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# Lesson 21: Curve Sketching

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### Lesson 21: Curve Sketching

1. 1. Section 4.4 Curve Sketching V63.0121.002.2010Su, Calculus I New York University June 10, 2010Announcements Homework 4 due Tuesday . . . . . .
2. 2. Announcements Homework 4 due Tuesday . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 2 / 45
3. 3. Objectives given a function, graph it completely, indicating zeroes (if easy) asymptotes if applicable critical points local/global max/min inflection points . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 3 / 45
4. 4. Why? Graphing functions is like dissection . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
5. 5. Why? Graphing functions is like dissection … or diagramming sentences . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
6. 6. Why? Graphing functions is like dissection … or diagramming sentences You can really know a lot about a function when you know all of its anatomy. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
7. 7. 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). Example Here f(x) = x3 + x2 , and f′ (x) = 3x2 + 2x. f .(x) .′ (x) f . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 5 / 45
8. 8. Testing for Concavity Theorem (Concavity Test) If f′′ (x) > 0 for all x in (a, b), then the graph of f is concave upward on (a, b) If f′′ (x) < 0 for all x in (a, b), then the graph of f is concave downward on (a, b). Example Here f(x) = x3 + x2 , f′ (x) = 3x2 + 2x, and f′′ (x) = 6x + 2. .′′ (x) f f .(x) .′ (x) f . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 6 / 45
9. 9. Graphing ChecklistTo 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! . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 7 / 45
10. 10. Outline Simple examples A cubic function A quartic function More Examples Points of nondifferentiability Horizontal asymptotes Vertical asymptotes Trigonometric and polynomial together Logarithmic . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 8 / 45
11. 11. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
12. 12. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. (Step 0) 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. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
13. 13. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
14. 14. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
15. 15. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
16. 16. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
17. 17. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
18. 18. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
19. 19. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . − . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
20. 20. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . ↗− . . 1 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
21. 21. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . ↗− . . 1 ↘ . 2 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
22. 22. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . ↗− . . 1 ↘ . 2 . ↗ . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
23. 23. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . ↗− . . 1 ↘ . 2 . ↗ . f .(x) m . ax . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
24. 24. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ 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 . ↗− . . 1 ↘ . 2 . ↗ . f .(x) m . ax m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
25. 25. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
26. 26. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f . ./2 1 f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
27. 27. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − .′′ (x) f . ./2 1 f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
28. 28. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . ./2 1 f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
29. 29. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
30. 30. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 . ⌣ f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
31. 31. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 . ⌣ f .(x) I .P . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
32. 32. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
33. 33. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. − . . . . + − . . + .′ (x) f . ↗− ↘ . . 1 . ↘ . 2 . ↗ . m . onotonicity . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
34. 34. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ . ⌢ 1/2 . . ⌣ . ⌣ c . oncavity . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
35. 35. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . − . 1 . 1/2 2 . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
36. 36. Combinations of monotonicity and concavity I .I I . . I .II I .V . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
37. 37. Combinations of monotonicity and concavity . decreasing, concave down I .I I . . I .II I .V . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
38. 38. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
39. 39. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . decreasing, concave up . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
40. 40. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . . decreasing, increasing, concave up concave up . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
41. 41. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 − . 1/2 2 . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
42. 42. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 − 1 2 . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
43. 43. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
44. 44. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
45. 45. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
46. 46. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
47. 47. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
48. 48. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
49. 49. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
50. 50. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
51. 51. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 (Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many other x→±∞ points on the graph are evident. . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
52. 52. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
53. 53. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
54. 54. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. 0 .. . x2 4 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
55. 55. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . x2 4 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
56. 56. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . x2 4 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
57. 57. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
58. 58. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . 0 .. . x − 3) ( 3 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
59. 59. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . 0 .. . x − 3) ( 3 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
60. 60. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . 0 .. . x − 3) ( 3 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
61. 61. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
62. 62. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . 0 .. 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
63. 63. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
64. 64. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
65. 65. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f 0 . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
66. 66. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
67. 67. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
68. 68. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 ↗ . . f .(x) . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
69. 69. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 ↗ . . f .(x) m . in . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
70. 70. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
71. 71. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
72. 72. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: 0 .. 1 . 2x 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
73. 73. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. 1 . 2x 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
74. 74. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + 1 . 2x 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
75. 75. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
76. 76. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . 0 .. . −2 x 2 . . . . . . .V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45