Transcript 3 2 Mean Value Th

3-2: Mean Value Theorem
Objectives:
1. To understand and
apply the Mean
Value and Rolle’s
Theorems
Assignment:
• P. 176-178: 1-25 odd,
30, 35-37, 39-45 odd,
53, 67, 73-75
Warm-Up M
Draw a
continuous and
differentiable
function from
point 𝐴 to point
𝐵.
Warm-Up M
Draw the secant
line from point 𝐴
to point 𝐵.
Warm-Up M
Find all lines
tangent to your
curve that are
parallel to the
your secant line.
Warm-Up M
This is the
Mean Value
Theorem
Warm-Up R
Draw a continuous
and differentiable
function from point
𝐴 to point 𝐵 such
that 𝐴 and 𝐵 would
have the same 𝑦coordinate.
Warm-Up R
Draw the secant
line from point 𝐴
to point 𝐵.
Warm-Up R
What does this
imply about the
curve between 𝐴
and 𝐵?
Warm-Up R
This is my
theorem!
Michel Rolle, c. 1691
Objective 1
You will be able to understand
and apply the Mean Value and
Rolle’s Theorem
Rolle’s Theorem
Let 𝑓 be continuous on the closed interval
𝑎, 𝑏 and differentiable on the open interval
𝑎, 𝑏 . If 𝑓(𝑎) = 𝑓(𝑏), then there exists at
least one number 𝑐 in 𝑎, 𝑏 such that
𝑓′(𝑐) = 0.
I totally hate
calculus!
In other words, there
must be at least one
horizontal tangent line
in the interval.
Michel Rolle, c. 1691
Exercise 1
Find the 𝑥-intercepts of 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2
and show that 𝑓′(𝑥) = 0 at some point
between the two 𝑥-intercepts.
Exercise 2
Let 𝑓(𝑥) = 𝑥 4 − 2𝑥 2 . Find all values of 𝑐 in
the interval (−2, 2) such that 𝑓′(𝑐) = 0.
Mean Value Theorem
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 𝑓 𝑐 =
𝑓 𝑏 −𝑓 𝑎
𝑏−𝑎
.
At some point, the
instantaneous rate of
change is equal to the
average rate of change.
Exercise 3
4
,
𝑥
Given 𝑓(𝑥) = 5 − find all values of 𝑐 in the
open interval (1, 4) such that
𝑓′ 𝑐 =
𝑓 4 −𝑓(1)
.
4−1
Exercise 4
Two stationary patrol cars equipped with
radar are 5 miles apart a highway As a truck
passes the first patrol car, its speed is
clocked at 55 miles per hour. Four minutes
later, when the truck passes the second
patrol car, its speed is clocked at 50 miles
per hour. Prove that the truck must have
exceed the speed limit (55 mph) at some
time during the 4 minutes.
Exercise 5: AP 2008
The function 𝑓 is
continuous for
−2 ≤ 𝑥 ≤ 2 and
𝑓 −2 = 𝑓 2 = 0. If
there is no 𝑐, where
− 2 < 𝑐 < 2, for which
𝑓′ 𝑐 = 0, which of the
following statements
must be true?
(A) For −2 < 𝑘 < 2,
𝑓′ 𝑘 > 0.
(B) For −2 < 𝑘 < 2,
𝑓′ 𝑘 < 0.
(C) For −2 < 𝑘 < 2, 𝑓′ 𝑘
exists.
(D) For −2 < 𝑘 < 2, 𝑓′ 𝑘
exists, but 𝑓′ is not
continuous.
(E) For some 𝑘, where
−2 < 𝑘 < 2, 𝑓′ 𝑘 does
not exist.
3-2: Mean Value Theorem
Objectives:
1. To understand and
apply the Mean
Value and Rolle’s
Theorems
Assignment:
• P. 176-178: 1-25 odd,
30, 35-37, 39-45 odd,
53, 67, 73-75