Lesson 21: Curve Sketching
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The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
Lesson 21: Curve Sketching
- 1. Section 4.4 Curve Sketching V63.0121.002.2010Su, Calculus I New York University June 10, 2010 Announcements Homework 4 due Tuesday . . . . . .
- 2. Announcements Homework 4 due Tuesday . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 2 / 45
- 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. Why? Graphing functions is like dissection . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
- 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. 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. 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. 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. 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! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 7 / 45
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
- 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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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 . 0 .. 0 .. .′′ (x) f 0 . 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 85. 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 86. 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) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 87. 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) I .P . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 88. 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) I .P I .P . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
- 89. Step 3: Grand Unified Sign Chart . Remember, f(x) = x4 − 4x3 + 10. − 0 . .. − . − 0 + . .. . .′ (x) f ↘ 0 . . ↘ . ↘ 3 ↗ . . . m .′′ onotonicity . + .. + 0 − . − .. . + . + 0+ + f . (x) . . ⌣ 0 . ⌢ 2 . . ⌣ . ⌣ c . oncavity 1. .0 − −. . .6 . 17 f .(x) 0 . 2 . 3 . s . hape I .P I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
- 90. Step 3: Grand Unified Sign Chart . Remember, f(x) = x4 − 4x3 + 10. − 0 . .. − . − 0 + . .. . .′ (x) f ↘ 0 . . ↘ . ↘ 3 ↗ . . . m .′′ onotonicity . + .. + 0 − . − .. . + . + 0+ + f . (x) . . ⌣ 0 . ⌢ 2 . . ⌣ . ⌣ c . oncavity 1. .0 − −. . .6 . 17 f .(x) . .0 2 . 3 . s . hape I .P I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
- 91. Step 3: Grand Unified Sign Chart . Remember, f(x) = x4 − 4x3 + 10. − 0 . .. − . − 0 + . .. . .′ (x) f ↘ 0 . . ↘ . ↘ 3 ↗ . . . m .′′ onotonicity . + .. + 0 − . − .. . + . + 0+ + f . (x) . . ⌣ 0 . ⌢ 2 . . ⌣ . ⌣ c . oncavity 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . 3 . s . hape I .P I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
- 92. Step 3: Grand Unified Sign Chart . Remember, f(x) = x4 − 4x3 + 10. − 0 . .. − . − 0 + . .. . .′ (x) f ↘ 0 . . ↘ . ↘ 3 ↗ . . . m .′′ onotonicity . + .. + 0 − . − .. . + . + 0+ + f . (x) . . ⌣ 0 . ⌢ 2 . . ⌣ . ⌣ c . oncavity 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . 3 s . hape I .P I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
- 93. Step 3: Grand Unified Sign Chart . Remember, f(x) = x4 − 4x3 + 10. − 0 . .. − . − 0 + . .. . .′ (x) f ↘ 0 . . ↘ . ↘ 3 ↗ . . . m .′′ onotonicity . + .. + 0 − . − .. . + . + 0+ + f . (x) . . ⌣ 0 . ⌢ 2 . . ⌣ . ⌣ c . oncavity 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
- 94. Step 4: Graph y . .(x) = x4 − 4x3 + 10 f . 0, 10) ( . . . x . . . 2, −6) ( . 3, −17) ( 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P . in m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
- 95. Step 4: Graph y . .(x) = x4 − 4x3 + 10 f . 0, 10) ( . . . x . . . 2, −6) ( . 3, −17) ( 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P . in m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
- 96. Step 4: Graph y . .(x) = x4 − 4x3 + 10 f . 0, 10) ( . . . x . . . 2, −6) ( . 3, −17) ( 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P . in m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
- 97. Step 4: Graph y . .(x) = x4 − 4x3 + 10 f . 0, 10) ( . . . x . . . 2, −6) ( . 3, −17) ( 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P . in m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
- 98. Step 4: Graph y . .(x) = x4 − 4x3 + 10 f . 0, 10) ( . . . x . . . 2, −6) ( . 3, −17) ( 1. .0 − −. . .6 . 17 f .(x) . .0 . 2 . . . . 3 s . hape I .P I .P . in m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
- 99. 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 21 / 45
- 100. Graphing a function with a cusp Example √ Graph f(x) = x + |x| . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 22 / 45
