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lecture20 - Lecture 20 1 Lecture 20(Sec 4.2 Part I...

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Lecture 20 1 Lecture 20: (Sec. 4.2, Part I) Applications of the Second Derivative: Con- cavity ex. Total sales S (in hundreds of dollars) of a prod- uct are related to the amount of money x spent on ad- vertising according to the function S ( x ) = 0 . 002 x 3 + 0 . 6 x 2 + x + 500 where x is measured in hundreds of dollars and 0 x 200. Consider the graph of S ( x ) and its tangent lines:
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Lecture 20 2 Def. Let function f be differentiable on an open interval ( a,b ). The graph of f is 1) concave upward (concave up) on ( a,b ) if 2) concave downward (concave down) on ( a,b ) if NOTE: f is concave upward (downward) at a point x = c if
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Lecture 20 3 Also note the following: 1) f is concave upward if 2) f is concave downward if
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Lecture 20 4 ex. Let f ( x ) = x 3 1) Sketch the graph of f ( x ). 2) Sketch the graph of f ′′ ( x ).
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Lecture 20 5 Theorem Let f be a function whose second derivative exists on interval ( a,b ). 1) If f ′′ ( x ) > 0 for each value of x in ( a,b ), then 2) If f ′′ ( x ) < 0 for each value of x in ( a,b ), then To find the intervals on which f is concave
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