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# Lesson 19: Curve Sketching

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### Lesson 19: Curve Sketching

1. 1. Section 4.4 Curve Sketching V63.0121.006/016, Calculus I New York University April 1, 2010 . . . . . .
2. 2. Second-chance Midterm: Tomorrow in Recitation 12 free-response questions, no multiple choice Covers all sections so far, through today Your score on this exam will replace your midterm score . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 2 / 47
3. 3. . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 3 / 47
4. 4. Quiz 3 tomorrow in recitation Section 2.6: implicit differentiation Section 2.8: linear approximation and differentials Section 3.1: exponential functions Section 3.2: logarithms Section 3.3: derivatives of logarithmic and exponential functions Section 3.4: exponential growth and decay Section 3.5: inverse trigonometric functions . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 4 / 47
5. 5. Outline The Procedure Simple examples A cubic function A quartic function More Examples Points of nondifferentiability Horizontal asymptotes Vertical asymptotes Trigonometric and polynomial together Logarithmic . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 5 / 47
6. 6. Objective Given a function, graph it completely, indicating zeroes asymptotes if applicable critical points local/global max/min inflection points . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 6 / 47
7. 7. Objective Given a function, graph it completely, indicating zeroes asymptotes if applicable critical points local/global max/min inflection points . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 6 / 47
8. 8. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 7 / 47
9. 9. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 8 / 47
10. 10. 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! . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 9 / 47
11. 11. Outline The Procedure Simple examples A cubic function A quartic function More Examples Points of nondifferentiability Horizontal asymptotes Vertical asymptotes Trigonometric and polynomial together Logarithmic . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 10 / 47
12. 12. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 11 / 47
13. 13. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 11 / 47
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: . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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) . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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 . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
25. 25. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
26. 26. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
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) . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
32. 32. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 13 / 47
33. 33. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
34. 34. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. − . . . . + − . . + .′ (x) f . ↗− ↘ . . 1 . ↘ . 2 . ↗ . m . onotonicity . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
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) . ⌢ . ⌢ 1/2 . . ⌣ . ⌣ c . oncavity . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
36. 36. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
37. 37. Combinations of monotonicity and concavity I .I I . . I .II I .V . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
38. 38. Combinations of monotonicity and concavity . decreasing, concave down I .I I . . I .II I .V . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
39. 39. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
40. 40. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . decreasing, concave up . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
41. 41. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
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 − . 1/2 2 . s . hape of f m . ax I .P m . in . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
45. 45. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
50. 50. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
51. 51. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 18 / 47
52. 52. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 18 / 47
53. 53. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
54. 54. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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 . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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 . . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
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) . . . . . . V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
70. 70. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 19 / 47
71. 71. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) . . . . . . .
72. 72. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: . . . . . . .
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 . . . . . . .
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 . . . . . . .
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 . . . . . . .
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 . . . . . . .
77. 77. 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 . . . . . . .
78. 78. 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 . . . . . . .
79. 79. 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 . . . . . . .
80. 80. 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 . . . . . . .
81. 81. 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 . . + .. + 0 − . − 0 .. .′′ (x) f 0 . 2 . f .(x) . . . . . .
82. 82. 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 . . + .. + 0 − . − 0 .. . + + .′′ (x) f 0 . 2 . f .(x) . . . . . .
83. 83. 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 . . + .. + 0 − . − 0 .. . + + .′′ (x) f . . ⌣ 0 2 . f .(x) . . . . . .
84. 84. 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 . . + .. + 0 − . − 0 .. . + + .′′ (x) f . . ⌣ 0 . ⌢ 2 . f .(x) . . . . . .