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when f’(x) = 0 or f’(x) is undefined

when f’(x) = 0 or f’(x) is undefined. x = -1 (horizontal tangent). upward U shape . x = 0 (cusp). upward U shape . Increasing: x < -1 OR -1 < x < 0. downward U shape . Decreasing: x > 0. downward U shape .

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when f’(x) = 0 or f’(x) is undefined

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  1. when f’(x) = 0 or f’(x) is undefined x = -1 (horizontal tangent) upward U shape x = 0 (cusp) upward U shape Increasing: x < -1 OR -1 < x < 0 downward U shape Decreasing: x > 0 downward U shape f’’(x) = 0 when x = -1 or x = 1.5 (the graph switches from upward to downward U shape) f’’(x) > 0 when -1 < x < 0 or 0 < x < 1.5 (the graph is upward U shape) f’’(x) < 0 when x < -1 or x > 1.5 (the graph is upward U shape) Work on questions for 1-2 minutes

  2. 4.4 Concavity • Identify properties of a graph including intervals of increase and decrease, turning points, critical points, points of inflection, concavity and asymptotes. • Identify the points of inflection and concavity of a function. • Use the critical points, intervals of increase and decrease and asymptotes, points of inflection and concavity to graph a function. Please Read

  3. Identify properties of a graph including intervals of increase and decrease, turning points, critical points, points of inflection, concavity and asymptotes. Describe the concavity and intervals of increase/decrease of the function

  4. concave down concave up f “ (x) > 0 Concave up an upward U shape. -1 < x < 0 and 0 < x < 1.5 concave up f “ (x) < 0 Concave down downward shape. x < -1 and 1.5 < x concave down f “ (x) = 0 momentarily linear. (Has linear r.o.c for an instant) point of inflection. points of inflection (-1 , 1) and (1.5, -1.8) Please Copy

  5. Identify the points of inflection and concavity of a function. • Use the critical points, intervals of increase and decrease and asymptotes, points of inflection and concavity to graph a function.. Example 11: Curve sketching with concavity Completely analyze the function. Use the information to sketch the function. Solution is on the next slides On your own: 1) determine intercepts, asymptotes and domain 2) find f’(x) and use it to find critical #’s, intervals of increase and decrease, coordinates of minimums, maximums and horizontal tangents. 3) find f’’(x) and use to find concavity and points of inflection

  6. Completely analyze the function. Use the information to sketch. Analyze the function to find intercepts, domain and asymptotes y – intercept: f(0) = 0 x – intercepts: the x-intercepts are asymptotes: none domain: Analysis of the derivatives is on next slide.

  7. Completely analyze the function. Use the information to sketch. Analyze the derivative to find critical numbers, intervals of inc/dec and the coorindates of minimums, maximums and horizontal tangents critical numbers: interval chart: intervals of increasing for: inc and dec: decreasing for: , min/max/hor tan: local min (switches dec to inc)local max: (switches dec to inc)locak min: (switches dec to inc) continued

  8. Completely analyze the function. Use the information to sketch. Analyze the 2nd derivative to find concavity and points of inflection Possible pts of: inflection interval chart: intervals of concave up for: inc and dec: concave down for: POI: continued

  9. Completely analyze the function. Use the information to sketch. Graph (0,0) (2,0) (-2,0) (,-4) (,-4) you don’t have to put the coordinates

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