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# Lesson 21: Curve Sketching (Section 10 version)

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The increasing/decreasing test and test for concavity allow us to spread out all the information we need about a function to reconstruct its graph.

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### Lesson 21: Curve Sketching (Section 10 version)

1. 1. Section 4.4 Curve Sketching I V63.0121, Calculus I March 30, 2009 Announcements Quiz 4 this week (Sections 2.5–3.5) Ofﬁce hours this week: M 1–2, T 1–2, W 2–3, R 9–10 . . Image credit: Fast Eddie 42 . . . . . .
2. 2. Ofﬁce Hours and other help In addition to recitation Day Time Who/What Where in WWH M 1:00–2:00 Leingang OH 718/618 3:30–4:30 Katarina OH 707 5:00–7:00 Curto PS 517 T 1:00–2:00 Leingang OH 718/618 4:00–5:50 Curto PS 317 W 1:00–2:00 Katarina OH 707 2:00–3:00 Leingang OH 718/618 R 9:00–10:00am Leingang OH 718/618 5:00–7:00pm Maria OH 807 F 2:00–4:00 Curto OH 1310 . . . . . .
3. 3. CIMS/NYU professor wins Abel Prize Mikhail Gromov, born 1943 in Russia contributions to geometry and topology discovered the pseudoholomorphic curve Abel Prize is the highest in mathematics . . . . . .
4. 4. On the problems assigned from Section 2.8 Announcements were made in class but not online Resubmit your Problem Set 6 on Wednesday, April 1 with Problem Set 8 We will pick two additional problems to grade from Problem Set 8 If the scores on the makeup problems from PS 8 exceed the scores of the Section 2.8 problems from PS 6, the makeup scores will be substituted. This also takes care of the problematic problem 2.8.28. This offer is only good this week. . . . . . .
5. 5. Outline The Procedure The examples A cubic function A quartic function . . . . . .
6. 6. 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 in (x, y) such that f(y) − f(x) = f′ (c) > 0. y−x So f(y) − f(x) = f′ (c)(y − x) > 0. . . . . . .
7. 7. Theorem (Concavity Test) If f′′ (x) > 0 for all x in I, then the graph of f is concave upward on I If f′′ (x) < 0 for all x in I, then the graph of f is concave downward on I Proof. Suppose f′′ (x) > 0 on I. This means f′ is increasing on I. Let a and x be in I. The tangent line through (a, f(a)) is the graph of L(x) = f(a) + f′ (a)(x − a) f(x) − f(a) = f′ (b). By MVT, there exists a b between a and x with x−a So f(x) = f(a) + f′ (b)(x − a) ≥ f(a) + f′ (a)(x − a) = L(x) . . . . . .
8. 8. Graphing Checklist To graph a function f, follow this plan: 0. Find when f is positive, negative, zero, not deﬁned. 1. Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. 2. Find f′′ and form its sign chart. Conclude concave up/concave down and inﬂection. 3. Put together a big chart to assemble monotonicity and concavity data 4. Graph! . . . . . .
9. 9. Outline The Procedure The examples A cubic function A quartic function . . . . . .
10. 10. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. . . . . . .
11. 11. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. First, let’s ﬁnd the zeros. We can at least factor out one power of x: f(x) = x(2x2 − 3x − 12) so f(0) = 0. The other factor is a quadratic, so we the other two roots are √ √ 3 ± 32 − 4(2)(−12) 3 ± 105 x= = 4 4 It’s OK to skip this step for now since the roots are so complicated. . . . . . .
12. 12. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: . . . . . . .
13. 13. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: . . . −2 x 2 . . x . +1 − .1 .′ (x) f . . − 2 . .1 f .(x) . . . . . .
14. 14. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . . x . +1 − .1 .′ (x) f . . − 2 . .1 f .(x) . . . . . .
15. 15. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f . . − 2 . .1 f .(x) . . . . . .
16. 16. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f .. + . − 2 . .1 f .(x) . . . . . .
