- The document discusses a math class lecture on March 14, 2008 that covered topics including the Mean Value Theorem, Rolle's Theorem, and using derivatives to determine if a function is increasing or decreasing on an interval. - It provides announcements about an upcoming midterm being graded, problem sessions, and office hours. It also announces Pi day contests happening at 3:14 PM and 4 PM to recite digits of Pi and eat pie. - The outline previews that the lecture will cover the Mean Value Theorem, Rolle's Theorem, why the MVT is useful, and using derivatives to sketch graphs and test for extremities. It also introduces the mathematician Pierre de Fermat.

1. Section 4.3
The Mean Value Theorem
and the shape of curves
Math 1a
March 14, 2008
Announcements
◮ Midterm is graded.
◮ Problem Sessions Sunday, Thursday, 7pm, SC 310
◮ Office hours Tues, Weds, 2–4pm SC 323
.
Image: Flickr user Jimmywayne32
. . . . . . .

2. Announcements
◮ Midterm is graded
◮ Problem Sessions Sunday, Thursday, 7pm, SC 310
◮ Office hours Tues, Weds, 2–4pm SC 323
. . . . . .

3. Happy Pi Day!
3:14 PM Digit recitation contest! Recite all the digits you know of π
(in order, please). Please let us know in advance if you’ll
recite π in a base other than 10 (the usual choice), 2, or 16.
Only positive integer bases allowed – no fair to memorize π
in base π /(π − 2)...
4 PM — Pi(e) eating contest! Cornbread are square; pie are round.
You have 3 minutes and 14 seconds to stuff yourself with as
much pie as you can. The leftovers will be weighed to
calculate how much pie you have eaten.
Contests take place in the fourth floor lounge of the Math
Department. .
.
Image: Flickr user Paul Adam Smith
. . . . . .

4. Outline
Recall: Fermat’s Theorem and the Closed Interval Method
The Mean Value Theorem
Rolle’s Theorem
Why the MVT is the MITC
The Increasing/Decreasing Test
Using the derivative to sketch the graph
Tests for extemity
The First Derivative Test
The Second Derivative Test
. . . . . .

5. Fermat’s Theorem
Definition
Let f be defined near a. a is a local maximum of f if
f(x) ≤ f(a)
for all x in an open interval containing a.
Theorem
Let f have a local maximum at a. If f is differentiable at a, then
f′ (a) = 0.
. . . . . .

6. The Closed Interval Method
Let f be a continuous function defined on a closed interval [a, b].
We are in search of its global maximum, call it c. Then:
This means to find the
◮ Either the maximum
maximum value of f on [a, b],
occurs at an endpoint of
we need to check:
the interval, i.e., c = a
or c = b, ◮ a and b
◮ Or the maximum occurs ◮ Points x where f′ (x) = 0
inside (a, b). In this ◮ Points x where f is not
case, c is also a local differentiable.
maximum.
◮ Either f is
differentiable at c, in
which case f′ (c) = 0
by Fermat’s Theorem.
◮ Or f is not
differentiable at c.
. . . . . .

7. The Closed Interval Method
Let f be a continuous function defined on a closed interval [a, b].
We are in search of its global maximum, call it c. Then:
This means to find the
◮ Either the maximum
maximum value of f on [a, b],
occurs at an endpoint of
we need to check:
the interval, i.e., c = a
or c = b, ◮ a and b
◮ Or the maximum occurs ◮ Points x where f′ (x) = 0
inside (a, b). In this ◮ Points x where f is not
case, c is also a local differentiable.
maximum.
◮ Either f is The latter two are both called
differentiable at c, in critical points of f. This
which case f′ (c) = 0 technique is called the
by Fermat’s Theorem. Closed Interval Method.
◮ Or f is not
differentiable at c.
. . . . . .

8. Meet the Mathematician: Pierre de Fermat
◮ 1601–1665
◮ Lawyer and number
theorist
◮ Proved many theorems,
didn’t quite prove his
last one
. . . . . .

9. Outline
Recall: Fermat’s Theorem and the Closed Interval Method
The Mean Value Theorem
Rolle’s Theorem
Why the MVT is the MITC
The Increasing/Decreasing Test
Using the derivative to sketch the graph
Tests for extemity
The First Derivative Test
The Second Derivative Test
. . . . . .

10. Rolle’s Theorem
Theorem (Rolle’s Theorem)
Let f be continuous on [a, b]
and differentiable on (a, b).
Suppose f(a) = f(b) = 0.
Then there exists a point
c ∈ (a, b) such that f′ (c) = 0. . .
• .
•
a
. b
.
. . . . . .

