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MAT 1235 Calculus II Section 6.5 Exponential Growth and Decay http://myhome.spu.edu/lauw
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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
