Sec 2 Cont: Mean Value Theorem (MVT)

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Presentation transcript:

Sec 2 Cont: Mean Value Theorem (MVT)

Review: Extreme Value Theorem Critical Numbers are any ________ or ________. Critical Numbers can be found by using the equation ________. The EVT finds ___________ maximums and minimums. To find any extreme using the EVT, you must test the ________ ________ , where the function is ___________ and the _____ ________.

Example 1: Extreme Value Theorem The volume of a 3-dimensional figure is given by the function y = x³ - 9x + 1 where the length of each side (x) is between .2 and 3. What length will give the greatest volume for the figure?

Review: Rolle’s Theorem For Rolle’s Theorem to work the function must be ____________ and ____________. If f(a) = f(b), the there must be a ________ _________ on the interval (a, b). With Rolle’s Theorem, the _________ are not tested because the function is differentiable on an _________ interval.

Ex 2: Rolle’s Theorem Let f(x) = x² - 5x + 4 on the interval [1, 4]. A. Can Rolle’s Theorem be applied? B. If so, find the value of c at which there is a critical point.

Mean Value Theorem (MVT) If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number, c, in (a, b) such that The MVT says that the slope of a tangent line on a curve is equal to the slope of the secant line on the same curve at a particular point.

Ex 3: Slope of the Tangent Line What value of c in the open interval (0, 4) satisfies the MVT for ?

Ex 4: MVT Given , find all values of c in the open interval (1,4) such that

Ex 2: Finding an Instantaneous Rate of Change Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first car, its speed is clocked at 55 mph. Four minutes later, the truck passes the 2nd patrol car at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.

HOMEWORK Pg 172 #27 - 30, 31 – 38 odds, 53 - 56