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03/16/2016 Math 131: Essential of Calculus – Unit 15 Applications of the 2nd Derivative – W 3/16/2015 1. Review: Finding intervals of increase/decreases and relative extrema 2. The meaning of (the sign of) the 2nd Derivative • f ′′ ( x ) > 0 implies tangent line slopes are increasing which implies curve is concave up • f ′′ ( x ) < 0 implies tangent line slopes are decreasing which implies curve is concave down 3. Finding Intervals of Concavity using 2nd Derivative 4. 2nd Derivative Test for Relative Extrema 5. Inflection Points – where concavity changes Review Problem Find the intervals of increase and decrease for this function. Express in interval notation. x y= 2 x +1 a. What do you differentiate? Why? b. What are critical numbers? How do you find them? c. Why do you plot the critical numbers on a number line? d. How and why do you test values of derivative on a Sign Line? Now find and identify the rel. maximums and rel. minimums a. What are your candidates for relative maximums (a cap) and relative minimums (a cup)? b. How do you determine whether you have a cap, a cup or a twist? The 2nd Derivative Again consider the function f ( x ) = Look at the tangent lines – what do you see? x x +1 2 What’s happening here? f e ? d a b Calculate ? 2nd Derivatives & Concavity If f ′′ ( x ) > 0 then f ′ ( x ) is increasing so the curve is concave up. If f ′′ ( x ) < 0 then f ′ ( x ) is decreasing so the curve is concave down c d2y What do you notice? dx 2 Finding Intervals of Concavity If f ′′ ( x ) is continuous (a continuous 2nd derivative!) by the IVT it cannot change sign without first becoming 0 (or undefined)! 1. Take the 2nd Derivative 2. Find all 2nd derivative critical numbers: f ′′ ( c ) = 0 or f ′′ ( c ) DNE 3. Sketch a 2nd derivative sign line & determine intervals of concavity 4. Express intervals of concavity in interval notation 5. Give x-y coordinates of any inflection points (if any) Note: This is the same method used with the 1st derivative to find intervals of increase/decrease 1 03/16/2016 2nd Derivative Test for Relative Extrema Exercise Find intervals of increase/decrease and intervals of concavity for x2 3 2 1. f ( x ) = x + 3 x + 1 2. f ( x ) = 2 x +3 a. Find (and ) b. Identify all 1st (and 2nd) derivative critical numbers c. Sketch the sign line and determine all intervals of increase/decrease (concave up concave down) d. Express intervals of increase/decrease (concavity) in interval notation e. Give x-y coordinates of inflection points (if any) Let (c, f(c)) be a critical point (i.e. f’(c) = 0 or f’(c) DNE). Then … If f”(c) > 0 is concave up then (c, f(c)) is a relative minimum (“cup”) If f”(c) < 0 is concave down then (c, f(c)) is a relative maximum (“cap”) Otherwise the 2nd Derivative test fails (“twist”) Counter-example: y = x3 has a critical point at x = 0 Question: To test for a relative min/max which “test” should you use? 1st Derivative OR 2nd Derivative Test? Inflection Points An inflection point for a curve f(x) occurs where the concavity changes Continuing with the sign line created, find intervals of “concave up” and “concave down” and identify points where concavity changes! Go back to a previous exercise and locate all inflection points! 1 Question! x = 0 is a critical number for f ( x ) = x . Why? Is x = 0 an inflection point too? Question! x = 0 is also a critical number for y = x4. Is it an inflection point? Max/Min Examples Use the 2nd Derivative Test to identify the relative maximums and/or minimums for the functions 1. f ( x ) = x4 − 2 x2 + 3 2. f ( x) = x + 1 x For Friday – Onto Unit 16 2