1. Applications - Business - Economics - Life Sciences Lesson 6.6

2. Compound Interest Recall continuous compounding formula A = future value P 0 = initial, one time investment Consider, instead, periodic deposits into the account  Referred to as an annuity

3. Future Value The future value of a periodic investment represented by f(t) Try for $1200 per year for 7 years  Yielding 6% compounded continuously Note (7 – t)

4. Present Value The present value is the amount that must be invested in order to generate a periodic pay out for a specified number of years You wish to receive $1800 per year for 10 years from an investment  The account receives 6.5% interest continually  What must be the initial investment?

5. Cost and Revenue Consider Cost function C(x) Revenue function R(x) Then profit or Earnings E(x) = R(x) – C(x) at any point in time What if we are given the rate of change of these functions  Called the marginal cost, revenue, earnings

6. Cost and Revenue What if we are given the rate of change of these functions  Called the marginal cost, revenue Net earnings can be interpreted as the area between these two curves R'(x) C'(x)

7. Try It Out Given rate of change (marginal) functions for Cost and Revenue Find the total revenue/earnings for the first 10 months

8. Consumer's and Producer's Surplus Let p = D(q) represent a demand function  p = price consumers are willing to pay  q = number of units purchased q0q0 Total willingness to spend - Actual Expenditure = Consumer's Surplus p = D(q)

9. Consumer's and Producer's Surplus Consumer's surplus given by p = D(q) q0q0

10. Consumer's and Producer's Surplus q0q0 Actual consumer expenditure for q 0 units - Total amt producers receive when q 0 units supplied Producer's Surplus = p = S(q)

11. Assignment A Lesson 6.6A Page 414 Exercises 5 – 27 odd

12. Survival and Renewal Consider a situation where f(t) gives us the proportion (fraction) of an initial population remaining after time t  We know an initial population = p 0  And we know a renewal rate r(t) At the end of t months, we know we have

13. Survival and Renewal Example: The fraction of people residing t years after they area is given by Current population is 20,000. New townspeople are arriving at rate of 500 /yr What will the population be 10 years from now?

14. Assignment B Lesson 6.6B Page 415 Exercises 29 – 53 odd