0
Section	4.3
                            The	Mean	Value	Theorem
                             and	the	shape	of	curves

     ...
Announcements




   ◮   Midterm	is	graded
   ◮   Problem	Sessions	Sunday, Thursday, 7pm, SC 310
   ◮   Office	hours	Tues, ...
Happy	Pi	Day!


3:14	PM Digit	recitation	contest! Recite	all	the	digits	you	know	of π
        (in	order, please). Please	l...
Outline

  Recall: Fermat’s	Theorem	and	the	Closed	Interval	Method

  The	Mean	Value	Theorem
     Rolle’s	Theorem

  Why	t...
Fermat’s	Theorem



  Definition
  Let f be	defined	near a. a is	a local	maximum of f if

                               f(x...
The	Closed	Interval	Method
   Let f be	a	continuous	function	defined	on	a	closed	interval [a, b].
   We	are	in	search	of	it...
The	Closed	Interval	Method
   Let f be	a	continuous	function	defined	on	a	closed	interval [a, b].
   We	are	in	search	of	it...
Meet	the	Mathematician: Pierre	de	Fermat




   ◮   1601–1665
   ◮   Lawyer	and	number
       theorist
   ◮   Proved	many	...
Outline

  Recall: Fermat’s	Theorem	and	the	Closed	Interval	Method

  The	Mean	Value	Theorem
     Rolle’s	Theorem

  Why	t...
Rolle’s	Theorem



  Theorem	(Rolle’s	Theorem)
  Let f be	continuous	on [a, b]
  and	differentiable	on (a, b).
  Suppose f...
Rolle’s	Theorem


                                                         c
                                             ...
Rolle’s	Theorem


                                                                   c
                                   ...
The	Mean	Value	Theorem



 Theorem	(The	Mean	Value
 Theorem)
 Let f be	continuous	on [a, b]
 and	differentiable	on (a, b)....
The	Mean	Value	Theorem



 Theorem	(The	Mean	Value
 Theorem)
 Let f be	continuous	on [a, b]
 and	differentiable	on (a, b)....
The	Mean	Value	Theorem



 Theorem	(The	Mean	Value                              c
                                        ...
Proof	of	the	MVT
  Proof.
  The	line	connecting (a, f(a)) and (b, f(b)) has	equation

                                   f...
Question
On	a	toll	road	a	driver	takes	a	time	stamped	toll-card	from	the
starting	booth	and	drives	directly	to	the	end	of	...
Question
On	a	toll	road	a	driver	takes	a	time	stamped	toll-card	from	the
starting	booth	and	drives	directly	to	the	end	of	...
Outline

  Recall: Fermat’s	Theorem	and	the	Closed	Interval	Method

  The	Mean	Value	Theorem
     Rolle’s	Theorem

  Why	t...
Why	the	MVT is	the	MITC


  Theorem
  Let f′ = 0 on	an	interval (a, b).




                                      .   .   ...
Why	the	MVT is	the	MITC


  Theorem
  Let f′ = 0 on	an	interval (a, b). Then f is	constant	on (a, b).




                ...
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...
Outline

  Recall: Fermat’s	Theorem	and	the	Closed	Interval	Method

  The	Mean	Value	Theorem
     Rolle’s	Theorem

  Why	t...
The	Increasing/Decreasing	Test

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

   Theorem	(The	Increasing/Decreasing	Test)
   If f′ > 0 on (a, b), then f is	increasing	o...
Example
Find	the	intervals	of	monotonicity	of f(x) = 2/3x − 5.




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

Solution
f′ (x) = 2/3 is	always	positive, so f is	increasi...
Example
Find	the	intervals	of	monotonicity	of f(x) = x2 − 1.




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

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

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

Solution
f′ (x) = 2x, which	is	positive	when x > 0 and	negat...
Example
Find	the	intervals	of	monotonicity	of f(x) = x2/3 (x + 2).




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

Solution
Write f(x) = x5/3 + 2x2/3 . Then

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

Solution
Write f(x) = x5/3 + 2x2/3 . Then

           ...
Outline

  Recall: Fermat’s	Theorem	and	the	Closed	Interval	Method

  The	Mean	Value	Theorem
     Rolle’s	Theorem

  Why	t...
The	First	Derivative	Test



   Let f be	continuous	on [a, b] and c in (a, b) a	critical	point	of f.
   Theorem
     ◮   I...
The	Second	Derivative	Test



   Let f, f′ , and f′′ be	continuous	on [a, b] and c in (a, b) a	critical
   point	of f.
   ...
Example
Find	the	local	extrema	of f(x) = x3 − x.




                                           .   .   .   .   .   .
Next	time: graphing	functions




                      .   .     .   .   .   .
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Lesson 17: The Mean Value Theorem and the shape of curves

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The Mean Value Theorem is the Most Important Theorem in Calculus because it relates information about the derivative of a function to information about the function itself.

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Transcript of "Lesson 17: The Mean Value Theorem and the shape of curves"

  1. 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. 2. Announcements ◮ Midterm is graded ◮ Problem Sessions Sunday, Thursday, 7pm, SC 310 ◮ Office hours Tues, Weds, 2–4pm SC 323 . . . . . .
  3. 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. 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. 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. 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. 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. 8. Meet the Mathematician: Pierre de Fermat ◮ 1601–1665 ◮ Lawyer and number theorist ◮ Proved many theorems, didn’t quite prove his last one . . . . . .
  9. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 20. Why the MVT is the MITC Theorem Let f′ = 0 on an interval (a, b). . . . . . .
  21. 21. Why the MVT is the MITC Theorem Let f′ = 0 on an interval (a, b). Then f is constant on (a, b). . . . . . .
  22. 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. 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. 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. 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. 26. Example Find the intervals of monotonicity of f(x) = 2/3x − 5. . . . . . .
  27. 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. 28. Example Find the intervals of monotonicity of f(x) = x2 − 1. . . . . . .
  29. 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. 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. 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. 32. Example Find the intervals of monotonicity of f(x) = x2/3 (x + 2). . . . . . .
  33. 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. 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. 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. 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. 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. 38. Example Find the local extrema of f(x) = x3 − x. . . . . . .
  39. 39. Next time: graphing functions . . . . . .
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