1. Composite Functions: Application The price per unit, p, for the product is p = 2000 – 10t, where t is the number of months past January 2010. Example 1: The monthly demand, D, for a product, is where p is the price per unit of the product. Write the monthly demand, D, as a function of t.

2. Composite Functions: Application Compute (D  p)(t) = D(p(t)). Note, D is a function of p, D(p)D(p) DpDp and p is a function of t. t p(t)p(t)

3. Composite Functions: Application (D  p)(t) = This is now a function of demand with respect to t, so can be relabeled,

4. Composite Functions: Application When will the monthly demand reach 6,250 units?

5. Composite Functions: Application 6250(2000 – 10t) = 5000000 12500000 – 62500t = 5000000 - 62500t = - 7500000 t = 120 months The monthly demand will reach 6,250 units in January 2005.

6. Composite Functions: Application Example 2:An observer on the ground is 300 feet away from the launching point of a balloon. The balloon is rising is rising at a rate of 10 feet per second. Let d = the distance (in feet) between the balloon and the observer. Let t = the time elapsed (in seconds) since the balloon was launched. Let x = the balloon's altitude (in feet). 300 feet x d

7. Composite Functions: Application (a)Express d as a function of x. Hint: Use the Pythagorean Theorem. 300 feet x d

8. Composite Functions: Application 300 feet x d (b)Express x as a function of t. x(t) = 10t The balloon is rising is rising at a rate of 10 feet per second. x = the balloon's altitude (in feet).

9. Composite Functions: Application 300 feet x d (c)Express d as a function of t. x(t) = 10t

10. Composite Functions: Application 300 feet x d (d) Use the result found in (c) to determine how long it takes from launching for the balloon to be 500 feet from the observer. It takes 40 seconds.

11. Composite Functions: Application