- 101. Graphing a function with a cusp Example √ Graph f(x) = x + |x| This function looks strange because of the absolute value. But whenever we become nervous, we can just take cases. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 22 / 45
- 102. Step 0: Finding Zeroes √ f(x) = x + |x| First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if x is positive. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
- 103. Step 0: Finding Zeroes √ f(x) = x + |x| First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if x is positive. Are there negative numbers which are zeroes for f? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
- 104. Step 0: Finding Zeroes √ f(x) = x + |x| First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if x is positive. Are there negative numbers which are zeroes for f? √ x + −x = 0 √ −x = −x −x = x2 x2 + x = 0 The only solutions are x = 0 and x = −1. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
- 105. Step 0: Asymptotic behavior √ f(x) = x + |x| lim f(x) = ∞, because both terms tend to ∞. x→∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
- 106. Step 0: Asymptotic behavior √ f(x) = x + |x| lim f(x) = ∞, because both terms tend to ∞. x→∞ lim f(x) is indeterminate of the form −∞ + ∞. It’s the same as x→−∞ √ lim (−y + y) y→+∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
- 107. Step 0: Asymptotic behavior √ f(x) = x + |x| lim f(x) = ∞, because both terms tend to ∞. x→∞ lim f(x) is indeterminate of the form −∞ + ∞. It’s the same as x→−∞ √ lim (−y + y) y→+∞ √ √ √ y+y lim (−y + y) = lim ( y − y) · √ y→+∞ y→∞ y+y y − y2 = lim √ = −∞ y→∞ y+y . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
- 108. Step 1: The derivative √ Remember, f(x) = x + |x|. To find f′ , first assume x > 0. Then d ( √ ) 1 f′ (x) = x+ x =1+ √ dx 2 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
- 109. Step 1: The derivative √ Remember, f(x) = x + |x|. To find f′ , first assume x > 0. Then d ( √ ) 1 f′ (x) = x+ x =1+ √ dx 2 x Notice f′ (x) > 0 when x > 0 (so no critical points here) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
- 110. Step 1: The derivative √ Remember, f(x) = x + |x|. To find f′ , first assume x > 0. Then d ( √ ) 1 f′ (x) = x+ x =1+ √ dx 2 x Notice f′ (x) > 0 when x > 0 (so no critical points here) lim f′ (x) = ∞ (so 0 is a critical point) x→0+ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
- 111. Step 1: The derivative √ Remember, f(x) = x + |x|. To find f′ , first assume x > 0. Then d ( √ ) 1 f′ (x) = x+ x =1+ √ dx 2 x Notice f′ (x) > 0 when x > 0 (so no critical points here) lim f′ (x) = ∞ (so 0 is a critical point) x→0+ lim f′ (x) = 1 (so the graph is asymptotic to a line of slope 1) x→∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
- 112. Step 1: The derivative √ Remember, f(x) = x + |x|. If x is negative, we have d ( √ ) 1 f′ (x) = x + −x = 1 − √ dx 2 −x Notice lim f′ (x) = −∞ (other side of the critical point) x→0− . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
- 113. Step 1: The derivative √ Remember, f(x) = x + |x|. If x is negative, we have d ( √ ) 1 f′ (x) = x + −x = 1 − √ dx 2 −x Notice lim f′ (x) = −∞ (other side of the critical point) x→0− lim f′ (x) = 1 (asymptotic to a line of slope 1) x→−∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
- 114. Step 1: The derivative √ Remember, f(x) = x + |x|. If x is negative, we have d ( √ ) 1 f′ (x) = x + −x = 1 − √ dx 2 −x Notice lim f′ (x) = −∞ (other side of the critical point) x→0− lim f′ (x) = 1 (asymptotic to a line of slope 1) x→−∞ ′ f (x) = 0 when 1 √ 1 1 1 1− √ = 0 =⇒ −x = =⇒ −x = =⇒ x = − 2 −x 2 4 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
- 115. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. .′ (x) f . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 116. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. 0 .. .′ (x) f . −4 . 1 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 117. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. 0 .. ∓. . ∞ .′ (x) f −4 . 1 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 118. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 .. ∓. . ∞ .′ (x) f −4 . 1 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 119. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ .′ (x) f −4 . 1 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 120. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f −4 . 1 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 121. