3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

3.4 Concavity & The Second Derivative Learning Targets 1. Determine intervals on which a function is concave upward or concave downward 2. Find inflection points of a function. 3. Find relative extrema of a function using Second Derivative Test. *no make‐up Monday today

Intro/Warm­up 1 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

Nov 7­8:41 AM 2 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

∫ Concave Upward or Concave Downward If f' is increasing on an interval, then the function is concave upward on that interval If f' is decreasing on an interval, then the function is concave downward on that interval.

note: concave upward means the graph lies above its tangent lines, and concave downwards means the graph lies below its tangent lines.

I.concave upward or concave downward 3 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

∫ Concave Upward or Concave Downward continued Test For Concavity: Let f be a function whose second derivative exists on an open interval I. 1. If f''(x) > 0 for all x in I, then the graph of f is concave upward on I. 2. If f''(x) < 0 for all x in I, then the graph of f is concave downward on I. 1. Determine the open intervals on which the graph of is concave upward or downward.

I.Concave Upward or Downward cont 4 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

∫∫∫ Second Derivative Test Second Derivative Test Let f be a function such that f'(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f''(c) > 0, then f has a relative minimum at (c, f(c)). 2. If f''(c) < 0, then f has a relative maximum at (c, f(c)). If f''(c) = 0, the test fails ‐ look at first derivative test instead. 4. Find the relative extrema for

III.Second Derivative Test 5 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

∫∫ Inflection Points of a Function Definition of Point of Inflection Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point.

Note!! 2. Graph {

II.inflection points of a function 6 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

∫∫∫ Relative Extrema of a Function one way theorem... Points of Inflection If (c, f(c)) is a point of inflection of the graph of f, then either f''(c) = 0 or f'' does not exist at x = c. 3. Determine the points of inflection for

III.relative extrema of a function 7 3.4 AP Calculus Concavity and the second derivative.notebook November 07, 2016

Your Homework AB Calculus 3.4 p.195 #1‐5,6‐54(x6),57‐60,63‐66,70,76,79,80,94

BC Calculus 10.3 p.727 #3‐6,15,26,27,36,41,47,52,60,61,62,72, 73,86,87,91,93,95,103,104

Homework 8