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3.2 The Second-Derivative Test. We continue developing a technique for sketching curves that uses the derivatives. What does the sign of the second derivative tell us about the graph?
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3.2 The Second-Derivative Test We continue developing a technique for sketching curves that uses the derivatives What does the sign of the second derivative tell us about the graph? Suppose the second derivative is positive. But, is the first derivative of the function . By examining the first derivative, the function is _________. On the other hand, is the _________ of the function . Thus, if the second derivative is positive, then the slope of the function is increasing. 1
If the slope increases, the curve remains above the tangent line in the • vicinity of the point and is called to be concave up at the point. • Definition: • A point at which the concavity changes is called an inflection point. • Concavity Theorem: For all x in the interval [a,b], the graph of a function f is • Concave up if • Concave down if Concave up = Increasing slope Concave down = Decreasing slope 2
Note: If is an inflection point, then , or does not exist. Converse is not (always) true: If at some point (x, f(x)), then this point is not necessarily an inflection point. Example: Identify intervals of concavity and inflection points. Solution: Even though the second derivative takes 0 value at x=0, this is not an inflection point since the second derivative does not change the sine, that is, the function is concave up everywhere. 3
Second-Derivative Test: • Suppose at some point x=c. • If , then f(c) is a minimum value; • If , then f(c) is a maximum value; • If , then the test fails (use the first-derivative test to identify if this is a minimum or maximum). • Summarized Procedure for Curve Sketching: • Find critical values and critical points; • Test the critical values (if they are minima or maxima): • use the second derivative test; • if the latter fails, use the first-derivative test; • Use the second derivative to find the intervals where the graph is concave up/down. • Determine the points of inflection; • Find asymptotes, easily determined intercepts, symmetry; • Plot all the critical and inflection points, asymptotes, intercepts and a few additional points, and sketch the curve. 4
Exercises: For each of the functions, find the extrema, the intervals where it concave up and concave down, inflection points, asymptotes, and then sketch the curve. 5
Homework Section 3.2: 3,7,9,13,15,23,27,29,31,37,39,41. 6