17. 17. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . + . − 2 . .1 f .(x) . . . . . .
18. 18. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . − 2 . .1 f .(x) . . . . . .
19. 19. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . ↗− 2 . . .1 f .(x) . . . . . .
20. 20. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . ↗− ↘ 2 . . .1 . f .(x) . . . . . .
21. 21. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . ↗− ↘ ↗ 2 . . .1 . . f .(x) . . . . . .
22. 22. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . ↗− ↘ ↗ 2 . . .1 . . f .(x) m . ax . . . . . .
23. 23. Monotonicity f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − − . .. . + . . −2 x 2 . − .. . . + + x . +1 − .1 .′ (x) f − .. . . + + . ↗− ↘ ↗ 2 . . .1 . . f .(x) m . ax m . in . . . . . .
24. 24. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . . . . . . .
25. 25. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f . f . .(x) 1/2 . . . . . .
26. 26. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f − .− . f . .(x) 1/2 . . . . . .
27. 27. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f − .− .+ + . f . .(x) 1/2 . . . . . .
28. 28. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f − .− .+ + . . ⌢ f . .(x) 1/2 . . . . . .
29. 29. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f − .− .+ + . . . ⌢ ⌣ f . .(x) 1/2 . . . . . .
30. 30. Concavity f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f − .− .+ + . . . ⌢ ⌣ f . .(x) 1/2 I .P . . . . . .
31. 31. One sign chart to rule them all . . . . . . .
32. 32. One sign chart to rule them all .′ (x) f − − . . .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m . onotonicity . . . . . .
33. 33. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘. ↗ 2 . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . . . . ⌢ ⌢ 1/2 ⌣ ⌣ c . oncavity . . . . . . .
34. 34. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . ⌢ 1/2 . . . ⌢ ⌣ ⌣ c . oncavity . − f . 61/2 .(x) −. . 20 7 .. . − 2 . .1 . . hape of f s 1/2 m . ax I .P m . in . . . . . .
35. 35. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . ⌢ 1/2 . . . ⌢ ⌣ ⌣ c . oncavity . − f . 61/2 .(x) −. . 20 7 .. . ..1 − 2 . . . hape of f s 1/2 m . ax I .P m . in . . . . . .
36. 36. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . ⌢ 1/2 . . . ⌢ ⌣ ⌣ c . oncavity . − f . 61/2 .(x) −. . 20 7 .. . . . 1 . 1/2 − 2 . . . hape of f s m . ax I .P m . in . . . . . .
37. 37. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . ⌢ 1/2 . . . ⌢ ⌣ ⌣ c . oncavity . − f . 61/2 .(x) −. . 20 7 .. . . . 1 . 1/2 . − 2 . . . hape of f s m . ax I .P m . in . . . . . .
38. 38. One sign chart to rule them all .′ (x) f − − .. .. . . + + . ↗− ↘ ↘ ↗ 2 . . .1 . . . m .′′ onotonicity f . (x) − .− − .− . .+ .+ + + . ⌢ 1/2 . . . ⌢ ⌣ ⌣ c . oncavity . − f . 61/2 .(x) −. . 20 7 .. . . . 1 . 1/2 . . − 2 . . . hape of f s m . ax I .P m . in . . . . . .
39. 39. Graph f .(x) . −1, 7) ( (√ ) . . 3− 4105 , 0 . 0, 0) ( . . . x (. ) √ . 1/2, −61/2) ( 3+ 105 . ,0 . 4 . . 2, −20) ( . . . . . .
40. 40. Graph f .(x) . −1, 7) ( (√ ) . . 3− 4105 , 0 . 0, 0) ( . . . x (. ) √ . 1/2, −61/2) ( 3+ 105 . ,0 . 4 . . 2, −20) ( . . . . . .
41. 41. Graph f .(x) . −1, 7) ( (√ ) . . 3− 4105 , 0 . 0, 0) ( . . . x (. ) √ . 1/2, −61/2) ( 3+ 105 . ,0 . 4 . . 2, −20) ( . . . . . .
