Welcome to Calculus. 
I'm Professor Greist, and we're about to begin Lecture 38, bonus material. 
Of those topics covered in our main lesson perhaps the most challenging is 
that of present value. To reiterate. 
Let's say, that I were to give you a thousand dollars in ten years, how much 
would it be worth to have right now? As an equivalent, assuming that you took 
that money and invested it at a fixed interest rate. 
Well, given what we know about such investments, it would be worth that 
amount of money $1000 times e to the negative rt where t is 10 years. 
That's how much it would be worth right now. 
But instead of some fixed amount of money to be given at some fixed future time. 
What happens if you have an income stream, a certain amount of money that's 
coming in overtime and perhaps varying? Well, to solve that more challenging 
problem, we integrate the present value element. 
Let's see how that works in a specific example. 
Which of the following income streams would be better for you in terms of 
having greater present value? Let's say, I give you the choice between 
earning income at a fixed rate. I of t is equal to sum constant C or I 
say, well, you get less money at the beginning, say 1 3rd C. 
But the amount of money that you're getting increases over time linearly. 
Let's say I of t equals 1 3rd C times quantity 1 plus rt. 
Which would you choose? In the second case, there would be a time 
in the future where you're making much more money than that fixed constant rate 
C. Well, let's compute the present values 
and find out which is better. We'll begin with the first income stream. 
I of t is a constant. In this case, the present value element 
is e to the negative r t times i of t d t. 
That is c e to the negative r t d t. If we integrate that, well, lets see, 
what are our bounds on integration? It wasn't said what sort of time frame we 
had. So, let's assume a very, very long 
lifespan and integrate as T goes from zero to infinity this present value 
element. This integral is not so hard. 
The constant comes out, we integrate e to the negative rt and get e to the negative 
rt over negative r. When we substitute in t equals 0 and take 
the limit to t goes to infinity, then we get simply that constant c over r. 
That is the present value of this infinite income stream. 
Let's see how this compares with an income stream that is linearly growing, 
but which starts off much lower. If we have i of t equals 1 3rd C times 1 
plus rt, then our present value element is that same income stream times d t 
times e to the negative rt. Now, integrating that is going to be a 
little bit more challenging, but not so bad. 
It breaks up in to two integrals. The first Is 1 3rd c, e to the negative 
rt dt, but that's really the same integral that we just did for the 
constant income stream. But the constant in front is 1 3rd c, 
therefore the first integral evaluates 2c over 3r. 
For the second integral, it's going to take a bit more work. 
Let's pull up the constant, Cr over 3 and then integrate what is leftover using 
integration by parts. We're going to let u d equal the t and dv 
be e to the negative rt dt. I'm going to leave some of the details to 
you at this point, to show that what we get is u times v. 
That is te over negative r times e to the negative rt, evaluated from zero to 
infinity. Well, what we get when we subtract the 
integral of vdu is adding the integral of 1 over r e to the negative rt dt. 
And that of course is a simple integral. The first term, the uv term, vanishes at 
those limits. And we're left with an integral that we 
can easily solve the net present value of the second income strain is C divided by 
3r plus Cr over by 3r squared that simplifies to 2 3rd C over by r. 
When we compare that to the fixed income stream, which had a present value of C 
over R, we see that this linearly increasing stream is worth only 2 3rd as 
much over time. That is not an obvious conclusion. 
Now, there are many, many other examples in finance and economics that work well, 
when argued in terms of elements. I'll leave it to you to think about what 
future value of an income stream might be.