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Thank you, `I_loveRegine` for picking me as a scribe for today , haha chhyyeaah as iff, buut thanks though for telling me because I wouldn't even bother checking who the scribe was. Well here's todays note. . .




EXPONENTIAL DECAY is when the function is decreasing.
EXPONENTIAL GROWTH is when the function is increasing.

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THE COMPOUND INTEREST FORMULA
A = P ( 1 + r/n ) ^tn

A is the amount at the end of the investment period.
P is the principle amount originally invested.
r is the percent rate of the return written as a decimal.
n is the number of times the prinpicle is compounded each year.
t is the time in years.

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INTEREST COMPOUND CONTINOUSLY

A = P e ^rt

A is the ampunt at the end of the investment period.
P is the principle amount originally invested.
r is the percent rate of the return written as a decimal.
t is the time in years.

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Y _t = Y _o e ^kt

Y_t original amount
Y_o final amount
k rate
t time

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EXAMPLES
1. If 1000 type A bacteria at present and of increase 0.025 per hour. How many bacteria in 24 hours?

Y_t= Y_oe^kt
Y _t = 1000e ^0.025(24)
*punch it in your calculator*
you should get . . . 1822 bacteria!!

2. A radioactive substance decay at a daily rate of 0.13. How long will i take for the substance to decompose half its size?
Y_t=Y_oe^kt
x = 2xe^-0.13t (divide both sides by "2x")
0.5 = e^-0.13t
ln0.5 = -0.13t lne (remember from regine's post that lne = 1)
ln0.5 = -0.13t (divide both sides by "-0.13t")
the answer is t = 5.33 days

3. If you can invest money at 6% compounded annually. How long will it take for investment to double?

A=P(1+r/n)^tn (n=1)
therefore, A=P(1+r)^t
2x = x(1+0.6)^t (divide both sides by "x")
2 = (1.06)^t
log² = t(log^1.06) (divide both sides by "log^1.06")
the answer is t = 11.9 years

4. forest has 500 rabbits on May 31^st. June 20^th there are 800. How many should be on June 28^th?

Given:
t_t = 800
Y^o = 500
t = 20 days
k = ?

First is you have to find "k"
Y_t=Y_oe^kt (use 20 for "t")
800 = 500e^20k (divide both sides by "500")
1.6 = e^20k
ln1.6 = 20k lne
ln1.6 = 20k (divide both sides by "20")
the answer is k = 0.0235

Now you know "k" you can solve the problems and find how much bunnies there are on June 28^th.

Y_t=Y_oe^kt (use 28 for "t")
Y_t = 500e^0.0235(28)
the answer is 965 bunnies. wow talk about energizer bunnies! -_-"


Hope you understand. That's all for tonight. Man, I'm tired! Do exercise 27. That's all for now, have a great weekend! The next scribe is ...... B00T5