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# lecture24hout - MATH 006 Calculus and Linear...

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Unformatted text preview: MATH 006 Calculus and Linear Algebra (Lecture 24) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 16 Outline 1 Increasing and Decreasing Functions 2 Critical Values 3 Local Extrema 4 First-Derivative Test 5 Application Albert Ku (HKUST) MATH 006 2 / 16 Increasing and Decreasing Functions Definition Suppose y = f ( x ). 1 f ( x ) is increasing on an interval a < x < b , if for any a < x 1 < x 2 < b , f ( x 1 ) < f ( x 2 ). 2 f ( x ) is decreasing on an interval a < x < b , if for any a < x 1 < x 2 < b , f ( x 2 ) < f ( x 1 ). Albert Ku (HKUST) MATH 006 3 / 16 Increasing and Decreasing Functions Example Determine the interval for which the function f ( x ) = x 2 is increasing. Solution From the graph of y = x 2 , it is obvious that the function is increasing when x > 0 and decreasing when x < 0. Notice that the slope of the tangent is positive when f is increasing, and is negative when f is decreasing. Albert Ku (HKUST) MATH 006 4 / 16 Increasing and Decreasing Functions Theorem For the interval a < x < b, 1 f ( x ) is an increasing function on a < x < b if and only if f ( x ) > on a < x < b....
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lecture24hout - MATH 006 Calculus and Linear...

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