1. More on Work & Applications to Economics (3/10/04) Last time we looked at work done when the force was variable. Another common situation is when the force (for example, the force of gravity) is constant but the distance moved is variable. Hence this is another example of our general principle that if we are multiplying quantities at least one of which is varying, the integral is a good tool to use.

2. An example of work done over variable distance How much work is done in emptying a full upright cylindrical water tank (radius 5 feet, height 20 feet) by pumping the water out the top? (Recall: Water weights 62.4 lbs/cu.ft.) Well, each horizontal disk of water weighs the same ((62.4)(25  )  h lbs), but each must be lifted a (different!) distance of (20 – h) feet. So

3. Application to Economics: Present & Future Value; Income Streams If I invest P dollars at an annual rate of r, compounded continuously, then the future value B of P after t years is B = P e r t. From this it is follows that the present value P of a future amount B is P = B e - r t. For example, if someone will pay me $1000 five years from now, and if the ongoing investment rate is 7%, then its present value is P = $1000 e –(0.07)(5) = $705.

4. Income Streams In most businesses (and homes), income arrives continuously, not in discrete packets. If the rate at which it arrives is a function P(t) in dollars/year, then between time t and time t +  t, the present value of that money will be P(t)  t e – r t. Adding up over the period from 0 years to M years gives

5. Example of Income Streams If income arrives at a rate of $5000/year over a ten year period, and if the investment rate averages 7%, the Present Value of this income stream is Likewise, the Future Value (in ten years) is

6. Assignment for Friday, Etc. Read Section 8.5 to the middle of page 380 On page 382, do #1, 3, 6, 7, and 9. We will meet from 12:20 to 1:15 on Friday. We will meet in the classroom on Monday, March 22 and there will be no assignment for that day. (But work in Hand-in #2!) Hand-in #2 is due on Wed, March 24. There will be additional homework for that day.