Question

what's the main difference between continuous compounding and exponential growth, and why does my text book say that it is better to express growth rates as if they are continuously compounded? What's the advantage---both give outrageous estimates to "what if" scenarios.

Answer 1

The big advantage to using continuous compounding to express growth rates is it avoids the problem of asymmetry in growth rates:

For exmaple, if we use the normal definition and $100 grows to $105 in one time period, that's a growth rate of $105/$100 - 1 = 5%

But if $105 decreases to $100, that's a growth rate of $100/$105 - 1 = -4.76%

The problem of asymmetry is those two growth rates, 5% and -4.75% are not equal up to a sign.

But if you use continuous compounding the growth rate in the first case is
ln(105/100) = 0.04879...

And the growth rate in the second is
ln(100/105) = -0.04879...

Those two growth rates are definitely the negative of each other.

Answer 2

Thanks so much! I had been wondering about this for a while, and now it makes sense. So just teaching about the basic formula of exponential functions serves what purpose? (If I am not going to use it, I mean: I used it in high school, but all of a sudden its not cut out for real life?)

Answer 3

Exponential growth is used all the time in the sciences and social sciences.

But continuous compounding doesn't appear much there. But it does appear and is used all the time in Finance.

Answer 4

Oh, okay...which makes sense for my course in terms of looking at the trade offs of natural resource production rates (and cost) in regards of environmental impact and sustainability. Thanks for your help!

Answer 5

james
can you tell the integration for
sin 4x * sin 3x

Answer 6

^Try asking that question on the actual page or the chat, and he might be able to answer. If you are interested in my help, i'd say no, automatically, and also by checking wolfram. See link: http://www.wolframalpha.com/input/?i=sin%283x%29+*sin%284x%29
You cannot multiply sine and cosine together (I'm pretty sure), so having different variables doesn't lift that rule either. You'd simply get, if any thing, sin(4x) *sin(3x).

Answer 7

^Sorry, you can't multiply sin(3x) *sin(4x) together is what I meant, because of the different numbers next to the variable x. But, if the numbers were the same, say 2x and 2x, your answer would conclude to be sin^2(2x). That is the only way.

Answer 8

no i knew the answer
sin (4x)*sin (3x) = 1/2 ( cos (x) + cos (7x) ) dx
then simply integrate it

Answer 9

Whoops, I read your problem wrong. Okay, so to integrate, you are doing the right thing.

Answer 10

thank you for caring

Answer 11

No prob :) Good luck!