

dt

y ce

dPkP

dt

Example2

Example 2

At (hour), size of the population is . Find if the relative growth constant is .

0( ) ktP t P e

Example 2

(4) ?

(8) ?

P

P

Example 2

Radioactive Decay

Radioactive substances decay by emitting radiation.

mass = Assumption: Rate of decay proportion to

its mass dmkm

dt

Radioactive Decay

Suppose , or Solution: Half-life : The time required for half of

any given quantity to decay.

0( ) ktm t m e

dmkm

dt

Example 3

The half-life of a radioactive substance is 25 years.

(a) A sample of has a mass of 60 mg. Find a formula for the mass of the sample after years.

(b) When will the mass reduced to 10 mg?

0( ) ktm t m e 64.68 .yr

0.0277