Transcript I wonder how mean this theorem really is? The Mean Value Theorem Lesson 4.2 This is Really Mean.

I wonder how
mean this
theorem really
is?
The Mean Value Theorem
Lesson 4.2
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
c
b
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
f (b)  f (a)
 f '(c)
ba
c
a
b
Mean Value Theorem
• Applied to a cubic equation
Note Geogebera
Example
Finding c
• Given a function f(x) = 2x3 – x2
 Find all points on the interval [0, 2] where
f (b)  f (a)
 f '(c)
ba
• Strategy
 Find slope of line from f(0) to f(2)
 Find f ‘(x)
 Set equal to slope … solve for x
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
We need to
prove it,
Rosco
Yuk! Yuk!
I think he was
speeding,
Enos
Assignment
• Lesson 4.2
• Pg 216
• Exercises 1 – 61 EOO