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Unformatted text preview: 18. Concavity and Graphing P. K. Lamm 10/27/11 (11:21) p. 1 / 8 Lecture Notes: 18. Concavity and Graphing These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. The Meaning of the Second Derivative in Graphing Recall: f > ⇒ f increasing , f < ⇒ f decreasing . Then it follows that f 00 > ⇒ ( f ) > ⇒ f increasing , f 00 < ⇒ ( f ) < ⇒ f decreasing . So what is the meaning of the sign of f 00 ? • For f 00 > 0 (i.e., for f increasing): – if f > 0, then the increasing function f has increasing slope, meaning that it increases more and more as x increases. The curve y = f ( x ) is bending upward as it increases. – if f < 0, then the decreasing function f has increasing slope, meaning that it decreases less and less as x increases. The curve y = f ( x ) is bending upward as it decreases. We say that f is concave up in both of the above circumstances. • For f 00 < 0 (i.e., for f decreasing): – if f > 0, then the increasing function f has decreasing slope, meaning that it increases less and less as x increases. Thus the curve y = f ( x ) is bending downward as it increases. – if f < 0, then the decreasing function f has decreasing slope, meaning that it decreases more and more as x increases. Thus the curve y = f ( x ) is bending downward as it decreases. We say that f is concave down in both of the above circumstances. Definition: We say the function f is concave up on some interval surrounding the point c if on that interval the graph of y = f ( x ) lies above the tangent line to the curve at c . We say the graph of f is concave down on some interval surrounding the point c if on that interval the graph of y = f ( x ) lies below the tangent line to the curve at c . 1 18. Concavity and Graphing P. K. Lamm 10/27/11 (11:21) p. 2 / 8 Theorem: Suppose f is continuous on I and twice differentiable on the interior of I . Then: (1) If f 00 ( x ) > 0 for all x ∈ I , the graph of f is concave up on I . (2) If f 00 ( x ) < 0 for all x ∈ I , the graph of f is concave down on I Example 1.1: Determine where f ( x ) = x 3 is concave up or down. For f ( x ) = x 3 we have f ( x ) = 3 x 2 and f 00 ( x ) = 6 x = ( , at x = 0 undef , at no values of x We’ve found only one place ( x = 0) where f 00 ( x ) can be zero. Because f 00 is continuous, this means that the only place where f 00 can change sign is at...
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 Fall '10
 KIHYUNHYUN
 Calculus, Derivative, Mathematical analysis, Convex function, P. K. Lamm

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