11. Rolle’s Theorem
c
.
.
•
Theorem (Rolle’s Theorem)
Let f be continuous on [a, b]
and differentiable on (a, b).
Suppose f(a) = f(b) = 0.
Then there exists a point
c ∈ (a, b) such that f′ (c) = 0. . .
• .
•
a
. b
.
. . . . . .

12. Rolle’s Theorem
c
.
.
•
Theorem (Rolle’s Theorem)
Let f be continuous on [a, b]
and differentiable on (a, b).
Suppose f(a) = f(b) = 0.
Then there exists a point
c ∈ (a, b) such that f′ (c) = 0. . .
• .
•
a
. b
.
Proof.
If f is not constant, it has a local maximum or minimum in (a, b).
Call this point c. Then by Fermat’s Theorem f′ (c) = 0.
. . . . . .

13. The Mean Value Theorem
Theorem (The Mean Value
Theorem)
Let f be continuous on [a, b]
and differentiable on (a, b).
Then there exists a point c in
(a, b) such that .
•
b
.
f(b) − f(a) . .
= f′ (c). •
a
.
b−a
. . . . . .

14. The Mean Value Theorem
Theorem (The Mean Value
Theorem)
Let f be continuous on [a, b]
and differentiable on (a, b).
Then there exists a point c in
(a, b) such that .
•
b
.
f(b) − f(a) . .
= f′ (c). •
a
.
b−a
. . . . . .

15. The Mean Value Theorem
Theorem (The Mean Value c
.
Theorem) .
•
Let f be continuous on [a, b]
and differentiable on (a, b).
Then there exists a point c in
(a, b) such that .
•
b
.
f(b) − f(a) . .
= f′ (c). •
a
.
b−a
. . . . . .

16. Proof of the MVT
Proof.
The line connecting (a, f(a)) and (b, f(b)) has equation
f(b) − f(a)
y − f (a ) = (x − a).
b−a
Apply Rolle’s Theorem to the function
f(b) − f(a)
g(x) = f(x) − (x − a).
b−a
Then g is continuous on [a, b] and differentiable on (a, b) since f
is. Also g(a) = 0 and g(b) = 0 (check both). So there exists a
point c ∈ (a, b) such that
f(b) − f(a)
0 = g′ (c) = f′ (c) − .
b−a
. . . . . .

17. Question
On a toll road a driver takes a time stamped toll-card from the
starting booth and drives directly to the end of the toll section.
After paying the required toll, the driver is surprised to receive a
speeding ticket along with the toll receipt. Which of the
following best describes the situation?
(a) The booth attendant does not have enough information to
prove that the driver was speeding.
(b) The booth attendant can prove that the driver was speeding
during his trip.
(c) The driver will get a ticket for a lower speed than his actual
maximum speed.
(d) Both (b) and (c).
Be prepared to justify your answer.
. . . . . .

18. Question
On a toll road a driver takes a time stamped toll-card from the
starting booth and drives directly to the end of the toll section.
After paying the required toll, the driver is surprised to receive a
speeding ticket along with the toll receipt. Which of the
following best describes the situation?
(a) The booth attendant does not have enough information to
prove that the driver was speeding.
(b) The booth attendant can prove that the driver was speeding
during his trip.
(c) The driver will get a ticket for a lower speed than his actual
maximum speed.
(d) Both (b) and (c).
Be prepared to justify your answer.
Answer
(b) and (c).
. . . . . .

19. Outline
Recall: Fermat’s Theorem and the Closed Interval Method
The Mean Value Theorem
Rolle’s Theorem
Why the MVT is the MITC
The Increasing/Decreasing Test
Using the derivative to sketch the graph
Tests for extemity
The First Derivative Test
The Second Derivative Test
. . . . . .

20. Why the MVT is the MITC
Theorem
Let f′ = 0 on an interval (a, b).
. . . . . .

21. Why the MVT is the MITC
Theorem
Let f′ = 0 on an interval (a, b). Then f is constant on (a, b).
. . . . . .

22. Why the MVT is the MITC
Theorem
Let f′ = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x < y. Then f is continuous
on [x, y] and differentiable on (x, y). By MVT there exists a point
z ∈ (x, y) such that
f(y) − f(x)
= f′ (z) = 0.
y−x
So f(y) = f(x). Since this is true for all x and y in (a, b), then f is
constant.
. . . . . .