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f ↗ . −4 . 1 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 122. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f ↗ . −4 ↘ 0 . 1. . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 123. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f ↗ . −4 ↘ 0 . 1. . ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 124. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f ↗ . −4 ↘ 0 . 1. . ↗ . f .(x) . max . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 125. Step 1: Monotonicity 1 1 + √ if x > 0 f′ (x) = 2 x 1 − √ 1 if x < 0 2 −x We can’t make a multi-factor sign chart because of the absolute value, but we can test points in between critical points. . + 0 −∓ . .. . . ∞ . + .′ (x) f ↗ . −4 ↘ 0 . .1 . . ↗ . f .(x) . max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
- 126. Step 2: Concavity If x > 0, then ( ) d 1 1 f′′ (x) = 1 + x−1/2 = − x−3/2 dx 2 4 This is negative whenever x > 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
- 127. Step 2: Concavity If x > 0, then ( ) d 1 1 f′′ (x) = 1 + x−1/2 = − x−3/2 dx 2 4 This is negative whenever x > 0. If x < 0, then ( ) ′′ d 1 −1/2 1 f (x) = 1 − (−x) = − (−x)−3/2 dx 2 4 which is also always negative for negative x. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
- 128. Step 2: Concavity If x > 0, then ( ) d 1 1 f′′ (x) = 1 + x−1/2 = − x−3/2 dx 2 4 This is negative whenever x > 0. If x < 0, then ( ) ′′ d 1 −1/2 1 f (x) = 1 − (−x) = − (−x)−3/2 dx 2 4 which is also always negative for negative x. 1 In other words, f′′ (x) = − |x|−3/2 . 4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
- 129. Step 2: Concavity If x > 0, then ( ) d 1 1 f′′ (x) = 1 + x−1/2 = − x−3/2 dx 2 4 This is negative whenever x > 0. If x < 0, then ( ) ′′ d 1 −1/2 1 f (x) = 1 − (−x) = − (−x)−3/2 dx 2 4 which is also always negative for negative x. 1 In other words, f′′ (x) = − |x|−3/2 . 4 Here is the sign chart: ′′ − . − −. . ∞ − . − . . (x) f . ⌢ . ⌢ . 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
- 130. Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| ′ . 1 + . + 0 −∓ . .. . . ∞ . + +. f . 1 (x) ↗ . ↗ . −1 ↘ 0 . 4. . ↗ . ↗m . . onotonicity − . ∞ − . − − . . . − ∞ − − . − − . f′′ . ∞ (x) . ⌢ . ⌢ . . ⌢ 0 . ⌢ . . oncavity ⌢c − . ∞ 0 .. .1 4 0 .. . ∞ + .(x) f . − . 1 −1 . .4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
- 131. Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| ′ . 1 + . + 0 −∓ . .. . . ∞ . + +. f . 1 (x) ↗ . ↗ . −1 ↘ 0 . 4. . ↗ . ↗m . . onotonicity − . ∞ − . − − . . . − ∞ − − . − − . f′′ . ∞ (x) . ⌢ . ⌢ . . ⌢ 0 . ⌢ . . oncavity ⌢c − . ∞ 0 .. .1 4 0 .. . ∞ + .(x) f . . . 1 − −1 . .4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
- 132. Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| ′ . 1 + . + 0 −∓ . .. . . ∞ . + +. f . 1 (x) ↗ . ↗ . −1 ↘ 0 . 4. . ↗ . ↗m . . onotonicity − . ∞ − . − − . . . − ∞ − − . − − . f′′ . ∞ (x) . ⌢ . ⌢ . . ⌢ 0 . ⌢ . . oncavity ⌢c − . ∞ 0 .. .1 4 0 .. . ∞ + .(x) f . . . 1 − . −1 . .4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
- 133. Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| ′ . 1 + . + 0 −∓ . .. . . ∞ . + +. f . 1 (x) ↗ . ↗ . −1 ↘ 0 . 4. . ↗ . ↗m . . onotonicity − . ∞ − . − − . . . − ∞ − − . − − . f′′ . ∞ (x) . ⌢ . ⌢ . . ⌢ 0 . ⌢ . . oncavity ⌢c − . ∞ 0 .. .1 4 0 .. . ∞ + .(x) f . . . 1 − . −1 . .4 . . 0 s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
- 134. Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| ′ . 1 + . + 0 −∓ . .. . . ∞ . + +. f . 1 (x) ↗ . ↗ . −1 ↘ 0 . 4. . ↗ . ↗m . . onotonicity − . ∞ − . − − . . . − ∞ − − . − − . f′′ . ∞ (x) . ⌢ . ⌢ . . ⌢ 0 . ⌢ . . oncavity ⌢c − . ∞ 0 .. .1 4 0 .. . ∞ + .(x) f . . . 1 − . −1 . .4 . . 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
- 135. Graph √ f(x) = x + |x| f .(x) . x . − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 136. Graph √ f(x) = x + |x| f .(x) . −1, 0) ( . . x . − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 137. Graph √ f(x) = x + |x| f .(x) .−1, 1) ( 4 4 . −1, 0) ( . . . x . − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 138. Graph √ f(x) = x + |x| f .(x) .−1, 1) ( 4 4 . −1, 0) ( . . . x . . 0, 0) ( − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 139. Graph √ f(x) = x + |x| f .(x) .−1, 1) ( 4 4 . −1, 0) ( . . . x . . 0, 0) ( − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 140. Graph √ f(x) = x + |x| f .(x) .−1, 1) ( 4 4 . −1, 0) ( . . . x . . 0, 0) ( − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 141. Graph √ f(x) = x + |x| f .(x) .−1, 1) ( 4 4 . −1, 0) ( . . . x . . 0, 0) ( − 0 . ∞ .. .1 4 0 .. . ∞ .(x) + f . . − . 1 . . .1 . . −4 0 . s . hape . . zero max min . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
