Transcript Sec 1 Cont: Mean Value Theorem (MVT)

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