mat 1235 calculus ii section 6.5 exponential growth and decay
TRANSCRIPT
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MAT 1235Calculus II
Section 6.5
Exponential Growth and Decay
http://myhome.spu.edu/lauw
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Homework and …
WebAssign HW 6.5
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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.
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Two Common Ways…
2 ways to introduce a mathematical fact…
1. Verification
2. Show (Prove)
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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
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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
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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
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Definitions
Differential Equation (D.E.): An equation involves derivatives
Initial Value Problem (IVP): A D.E. with an initial condition
Section 9
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Example 1
D.E.
IVP
dyky
dt
; (0) 2dy
ky ydt
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Theorem
The solution of
is
where c is some constant.
dyky
dt
kty ce
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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.
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Verificationkty ce
dy
dt
dyky
dt
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Separable Equations (10.3)
dyky
dt
kty ce
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Application Examples
Elementary, at pre-cal level.
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Population Model: Unlimited Growth
Size of Population = Assumption: Rate of change of
population proportion to its size
= relative growth rate
dPkP
dt
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Population Model: Unlimited Growth
Suppose , or Solution:
0( ) ktP t P e
kt
dyky
dt
y ce
dPkP
dt
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Example2
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Example 2
At (hour), size of the population is . Find if the relative growth constant is .
0( ) ktP t P e
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Example 2
(4) ?
(8) ?
P
P
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Example 2
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Radioactive Decay
Radioactive substances decay by emitting radiation.
mass = Assumption: Rate of decay proportion to
its mass dmkm
dt
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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
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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