Mean+Value+Theorem

=__The Mean Value Theorem__=

Before, we used the Intermediate Value Theorem to show that a continuous function on (a,b) must take on every value y between f(a) and f(b). Now we will discuss another interesting property of continuous functions using the Mean Value Theorem.

//**The Mean Value Theorem**// If f(x) is continuous and differentiable on the interval (a,b), then there must be some point c such that f '(c)=[f(b)-f(a)] / [b-a]. In other words, there must be a point on the curve where the slope of the tangent line equals the average slope, given by the secant line through a and b.

Willie is hurrying to present his latest scientific breakthrough at the local college. His position from home, as a function of time is given by s(t)=60(t) (1/2), where t is in hours and s(t) is in miles. The highway patrol saw him pass a point 1 mile from his house. Where is the FARTHEST distance they could have spotted him a second time to prove he had been speeding if the speed limit is 40mph?
 * Example One **

Solution: We will find the distance at which the average speed will be 40mph. The patrolmen will have had to see Willie a second time just before he reached this distance to prove he was speeding, as the average speed is decreasing with time. The patrolmen will need to set a speed trap sometime before he has traveled 89 miles to prove he was speeding.