Composite Functions: Application The price per unit, p, for the product is p = 2000 – 10t, where t is the number of months past January 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.
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)
Composite Functions: Application (D p)(t) = This is now a function of demand with respect to t, so can be relabeled,
Composite Functions: Application When will the monthly demand reach 6,250 units?
Composite Functions: Application 6250(2000 – 10t) = – 62500t = t = t = 120 months The monthly demand will reach 6,250 units in January 2005.
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
Composite Functions: Application (a)Express d as a function of x. Hint: Use the Pythagorean Theorem. 300 feet x d
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).
Composite Functions: Application 300 feet x d (c)Express d as a function of t. x(t) = 10t
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.
Composite Functions: Application