- 142. Example with Horizontal Asymptotes Example Graph f(x) = xe−x 2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 31 / 45
- 143. Example with Horizontal Asymptotes Example Graph f(x) = xe−x 2 Before taking derivatives, we notice that f is odd, that f(0) = 0, and lim f(x) = 0 x→∞ . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 31 / 45
- 144. Step 1: Monotonicity If f(x) = xe−x , then 2 ( ) f′ (x) = 1 · e−x + xe−x (−2x) = 1 − 2x2 e−x 2 2 2 ( √ )( √ ) = 1 − 2x 1 + 2x e−x 2 The factor e−x is always positive so it doesn’t figure into the sign of 2 f′ (x). So our sign chart looks like this: . + .. + 0 . − . √ √. . − 1 2x . 1/2 − . 0 .. . + . + √ √ 1 . + 2x − . 1/2 − . 0 .. . + 0 . − . .′ (x) f √ √. ↘ . − 1/2 ↗ . ↘ . f .(x) . . . . 1/2 min max . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 32 / 45
- 145. Step 2: Concavity If f′ (x) = (1 − 2x2 )e−x , we know 2 ( ) f′′ (x) = (−4x)e−x + (1 − 2x2 )e−x (−2x) = 4x3 − 6x e−x 2 2 2 = 2x(2x2 − 3)e−x 2 − . − . 0 .. . + . + 2 .x 0 . − . − . − . 0 . . + √ √ √. . 2x − 3 . 3/2 − . 0 .. . + . + . + √ √ √ . 2x + 3 − . 3/2 − . − .. . + 0 + 0 .. − . − .. 0 . + + .′′ (x) f . ⌢ √ . ⌣ ⌢ √3 . . ⌣ − 3/2 . 0 . f .(x) . . . . /2 IP IP IP . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 33 / 45
- 146. Step 3: Synthesis f(x) = xe−x 2 − . − 0 + . .. . + . − − .′ (x) f √ . . √. . 0 . ↘ . ↘ ↗ . . 1/2 . − ↗ ↘ . . 1/2. ↘ . m . onotonicity − . − .. . + 0+ + 0 − . + .. . − − 0 . − .. . + + .′′ (x) f . ⌢ √. ⌣ ⌣ . . . . √3 . − . 3/2 0 ⌢ ⌢ . /2 ⌣ c . oncavity √ √ − . 3 2e3 − . √1 .√1 . 33 f .(x) . . 2e 0 .. . √2e 2e . √ . √ √ . . − . . . . − 1/2 . 3/2 . . . 0 . . 1/2 . 3/2 . . s . hape IP min IP max IP . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 34 / 45
- 147. Step 4: Graph f .(x) (√ )( . 1/2, √1 √ √ ) 2e . 3 .(x) = xe−x 2 3/2, f . 2e3 . . x . . 0, 0) ( ( . √ √ ) . . − 3/2, − 2e3 ( √ 3 ) . − 1/2, − √1 2e √ √ − 2e3 √1 . 3 − . .√1 . 2e33 f .(x) . √2e 0 . 2e . √ . . . . . √ . . √. . − 3 . 1/2 . . − . 0 s . hape . . . /2 . .1/2 . 3/2 IP min IP max IP . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 35 / 45
- 148. Example with Vertical Asymptotes Example 1 1 Graph f(x) = + 2 x x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 36 / 45
- 149. Step 0 Find when f is positive, negative, zero, not defined. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
- 150. Step 0 Find when f is positive, negative, zero, not defined. We need to factor f: 1 1 x+1 f(x) = + = . x x2 x2 This means f is 0 at −1 and has trouble at 0. In fact, x+1 lim = ∞, x→0 x2 so x = 0 is a vertical asymptote of the graph. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
- 151. Step 0 Find when f is positive, negative, zero, not defined. We need to factor f: 1 1 x+1 f(x) = + = . x x2 x2 This means f is 0 at −1 and has trouble at 0. In fact, x+1 lim = ∞, x→0 x2 so x = 0 is a vertical asymptote of the graph. We can make a sign chart as follows: − . 0 .. . . + x . +1 − . 1 . + 0 .. . + .2 x 0 . − . .. . 0 + ∞ .. . + f .(x) − . 1 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
- 152. Step 0, continued For horizontal asymptotes, notice that x+1 lim = 0, x→∞ x2 so y = 0 is a horizontal asymptote of the graph. The same is true at −∞. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 38 / 45
- 153. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . 0 .. . + .3 x 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 154. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . 0 .. . + .3 x 0 . − . 0 .. . + ∞ .. − . .′ (x) f − . 2 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 155. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . 0 .. . + .3 x 0 . − .. . 0 . + ∞ .. − . .′ (x) f ↘ . 2 . − 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 156. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . 0 .. . + .3 x 0 . − .. . 0 . + ∞ .. − . .′ (x) f ↘ . 2 . − ↗ . 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 157. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . .. . 0 + .3 x 0 . − .. . 0 . + ∞ − .. . .′ (x) f ↘ . 2 . − ↗ . 0 ↘ . . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 158. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . .. . 0 + .3 x 0 . − .. . 0 . + ∞ − .. . .′ (x) f ↘ . 2 . − ↗ . 0 ↘ . . f .(x) m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 159. Step 1: Monotonicity We have 1 2 x+2 −f′ (x) = − =− 3 . x2 x3 x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . .. . 0 + .3 x 0 . − .. . 0 . + ∞ − .. . .′ (x) f ↘ . 2 . − ↗ . 0 ↘ . . f .(x) m . in V .A . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
