Lesson 18: Graphing

Slide 1: Section 4.3 Graphing functions Math 1a Introduction to Calculus March 17, 2008 Announcements ◮ Thank you for the evaluations! ◮ Problem Sessions Sunday, Thursday, 7pm, SC 310 ◮ Office hours Tues, Weds, 2–4pm SC 323 . . Image: Flickr user Cobalt123 . . . . . .

Slide 2: Announcements ◮ Thank you for the evaluations! ◮ Problem Sessions Sunday, Thursday, 7pm, SC 310 ◮ Office hours Tues, Weds, 2–4pm SC 323 . . . . . .

Slide 3: Outline The checklist Big example Your turn . . . . . .

Slide 4: 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 4. Graph! . . . . . .

Slide 5: Outline The checklist Big example Your turn . . . . . .

Slide 6: Big example 1 1 Let f(x) = + 2 . We will do a complete dissection of f. x x . . . . . .

Slide 7: Step 0 Find when f is positive, negative, zero, not defined. . . . . . .

Slide 8: Step 0 Find when f is positive, negative, zero, not defined. We need to factor f: 1 1 x+1 f(x) = + 2 = . x x 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. . . . . . .

Slide 9: Step 0 Find when f is positive, negative, zero, not defined. We need to factor f: 1 1 x+1 f(x) = + 2 = . x x 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 . . . . . .

Slide 10: 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 −∞. . . . . . .

Slide 11: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. . . . . . .

Slide 12: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x x The critical points are x = −2 and x = 0. We have the following sign chart: . + 0 .. . − . − . (x + 2) − . 2 − . 0 .. . + .3 x 0 . . . . . . .

Slide 13: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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 .(x) f . . . . . .

Slide 14: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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) . . . . . .

Slide 15: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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) . . . . . .

Slide 16: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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 ↘ . . .(x) f . . . . . .

Slide 17: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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 ↘ . . .(x) f . in m . . . . . .

Slide 18: Step 1 Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. We have 1 2 x+2 f ′ (x ) = − 2 − 3 = − 3 . x x 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) . in m .A V . . . . . .

Slide 19: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. . . . . . .

Slide 20: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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) . . . . . .

Slide 21: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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) . . . . . .

Slide 22: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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 .(x) f . . . . . .

Slide 23: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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 .(x) f . . . . . .

Slide 24: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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 .(x) f . . . . . .

Slide 25: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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 .(x) f . . . . . .

Slide 26: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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 ⌣ .(x) f . . . . . .

Slide 27: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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) .P I . . . . . .

Slide 28: Step 2 Find f′′ and form its sign chart. Conclude concave up/concave down and inflection points. We have 2 6 2 (x + 3 ) f′′ (x) = 3 + 4 = . x x 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) .P I .A V . . . . . .

Slide 29: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 . ign/value s . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . . − .. − 0 . + + ∞ − .. . − . . . oncavity c ⌢ . 3 − ⌣ . . 0 ⌣ . . . . . .

Slide 30: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . . − .. − 0 . + + ∞ − .. . − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ H . A . . . . . .

Slide 31: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . . − .. − 0 . + + ∞ − .. . − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ . A .✟ H . . . . . .

Slide 32: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . − . − .. 0 . + + .. . − ∞ − . . . oncavity c ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P H I . . . . . .

Slide 33: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . − . − .. 0 . + + .. . − ∞ − . . . oncavity c ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P . ✡ H I . . . . . .

Slide 34: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 .. 0 ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . − . − .. 0 . + + ∞ − .. . − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P . ✡ . in H I m . . . . . .

Slide 35: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 .. 0 ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . − . − .. 0 . + + ∞ − .. . − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P . ✡ . in . ✠ H I m . . . . . .

Slide 36: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 .. 0 ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . . − .. − 0 . + + ∞ − .. . − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P . ✡ . in . ✠ . H I m 0 . . . . . .

Slide 37: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 s . ign/value . ∞ . − − . 1 . − + . 0 + . . ∞ − .. . 0 . + ∞ − .. . . onotonicity m ↘ . 2 . − ↗ . 0 ↘ . . . − .. − 0 . + + .. . − ∞ − . . c . oncavity ⌢ . 3 − ⌣ . . 0 ⌣ . A . ✟ .P . ✡ . in . ✠ . H I m 0 . ✠ . . . . . .

Slide 38: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 .. 0 ∞ .. 0 . s . ign/value . ∞ . − − . 1 . − + 0 . + . . ∞ − .. . 0 . + ∞ .. − . . onotonicity m ↘ . 2 . − ↗ . 0 . ↘ . − . − .. 0 . + + ∞ .. . − − . ⌢ . 3 . ⌣ . ⌣ . oncavity c − . 0 . A . ✟ .P . ✡ . in . ✠ . H I m 0 . ✠.A V . . . . . .

Slide 39: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 .. 0 ∞ .. 0 . s . ign/value . ∞ . − − . 1 . − + 0 . + . . ∞ − .. . 0 . + ∞ .. − . . onotonicity m ↘ . 2 . − ↗ . . 0 ↘ . − . − .. 0 . + + ∞ .. − . − . ⌢ . 3 . ⌣ . ⌣ . oncavity c − . 0 . A . ✟ .P . ✡ . in . ✠ . H I m 0 . ✠.A V .✡ . . . . . .

Slide 40: Step 3 Put together a big chart! . 0 − . 2/9 − . 1/4 0 .. ∞ .. . 0 . ign/value s − . ∞ . − . 1 . − + 0 . + . . ∞ − .. . 0 . + ∞ .. − . m . onotonicity ↘ . 2 . − ↗ . 0 . ↘ . . − .. − 0 . + + ∞ .. . − − . ⌢ . 3 . ⌣ . ⌣ c . oncavity − . 0 . A . ✟ .P . ✡ . in . ✠ . H I m 0 . ✠V .A . ✡. A H . . . . . .

Slide 41: Step 4 -4 -3 -2 -1 1 2 2 1 1,0 3, 2, 9 4 . . . . . .

Slide 42: Outline The checklist Big example Your turn . . . . . .

Slide 43: Your turn Plot these functions (Group work): 1. f(x) = x4 − 4x3 + 10 3 2. f(x) = (x2 − 1)2/3 4 x3 3. f(x) = 2 3x + 1 4. f(x) = (2 − x2 )3/2 . . . . . . .

Slide 44: Graph of f(x) = x4 − 4x3 + 10 80 60 40 20 2 1 1 2 3 4 5 . . . . . .

Slide 45: 3 2 Graph of f(x) = (x − 1)2/3 4 1.5 1.0 0.5 2 1 1 2 . . . . . .

Slide 46: x3 Graph of f(x) = 2 3x + 1 0.6 0.4 0.2 2 1 1 2 0.2 0.4 0.6 . . . . . .

Slide 47: Graph of f(x) = (2 − x2 )3/2 2.5 2.0 1.5 1.0 0.5 1.0 0.5 0.5 1.0 . . . . . .