4.2MEAN VALUE

THEOREM

ROLLE’S THEOREM Exploration:

Sketch a rectangular coordinate plane on a piece of paper.

Label the points (1, 3) and (5, 3).Draw the graph of a differentiable function

that starts at (1, 3) and ends at (5, 3).WHAT DO YOU NOTICE ABOUT YOUR

GRAPH?

ROLLE’S THEOREM If

f is continuous on [a, b], f is differentiable on (a, b) AND f(a) = f(b)

Then there is at least one number c in (a, b) such that f’(c) = 0.

In other words, if the stipulations above hold true, there is at least one point where the derivative equals 0 (there exists at least one critical point in (a, b) ALSO f has at least an absolute max or min on (a, b)).

ROLLE’S THEOREM Example: Show that f(x) = x4 – 2x2

satisfies the stipulations to Rolle’s Theorem on [-2, 2] and find where Rolle’s Theorem holds true.• f is continuous on [-2, 2]

• f is differentiable on (-2, 2)

• f(-2) = f(2) = 8

044)(' 3 xxxf0)1(4 2 xx0)1)(1(4 xxx

04 x 01x 01x0x 1x 1x

MEAN VALUE THEOREM Consider the graph below:

ca b

Slope of tangent at c? f’(c)

Slope of secant from a to b?

ab

afbf

)()(

MEAN VALUE THEOREM If

f is continuous at every point on a closed interval [a, b]

f is differentiable at every point on its interior (a, b)

Then there is at least one point c in (a, b) at which

ab

afbfcf

)()(

)('

In other words, if the stipulations above hold true, there is at least one point where the instantaneous slope equals the average slope.

MEAN VALUE THEOREM Example:

a.) Show that the function f(x) = x2 + 2x – 1 on [0, 1] satisfies the Mean Value Theorem. b.) Find each value of c that satisfies the MVT.

a.) f is continuous and differentiable on [0, 1]

b.) 22)(' ccf

3)1(201

)0()1()()(

ff

ab

afbf

So, 322 c

2

1c

MEAN VALUE THEOREM Example:

Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) that satisfies the Mean Value Theorem.

2

4)('c

cf

13

14

14

)1()4()()(

ff

ab

afbf

So, 142

c2c

INCREASING/DECREASING FUNCTIONS

Let f be a function defined on an interval I and let x1 and x2 be any two points in I.

f increases on I if x1 < x2 results in f(x1) < f(x2)

f decreases on I if x1 > x2 results in f(x1) > f(x2)

More importantly relating to derivatives:

Let f be continuous on [a, b] and differentiable on (a, b):

o If f’ > 0 at each point of (a, b), then f increases on [a, b]

o If f’ < 0 at each point of (a, b), then f decreases on [a, b]

o If f’ = 0 at each point of (a, b), then f is constant on [a, b] (called monotonic)

INCREASING/DECREASING FUNCTIONS

Example: Find the open intervals on which

23

2

3)( xxxf is increasing or decreasing.

1.) Find critical points 033)(' 2 xxxf

0)1(3 xx03 x 01x0x 1x

Interval (-∞, 0) (0, 1) (1, ∞)

Test Value x = -1 x = ½ x = 2

f’(x) 6 > 0 -3/4 < 0 6 > 0

Conclusion Increasing Decreasing Increasing

ANTIDERIVATIVES Find a function F whose derivative is f(x) =

3x2

F(x) = x3

A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I.

To find an antiderivative of a power function, increase the exponent by 1 and divide the original coefficient by the new exponent.

What is the antiderivative of 4x5?(2/3)x6

Look at example 8 on pg. 201!!!