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- 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 . . . . . .

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