- 160. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + 0 .. . + .4 x 0 . 0 .. ∞ .. .′′ (x) f − . 3 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 161. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + 0 .. . + .4 x 0 . − . − .. 0 ∞ .. .′′ (x) f − . 3 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 162. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + 0 .. . + .4 x 0 . − . − .. 0 . + + ∞ .. .′′ (x) f − . 3 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 163. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f − . 3 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 164. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f . ⌢ . 3 − 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 165. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f . ⌢ . 3 − . ⌣ 0 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 166. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f . ⌢ . 3 − . ⌣ . . 0 ⌣ f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 167. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f . ⌢ . 3 − . ⌣ . . 0 ⌣ f .(x) I .P . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 168. Step 2: Concavity We have 2 6 2(x + 3) f′′ (x) = + = . x3 x4 x4 The critical points of f′ are −3 and 0. Sign chart: − . 0 .. . . + . x + 3) ( − . 3 . + .. . 0 + .4 x 0 . − . − .. 0 . + + ∞ + .. . + .′′ (x) f . ⌢ . 3 − . ⌣ . . 0 ⌣ f .(x) I .P V .A . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
- 169. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 170. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f H . A . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 171. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . H . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 172. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P H I . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 173. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P . H I . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 174. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P . . in H I m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 175. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P . . in . H I m . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 176. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P . . in . H I m 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 177. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − + . 1 . 0 . . + ∞s . . hape of f . A . .P . . in . H I m 0 . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 178. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − . 1 . + .0 . + ∞s . . hape of f . A . .P . . in . H I m 0 . .A . V . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 179. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − . 1 . + .0 . + ∞s . . hape of f . A . .P . . in . H I m 0 . .A . . V . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 180. Step 3: Synthesis . − .. . 0 . + ∞ − .. . .′ f ↘ . 2 . − ↗ . 0 ↘ . . m . onotonicity − . − .. 0 . + + ∞ + .. . + .′′ f . ⌢ . 3 − . ⌣ . . 0 ⌣ c . oncavity 0 . − . 2/9 − . 1/4 0 .. ∞ .. 0f .. . . − . ∞ . . 3 − − − . 2 − . 1 . + .0 . ∞s + . . hape of f . A . .P . . in . H I m 0 . .A . . A . V H . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
- 181. Step 4: Graph y . . x . . . . −3, −2/9) . −2, −1/4) ( ( . − − 0 . 2/9 . 1/4 .. 0 ∞ .. 0f .. . . − −− . ∞. . 3 . 2 . 1 . . . − − + 0 + ∞s . . hape of f . A . .P . . in . . . . A . H I m 0 V H . A . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 42 / 45
- 182. Problem Graph f(x) = cos x − x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 43 / 45
- 183. Problem Graph f(x) = cos x − x y . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 43 / 45
- 184. Problem Graph f(x) = x ln x2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 44 / 45
- 185. Problem Graph f(x) = x ln x2 y . . x . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 44 / 45
- 186. Summary Graphing is a procedure that gets easier with practice. Remember to follow the checklist. Graphing is like dissection—or is it vivisection? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 45 / 45