lecture18 - f ( x ) on each of the inter-vals, and use the...

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Lecture 18 1 Lecture 18: (Sec. 4.1, Part I) Increasing and Decreasing Functions ex. ex.
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Lecture 18 2 Def. A function f is increasing on an interval ( a, b ) if for any two values x 1 and x 2 in ( a, b ), when- ever x 1 < x 2 , then Def. A function f is decreasing on an interval ( a, b ) if for any two values x 1 and x 2 in ( a, b ), when- ever x 1 < x 2 , then NOTE: f is increasing (decreasing) at a point c if there is an interval ( a, b ) containing c so that f is increasing (decreasing) on ( a, b ). ex.
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Lecture 18 3 ex. For what intervals is f ( x ) = 4 x 2 increasing and decreasing? NOTE: the graph of f ( x ):
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Lecture 18 4 Theorem: Test for Increasing and Decreas- ing Functions Let f be diferentiable on the interval ( a, b ). 1) 2) 3) NOTE:
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Lecture 18 5 ex. For what intervals is f ( x ) = x 2 / 3 increasing and decreasing? NOTE:
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Lecture 18 6 Def. Suppose y = f ( x ) is defned at x = c . Then c is a critical point(number) oF f iF 1) or 2) NOTE:
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Lecture 18 7 To Determine the intervals on which a con- tinuous function is increasing or decreasing 1) Find the critical points of f and ±nd the open intervals determined by these points. 2) Determine the sign of
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Unformatted text preview: f ( x ) on each of the inter-vals, and use the Test for Increasing and Decreasing Functions to determine whether f is increasing or decreasing on each interval. Lecture 18 8 ex. Given f ( x ) = x 3 3 2 x 2 6 x + 3. 1) Find all critical points of f . What are the coordi-nates of each point? Lecture 18 9 2) Find the open intervals on which f is increasing and decreasing. Lecture 18 10 3) Sketch the graph of f ( x ) = x 3 3 2 x 2 6 x + 3 Lecture 18 11 ex. The position of a particle is given by the function s ( t ) = t 3 9 t 2 + 24 t where t is measured in minutes and s ( t ) is measured in yards. When is the particle moving forward? When is it moving backwards? Lecture 18 12 ex. Consider the following graph of the derivative of a function f (so the graph is y = f ( x )). On what intervals is f increasing and decreasing?...
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lecture18 - f ( x ) on each of the inter-vals, and use the...

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