The Mean Value Theoremand Rolle’s Theorem

Lesson 3.2

I wonder how mean this

theorem really is?

3.2 Bellwork

• Locate the COORDINATES of the absolute extrema of the function on the closed interval given.

• Verify the absolute extrema you found by graphing the function in an appropriate viewing window.

• Find the equation of the tangent line to the curve when x = 2.

• Graph the curve and the tangent line in an appropriate viewing window on your calculator.

Finding the equation of the tangent line.

The absolute maximum and absolute minimum are clearly shown in this viewing window.

You can also see that each critical point represents a local extrema for the graph.

(0,0) is the location of a relative maximum, also known as a local maximum of f(x).

(1,-1/2) is the location of a relative minimum, also known as a local minimum of f(x).

Notice that the slope of the tangent line clearly matches the slope of the curve at the point (2,2).

The Mean Value Theoremand Rolle’s Theorem

Lesson 3.2

I wonder how mean this

theorem really is?

This is Really Mean

Think About It

• Consider a trip of two hours that is 120 miles in distance … You have averaged 60 miles per hour

• What reading on your speedometer would you have expected to see at least once?

60

Rolle’s Theorem

• Given f(x) on closed interval [a, b] Differentiable on open interval (a, b)

• If f(a) = f(b) … then There exists at least one number

a < c < b such that f ’(c) = 0

f(a) = f(b)

a bc

Mean Value Theorem

• We can “tilt” the picture of Rolle’s Theorem Stipulating that f(a) ≠ f(b)

• Then there exists a c such that

a bc

( ) ( )'( )

f b f af c

b a

Mean Value Theorem

• Applied to a cubic equation

Note Geogebera Example

Note Geogebera Example

Finding c

• Given a function f(x) = 2x3 – x2 Find all points on the interval [0, 2] where

• Strategy Find slope of line from f(0) to f(2) Find f ‘(x) Set equal to slope … solve for x

( ) ( )'( )

f b f af c

b a

Modeling Problem• Two police cars are located at fixed points 6

miles apart on a long straight road. The speed limit is 55 mph A car passes the first point at 53 mph Five minutes later he passes the second at 48

mph Yuk! Yuk! I think he was

speeding, EnosWe need to

prove it, Rosco