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# Lesson 21: Curve Sketching (handout)

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### Transcript of "Lesson 21: Curve Sketching (handout)"

1. 1. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Sec on 4.4 Curve Sketching V63.0121.001: Calculus I Professor Ma hew Leingang New York University April 13, 2011 . . Notes Announcements Quiz 4 on Sec ons 3.3, 3.4, 3.5, and 3.7 this week (April 14/15) Quiz 5 on Sec ons 4.1–4.4 April 28/29 Final Exam Thursday May 12, 2:00–3:50pm I am teaching Calc II MW 2:00pm and Calc III TR 2:00pm both Fall ’11 and Spring ’12 . . Notes Objectives given a func on, graph it completely, indica ng zeroes (if easy) asymptotes if applicable cri cal points local/global max/min inﬂec on points . . . 1.
2. 2. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Why? Graphing func ons is like dissec on … or diagramming sentences You can really know a lot about a func on when you know all of its anatomy. . . Notes 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 f(x) f′ (x) f(x) = x3 + x2 f′ (x) = 3x2 + 2x . . . Notes 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 f′′ (x) f′ (x) f(x) f(x) = x3 + x2 f′ (x) = 3x2 + 2x . f′′ (x) = 6x + 2 . . . 2.
3. 3. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Graphing Checklist To graph a func on f, follow this plan: 0. Find when f is posi ve, nega ve, zero, not deﬁned. 1. Find f′ and form its sign chart. Conclude informa on about increasing/decreasing and local max/min. 2. Find f′′ and form its sign chart. Conclude concave up/concave down and inﬂec on. . . Notes Graphing Checklist To graph a func on f, follow this plan: 3. Put together a big chart to assemble monotonicity and concavity data 4. Graph! . . Notes Outline Simple examples A cubic func on A quar c func on More Examples Points of nondiﬀeren ability Horizontal asymptotes Ver cal asymptotes Trigonometric and polynomial together Logarithmic . . . 3.
4. 4. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes 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 quadra c, 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. . Notes 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: − .− + x−2 2 − + + x+1 −1 f′ (x) + − + ↗−1 ↘ 2 ↗ f(x) max min . . Notes Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . −− ++ f′′ (x) ⌢ 1/2 ⌣ f(x) IP . . . 4.
5. 5. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. + −. − + f′ (x) ↗−1 ↘ ↘ 2 ↗ monotonicity −− −− ++ ++ f′′ (x) ⌢ ⌢ 1/2 ⌣ ⌣ concavity 7 −61/2 −20 f(x) −1 1/2 2 shape of f max IP min . . Notes monotonicity and concavity increasing, decreasing, concave concave down down II I . III IV decreasing, increasing, concave concave up up . . Notes Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. + −. − + f′ (x) ↗−1 ↘ ↘ 2 ↗ monotonicity −− −− ++ ++ f′′ (x) ⌢ ⌢ 1/2 ⌣ ⌣ concavity 7 −61/2 −20 f(x) −1 1/2 2 shape of f max IP min . . . 5.
6. 6. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 f(x) Notes Step 4: Graph f(x) = 2x3 − 3x2 − 12x ( √ ) (−1, 7) 3− 105 4 ,0 (0, 0) . ( x√ ) (1/2, −61/2) 3+ 105 4 ,0 (2, −20) 7 −61/2 −20 f(x) −1 1/2 2 shape of f max IP min . . Notes Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 (Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many x→±∞ other points on the graph are evident. . . Notes Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. +0 . + + 4x2 0 − − 0+ (x − 3) 3 ′ −0 − 0 + f (x) ↘0 ↘ 3 ↗ f(x) min . . . 6.
7. 7. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: −0. + + 12x 0 − 0 +− x−2 2 f′′ (x) ++0 −− 0 ++ ⌣0 ⌢ 2 ⌣ f(x) IP IP . . Notes Step 3: Grand Uniﬁed Sign Chart . Remember, f(x) = x4 − 4x3 + 10. −0 − −0+ f′ (x) ↘0 ↘ ↘3↗ monotonicity f′′ (x) ++0 −− 0++ ++ ⌣0 ⌢ 2⌣ ⌣ concavity 10 −6 −17 f(x) 0 2 3 shape IP IP min . . y Notes Step 4: Graph f(x) = x4 − 4x3 + 10 (0, 10) . x (2, −6) (3, −17) 10 −6 −17 f(x) 0 2 3 shape IP IP min . . . 7.
