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125 3.3

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125 3.3

  1. 1. 3.3 Graphing Using the First and Second Derivatives • What does concave up or concave down look like? • What is an inflection point? • What does the second derivative tell us about the original function f? • What’s “The Second Derivative Test”?
  2. 2. A. What does concave up or concave down look like?• If I gave you a riddle that sounded like, “The rate of increase is increasing,” or “The slope of the tangent is increasing,” could you draw a picture of that?• That is CONCAVE UP!
  3. 3. • If I gave you a riddle that sounded like, “The rate of increase is decreasing,” or “The slope of the tangent is decreasing,” could you draw a picture of that?• That is CONCAVE DOWN!
  4. 4. B. What is an inflection point?• An inflection point is where the concavity changes (from up to down, or down to up).
  5. 5. C. What does the second derivativetell us about the original function f?• Recall that the first derivative gave us the increase/decrease for f, essentially the rate of change for f.• Well, the second derivative will give us the rate of change for the rate of change for f. In other words, the rate of increase of the rate of increase of f.• Hey! That is concavity!
  6. 6. • So, remember when you found values for which the first derivative equals zero or is undefined, and you found critical points?• When you take the second derivative and find values for which it is zero or undefined, you will be finding inflection points!• (That sounds like the beginning of a “sign diagram”? Right, you are!)
  7. 7. A second derivative sign diagram….• Find the second derivative.• Set its numerator and denominator equal to zero, factor, and solve to find inflection points.• Chop up the number line into pieces at the inflection points, and choose a test value for each interval.• Plug in each test value into the second derivative, only caring if it will be positive, negative, or zero.• Positive second derivative means f is concave up.• Negative second derivative means f is concave down.• A second derivative that is ZERO means that f is STRAIGHT (no concavity).
  8. 8. I might be asked to graph something…• What would it look like if f’ > 0 and f’’ > 0?• What about f’ > 0 and f” < 0?• What about f’ < 0 and f” < 0?• What about f’ < 0 and f” > 0?
  9. 9. Graph f ( x ) = x 3 − 9 x 2 + 24 x showing all relative extreme (max/min) pointsand all inflection points. (Let s do a sign diagram for f first, and then one for f .)
  10. 10. Besides the graph, I could have asked you for:• Critical numbers: 2, 4• Intervals of increase of f: (-inf, 2),(4, inf)• Intervals of decrease of f: (2, 4)• Relative min: (4, 16)• Relative max: (2, 20)• Inflection pts: (3, 18)• Intervals where f is concave up: (3, inf)• Intervals where f is concave down: (-inf, 3)
  11. 11. Graph f ( x ) = 18 x1/ 3 showing all relative extrema and inflection points.
  12. 12. Besides the graph, I could have asked you for:• Critical numbers: 0• Intervals of increase of f: (-inf, inf)• Intervals of decrease of f: none• Relative min: none• Relative max: none• Inflection pts: (0, 0)• Intervals where f is concave up: (-inf, 0)• Intervals where f is concave down: (0, inf)
  13. 13. You try : f ( x ) = x + 3 x − 9 x + 5 Graph, show all 3 2relative extrema and inflection points.
  14. 14. You try : f ( x ) = 9 x Graph, show all 4/3relative extrema and inflection points.
  15. 15. D. What’s “the second derivative test”?• You may think it silly that we use the second derivative to do all this stuff and THEN we have some rule called “THE second derivative test.”• (Consider you have a nice, smooth, normal, non- pointy, non-asymptote-ey curve,)• Imagine a relative maximum. What concavity does it have there? CONCAVE DOWN!• What about a relative minimum? CONCAVE UP!
  16. 16. THE second derivative test:If x = c is a critical number of f at which f ′′ is defined,(saying f ′′ is restricting this rule to only those nice,non - cuspy, non - asymptote - ey places), thenf ′′( c ) > 0 means that f has a relative minimum at x = c.f ′′( c ) < 0 means that f has a relative maximum at x = c.
  17. 17. Using the second derivative test:• Take the first derivative and find critical numbers.• Plug the critical numbers into the second derivative.• If the second derivative is negative, there is a rel. maximum at that x-value. (think concave down)• If the second derivative is positive, there is a rel minimum at that x-value. (think concave up)• If you get zero, then we don’t know what it is! It could be a max or a min or an inflection point!• Plug the critical numbers back into the original f function to find their partnering y-values.
  18. 18. Use the second - derivative test to find all relativeextrema points of f ( x ) = x 3 − 9 x 2 + 24 x(Polynomials will always have defined second derivatives.)
  19. 19. You try : Use the second - derivative test to find all relativeextrema points of f ( x ) = x 3 − 3 x 2 + 3 x + 4.

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