42. 42. Graph f .(x) . −1, 7) ( (√ ) . . 3− 4105 , 0 . 0, 0) ( . . . x (. ) √ . 1/2, −61/2) ( 3+ 105 . ,0 . 4 . . 2, −20) ( . . . . . .
43. 43. Graph f .(x) . −1, 7) ( (√ ) . . 3− 4105 , 0 . 0, 0) ( . . . x (. ) √ . 1/2, −61/2) ( 3+ 105 . ,0 . 4 . . 2, −20) ( . . . . . .
44. 44. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 . . . . . .
45. 45. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 We know f(0) = 10 and lim f(x) = +∞. Not too many other x→±∞ points on the graph are evident. . . . . . .
46. 46. Monotonicity f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. . . +0 + + . x2 4 0 . − − . . .. . 0+ . x − 3) ( 3 . .′ (x) f −0 − . .. . .. . 0+ ↘0 ↘ 3↗ .. .. . f .(x) m . in . . . . . .
47. 47. Concavity f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: −0 . .. . . + + 1 . 2x 0 . − − . . . 0 .. + . −2 x 2 . .′′ (x) f − .− . + .. .+ +0 0 .. + .. . . ⌣0 ⌢ ⌣ 2 . f .(x) I .P I .P . . . . . .
48. 48. Grand Uniﬁed Sign Chart . .′ (x) f −0 − −0+ . .. . . .. . ↘0 ↘ ↘3↗ .. . .. . m .′′ onotonicity f . (x) − .− . + .. .. . + . + +0 0+ + .. . . . ⌣0 ⌢ ⌣ ⌣ 2 . c . oncavity f .(x) − −. . .6 . 17 1. .0 0 . 2 . 3 . s . hape I .P I .P . inm . . . . . .
49. 49. Grand Uniﬁed Sign Chart . .′ (x) f −0 − −0+ . .. . . .. . ↘0 ↘ ↘3↗ .. . .. . m .′′ onotonicity f . (x) − .− . + .. .. . + . + +0 0+ + .. . . . ⌣0 ⌢ ⌣ ⌣ 2 . c . oncavity f .(x) − −. . .6 . 17 1. .0 ..0 2 . 3 . s . hape I .P I .P . inm . . . . . .
50. 50. Grand Uniﬁed Sign Chart . .′ (x) f −0 − −0+ . .. . . .. . ↘0 ↘ ↘3↗ .. . .. . m .′′ onotonicity f . (x) − .− . + .. .. . + . + +0 0+ + .. . . . ⌣0 ⌢ ⌣ ⌣ 2 . c . oncavity f .(x) − −. . .6 . 17 1. .0 .. . 0 2 . 3 . s . hape I .P I .P . inm . . . . . .
51. 51. Grand Uniﬁed Sign Chart . .′ (x) f −0 − −0+ . .. . . .. . ↘0 ↘ ↘3↗ .. . .. . m .′′ onotonicity f . (x) − .− . + .. .. . + . + +0 0+ + .. . . . ⌣0 ⌢ ⌣ ⌣ 2 . c . oncavity f .(x) − −. . .6 . 17 1. .0 .. . .. 0 2 . 3 s . hape I .P I .P . inm . . . . . .
52. 52. Grand Uniﬁed Sign Chart . .′ (x) f −0 − −0+ . .. . . .. . ↘0 ↘ ↘3↗ .. . .. . m .′′ onotonicity f . (x) − .− . + .. .. . + . + +0 0+ + .. . . . ⌣0 ⌢ ⌣ ⌣ 2 . c . oncavity f .(x) − −. . .6 . 17 1. .0 .. . ... 0 2 . 3 s . hape I .P I .P . inm . . . . . .
53. 53. Graph y . . 0, 10) ( . . x . . . . 2, −6) ( . 3, −17) ( . . . . . .