8. 8. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Outline Simple examples A cubic func on A quar c func on More Examples Points of nondiﬀeren ability Horizontal asymptotes Ver cal asymptotes Trigonometric and polynomial together Logarithmic . . Notes Graphing a function with a cusp Example √ Graph f(x) = x + |x| This func on looks strange because of the absolute value. But whenever we become nervous, we can just take cases. . . Notes 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 posi ve. Are there nega ve numbers which are zeroes for f? √ √ x + −x = 0 =⇒ −x = −x −x = x2 =⇒ x2 + x = 0 The only solu ons are x = 0 and x = −1. . . . 8.
9. 9. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes 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 x→−∞ √ as lim (−y + y) y→+∞ √ √ √ y+y lim (−y + y) = lim ( y − y) · √ y→+∞ y→∞ y+y y − y2 = lim √ = −∞ y→∞ y+y . . Notes 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 No ce f′ (x) > 0 when x > 0 (so no cri cal points here) lim+ f′ (x) = ∞ (so 0 is a cri cal point) x→0 lim f′ (x) = 1 (so the graph is asympto c to a line of slope 1) x→∞ . . Notes Step 1: The derivative √ Remember, f(x) = x + |x|. If x is nega ve, we have d ( √ ) 1 f′ (x) = x + −x = 1 − √ dx 2 −x No ce lim− f′ (x) = −∞ (other side of the cri cal point) x→0 lim f′ (x) = 1 (asympto c 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 . . . 9.
10. 10. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 1: Monotonicity  1 1 + √  if x > 0 ′ f (x) = 2 x 1 − √ 1  if x < 0 2 −x We can’t make a mul -factor sign chart because of the absolute value, but we can test points in between cri cal points. + 0− ∞ + f′ (x) . ↗ − 1↘ 0 4 ↗ f(x) max min . . Notes Step 2: Concavity ( ) d 1 1 If x > 0, then f′′ (x) = 1 + x−1/2 = − x−3/2 This is dx 2 4 nega ve whenever x > 0. ( ) d 1 1 If x < 0, then f′′ (x) = 1 − (−x)−1/2 = − (−x)−3/2 dx 2 4 which is also always nega ve for nega ve x. 1 In other words, f′′ (x) = − |x|−3/2 . 4 Here is the sign chart: −− −∞ −− f′′ (x) . ⌢ 0 ⌢ f(x) . . Notes Step 3: Synthesis Now we can put these things together. √ f(x) = x + |x| +1 + 0− ∞ + f′ +1 (x) . ↗ ↗ − 1↘ 0 ↗ ↗monotonicity f′′ −∞ −− −− 4 −∞ −− −∞ (x) ⌢ ⌢ 1 ⌢0 ⌢ ⌢concavity −∞ 0 4 0 +∞f(x) −1 −4 0 1 shape zero max min . . . 10.
11. 11. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Graph √ f(x) = x + |x| (− 1 , 1 ) 4 4 (−1, 0) . x (0, 0) 1 −∞ 0 4 0 +∞ x −1 −1 0 shape 4 zero max min . . Example with Horizontal Notes Asymptotes Example Graph f(x) = xe−x 2 . . Notes Step 1: Monotonicity . . . 11.
12. 12. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 2: Concavity . . Notes Step 3: Synthesis . . Notes Step 4: Graph f(x) x . . . . 12.
13. 13. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Example with Vertical Asymptotes Example 1 1 Graph f(x) = + x x2 . . Notes Step 0 . . Notes Step 1: Monotonicity . . . 13.
14. 14. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 2: Concavity . . Notes Step 3: Synthesis . . Notes Step 4: Graph y . x . . . 14.
15. 15. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Trigonometric and polynomial Notes together Problem Graph f(x) = cos x − x . . Notes Step 0: intercepts and asymptotes . . Notes Step 1: Monotonicity . . . 15.
16. 16. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 2: Concavity . . Notes Step 3: Synthesis . . Notes Step 4: Graph f(x) = cos x − x y . x . . . 16.
17. 17. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Logarithmic Problem Graph f(x) = x ln x2 . . Step 0: Intercepts and Notes Asymptotes . . Notes Step 1: Monotonicity . . . 17.
18. 18. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Step 2: Concavity . . Notes Step 3: Synthesis . . Notes Step 4: Graph y . x . . . 18.
19. 19. . V63.0121.001: Calculus I . Sec on 4.4: Curve Sketching . April 13, 2011 Notes Summary Graphing is a procedure that gets easier with prac ce. Remember to follow the checklist. . . Notes . . Notes . . . 19.