Calculus Integrating Exponential Functions
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Integrating Exponential Functions

 

 While a few people do math for it's own sake (yikes!),
 most people just use math to help describe things that happen in real life.
 Lots of things can be described with expressions like 3X 2 + 2X -5,
 and we've already talked about doing derivatives and integrals of stuff like that.
 Sometimes though the expression that fits what's going on
 looks more like 2 X, where the variable is the exponent.
 In expressions like this the variable is usually time,
 and something is getting bigger or smaller as time passes.
 
 The big deal about this variable as exponent form is 
 that the amount of change that happens depends on how much of the stuff 
 that's getting bigger or smaller is already there.
 
 One good example of this is a savings account. 
 The interest that's earned is the amount of change in the balance.
 The more money in the account, the more interest that's added.
 
 Say you find a savings account that pays 10% interest a year.
 This being a great rate, you deposit $100.
 The first year you get $10 in interest which you leave in the account.
 The second year you get interest on the $110, so you get $11 interest.
 You keep letting the interest stay in the account, 
 and it keeps paying 10% interest.
 
 In the following five years the interest amounts are:
 12.10, 13.31, 14.64, 16.10, 17.72.
 Each year the interest is 10% of the amount in the account.
 
 Bacteria cultures in a serer pipe or bad restaurant grow the same way. 
 
 Now when math types write the equations to describe this stuff, 
 they could start with 2X or 9X or <any number>X that they wanted
 and adjust the constants to make the graph work right for that particular curve.
 
 Sooner or later, we are going to want to find a derivative or integral 
 of whatever we equation we came up with.
 So if 5 X is easier to use than 8X or anything else, we'd use it right?
 Well math types tried to figure out if there was some exponent base,
 5, or 12, or whatever that was easier to use than all the others. 
 It was a long shot, but what the hey, 
 it was ancient times before the internet and pizza delivery.
 What else did they have to do?
 
 They might have come up with a table of derivatives like this
 (no you're not supposed to know how to figure these out yet)
 
F(x)

F'(x)
1.50X 1.50X x 0.405465
1.75X  1.75X x 0.559616
2.00X  2.00X x 0.693147
2.25X 2.25X x 0.810930
2.50X 2.50X x 0.916291
2.75X 2.75X x 1.011601
3.00X 3.00X x 1.098612
3.25X 3.25X x 1.178655
3.50X 3.50X x 1.252763
3.75X 3.75X x 1.321756
4.00X 4.00X x 1.386294
 
 See it?
 It looks like there might be a number between 2.5 and 2.75 (closer to 2.75)
 where the derivative of (that number)X = (that number)X times one!
 That would mean that the derivative of (that number)X would be ITSELF!
 That would mean that the integral of (that number)X would be ITSELF TOO!
 Hey, you can't get any easier than that.
 
 So did they ever find the magic number?
 Well, yes and no ...
 They got close. 
 It turns out to be an irrational number (like pi) that starts with 2.71828...
 Because this number is SO important (and so long) we use a letter to stand for it.
 The letter is "e."
 So the derivative of eX is eX,
 and the integral of eX is eX
 When you write these equations, 
 don't forget the little "dX" and "+ C" things that go along with Calculus.
 

 
 
 Remember that the "+ C" part is only needed when we do an indefinite integral.
 If we have actual numbers by our integral, we can lose the + C.
 

 
 Examples:

 
 The general form of this is ...

 
 
 Or if you really want to see something that looks nasty ...

 
 
 All that last one means is that if we have some nasty thing for an exponent,
 the derivative is the original function times the derivative of the exponent.
 
 It's a different looking kind of chain rule inside/outside thing.
 
 If that explanation is too confusing, look again at the first two examples above.
 
 All this also means that if we are finding an integral
 of one of these that has a nasty exponent,
 we need the derivative of the nasty exponent
 sitting next to the e(exponent) to stuff back in.
 
 Some people are never satisfied, and want a proof.
 If you don't want to see it, the page is over.
 I repeat, this is a nasty proof. 
 You probably don't need it.
 
 I'd turn back if I were you ...
 
 Still here?
 
 OK, here goes.
 
 First, where does "e" come from?
 
 We were talking about things that grow based on how big they are at the moment,
 and we said that this turned out to be an exponential term
 like 2X or in general aX.
 
 In an exponential GROWTH equation, the a would be the original amount,
 plus the interest percent. 
 In our savings account example above, a would be 1.1.
 1 for 100% of the original amount, plus .1 for the 10% it grows each time period.
 So it is more understandable as what it is, you can rewrite aX as:
 

 
 
 If instead of calculating the interest once a year 
the bank calculates it twice a year, 
 the interest percent would be half as much
 and the number of periods would be doubled.
 If the bank calculated interest once every three months, 
 the interest percent would be one fourth as much 
 and the number of periods would be four times as much.
 Interest rate and time periods have an inverse relationship.
 If we write this idea with one variable, call it "n",
 the equation would look something like:
 

 
 Now in real life, stuff like bacteria don't call "time out" every now and then,
 take roll, and then grow.
 They grow pretty much continuously.
 That means the time periods are extremely, extremely tiny.
 To try to build this into an equation, we have to make the time period smaller 
 and smaller and smaller until we have ...
 
 A LIMIT PROBLEM!!!
 A limit Problem, why did it have to be a limit problem.
 This one is:

 
 
 Now if we try to take the limit of the exponent first, we're in trouble!
 So we need to get rid of the exponent.
 How?
 We multiply it out!
 How?
 By using a little trick from Algebra called the binomial theorem.
 It says that if we have (a + b) n, which we really do,
 this multiplies out to the form:

 
 
 As n goes to infinity, the number of terms we get from this goes to infinity.
 Our only hope, is that after the first few terms, 
 the value of each terms gets to small to matter.
 
 Well guess what?
 
 That's just what happens!
 
 Putting in 1 and 1/n for a and b, we get:

 
 
 a bit of combining gets us:

 
 
 canceling n's

 
 
 now as n approaches infinity, 
 terms with n in the denominator get closer and closer to zero.
 That gives us:

 
 If you really want to get technical ...

 
 
 Adding on each additional 1/factorial term
 adds way less and less to the total value.
 The total of all the terms adds up to ... wait for it ... 2.71828... also known as "e."
 
 Since the number we get when we add the terms stays at about 2.71828... 
 no matter how many terms we add,
 the series of terms is said to "converge" on that value.
 This convergence is like an asymptote.
 
 So big deal! e is the answer to some continuous growth function,
 but why is it that e is it's own derivative????
 
 OK. Here goes. This is the big one ...
 

 
 
 Now, going through the same drill we just did ...

   

 
 simplifying:

 
 
 canceling n's ...

 
 
 take the limit as n approaches infinity ...

 
 
 So we get a string of terms for eX ...

 
 
 Take the derivative of this ...

 
 
 Drop the leading zero from the derivative ...

 
 

 AND WE GET THE ORIGINAL FUNCTION!!!!!!!!
 

   copyright 2005 Bruce Kirkpatrick
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