23. Outline
Recall: Fermat’s Theorem and the Closed Interval Method
The Mean Value Theorem
Rolle’s Theorem
Why the MVT is the MITC
The Increasing/Decreasing Test
Using the derivative to sketch the graph
Tests for extemity
The First Derivative Test
The Second Derivative Test
. . . . . .

24. The Increasing/Decreasing Test
Theorem (The Increasing/Decreasing Test)
If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b),
then f is decreasing on (a, b).
. . . . . .

25. The Increasing/Decreasing Test
Theorem (The Increasing/Decreasing Test)
If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b),
then f is decreasing on (a, b).
Proof.
It works the same as the last theorem. Pick two points x and y in
(a, b) with x < y. We must show f(x) < f(y). By MVT there exists
a point c ∈ (x, y) such that
f(y) − f(x)
= f′ (c) > 0.
y−x
So
f(y) − f(x) = f′ (c)(y − x) > 0.
. . . . . .

26. Example
Find the intervals of monotonicity of f(x) = 2/3x − 5.
. . . . . .

27. Example
Find the intervals of monotonicity of f(x) = 2/3x − 5.
Solution
f′ (x) = 2/3 is always positive, so f is increasing on (−∞, ∞).
. . . . . .

28. Example
Find the intervals of monotonicity of f(x) = x2 − 1.
. . . . . .

29. Example
Find the intervals of monotonicity of f(x) = x2 − 1.
Solution
f′ (x) = 2x, which is positive when x > 0 and negative when x is.
. . . . . .

30. Example
Find the intervals of monotonicity of f(x) = x2 − 1.
Solution
f′ (x) = 2x, which is positive when x > 0 and negative when x is.
We can draw a number line:
−
. 0
.. .
+ .′
f
0
.
. . . . . .

31. Example
Find the intervals of monotonicity of f(x) = x2 − 1.
Solution
f′ (x) = 2x, which is positive when x > 0 and negative when x is.
We can draw a number line:
−
. ..
0 .
+ .′
f
↘
. 0
. ↗
. f
.
. . . . . .

32. Example
Find the intervals of monotonicity of f(x) = x2/3 (x + 2).
. . . . . .

33. Example
Find the intervals of monotonicity of f(x) = x2/3 (x + 2).
Solution
Write f(x) = x5/3 + 2x2/3 . Then
f′ (x) = 5 x2/3 + 4 x−1/3
3 3
= 1 x−1/3 (5x + 4)
3
The critical points are 0 and and −4/5.
−
. ×
.. .
+
. −1/3
x
0
.
−
. 0
.. .
+
.x+4
5
−
. 4/5
. . . . . .

34. Example
Find the intervals of monotonicity of f(x) = x2/3 (x + 2).
Solution
Write f(x) = x5/3 + 2x2/3 . Then
f′ (x) = 5 x2/3 + 4 x−1/3
3 3
= 1 x−1/3 (5x + 4)
3
The critical points are 0 and and −4/5.
−
. ×
.. .
+
. −1/3
x
0
.
−
. 0
.. .
+
.x+4
5
−
. 4/5
.
+ 0 − ×
.. . . . .
+ .′ (x)
f
↗
. − ↘ .
. 4/5 . 0 ↗
. f
.(x)
. . . . . .

35. Outline
Recall: Fermat’s Theorem and the Closed Interval Method
The Mean Value Theorem
Rolle’s Theorem
Why the MVT is the MITC
The Increasing/Decreasing Test
Using the derivative to sketch the graph
Tests for extemity
The First Derivative Test
The Second Derivative Test
. . . . . .

36. The First Derivative Test
Let f be continuous on [a, b] and c in (a, b) a critical point of f.
Theorem
◮ If f′ (x) > 0 on (a, c) and f′ (x) < 0 on (c, b), then f(c) is a
local maximum.
◮ If f′ (x) < 0 on (a, c) and f′ (x) > 0 on (c, b), then f(c) is a
local minimum.
◮ If f′ (x) has the same sign on (a, c) and (c, b), then (c) is not a
local extremum.
. . . . . .

37. The Second Derivative Test
Let f, f′ , and f′′ be continuous on [a, b] and c in (a, b) a critical
point of f.
Theorem
◮ If f′′ (c) < 0, then f(c) is a local maximum.
◮ If f′′ (c) > 0, then f(c) is a local minimum.
◮ If f′′ (c) = 0, the second derivative is inconclusive (this does
not mean c is neither; we just don’t know yet).
. . . . . .

38. Example
Find the local extrema of f(x) = x3 − x.
. . . . . .

39. Next time: graphing functions
. . . . . .