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Unformatted text preview: Application 3 The DERIVATIVE in Sketching Graphs of Functions/ Relations ANALYSIS OF FUNCTIONS I: INCREASE, DECREASE and CONCAVITY OBJECTIVES : • define increasing and decreasing functions; • define concavity and direction of bending that is concave upward or concave downward; and • determine the point of inflection. . 2 4 increasing decreasing increasing constant The term increasing, decreasing, and constant are used to describe the behavior of a function as we travel left to right along its graph. An example is shown below. INCREASING and DECREASING FUNCTIONS Definition 4.1.1 (p. 233) he following definition, which is illustrated in igure 4.1.2, expresses these intuitive ideas precisel Figure 4.1.2 (p. 233) y x • • Each tangent line has positive slope; function is increasing y x • • Each tangent line has negative slope; function is decreasing y x Each tangent line has zero slope, function is constant • • Theorem 4.1.2 (p. 233) . g sin decrea and g sin increa is 3 x 4 x f(x) which on intervals the Find . 1 2 + = y ( 29 ( 29 ( 29 ( ] ( 29 [ 29 + ∞ ⇒ ∞ ⇒ < < = = 2, on increasing is f 2 x when x ' f ,2 on decreasing is f 2 x when x ' f thus 2 x 2 4 x 2 x ' f1 2 3 x 3 x 4 x ) x ( f 2 + = increasing decreasing EXAMPL E : . g sin decrea and g sin increa is x f(x) which on intervals the Find . 2 3 = 3 x ) x ( f = ( 29 ( 29 ( ] ( 29 [ 29 + ∞ ⇒ ∞ ⇒ < = 0, on increasing is f x when x ' f ,0 on increasing is f x when x ' f thus x 3 x ' f 2 increasing increasing y4 3 4 x3 CONCAVI TY Although the sign of the derivative of f reveals where the graph of f is increasing or decreasing , it does not reveal the direction of the curvature . Fig. 4.1.8 suggests two ways to characterize the concavity of a differentiable f on an open interval : • f is concave up on an open interval if its tangent lines have increasing slopes on that interval and is concave down if they have decreasing slopes. • f is concave up on an open interval if its graph lies above its tangent lines and concave down if it lies below its tangent lines. • • • • • concave up y x • • • • • concave down x y increasing slopes decreasing slopes Figure 4.1.8 Formal definition of the “concave up” and “concave down ”. Definition 4.1.3 (p....
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 Summer '11
 Ma'amRosarioExconde
 Critical Point, Derivative, Differential Calculus, Stationary point

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