Transcript Document

4.3 The Mean Value Theorem
and Monotonicity
Mon Dec 1
Do Now
Find the critical points of
1)
f (x) = x - 3x +1
2)
f (x) = x + 6x - 2
3
4
2
Mean Value Theorem
• Assume that f(x) is continuous on the closed
interval [a, b] and differentiable on (a, b).
Then there exists at least one value c in (a, b)
such that
f (b) - f (a)
f '(c) =
b-a
• Note: This is the general version of Rolle’s
Theorem
Increasing / Decreasing
Behavior of Functions
• Let f(x) be a differentiable function on an
open interval (a, b)
• If f’(x) > 0, then f(x) is increasing on (a, b)
• If f’(x) < 0, then f(x) is decreasing on (a, b)
• If a function is increasing or decreasing on
(a,b), it is considered to be monotonic
First Derivative Test
• We can use the first derivative to test whether
a critical point of f(x) is a max or min.
• If f’(x) changes from positive to negative, it
has a local maximum
• If f’(x) changes from negative to positive, it
has a local minimum
• We use a table to complete this test
Ex
• Find the local min and max for the function
f (x) = x - 27x - 20
3
Ex 2
• Analyze the critical points of the function
f (x) = x - 8x
4
2
Ex 3
• Analyze the critical points of the function
f (x) = x - x + x
1
3
3
2
Ex 4
• Analyze the critical points of the function on the
2
interval (0, pi)
f (x) = cos x + sin x
You try:
• Analyze the critical points of each
function
f (x) = x - 4 x + 2
3
• 1)
• 2)
f (x) = x + 2x +1
3
2
Closure
• Analyze the critical points of
f (x) = x + x
4
• HW: p.232 #1-61 odds
3
4.3 MVT and Mins/Maxs
Tues Dec 2
• Do Now
• Use the 1st Derivative Test to find the
local min and max
3
f (x) = x -12x - 4
• 1)
• 2)
f (x) = x + x
3
-3
HW Review: p.232
Practice
• 4.3 worksheet
Closure
• How can you use the 1st derivative to
find the local minimums and maximums
of a given function?
• HW: none