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# 4.1_ - f ’(x = 0 or f ’(x DNE • Relative Extrema A...

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Math 1431 Section 4.1 First Derivative Lecture Notes for class 4.1 Applications of the First Derivative Determining intervals where a function increases or decreases from a graph Identifying Critical Numbers of a function Determining intervals where a function increases or decreases from an algebraic definition Identifying Relative (Local) Extrema

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When no graph is given The procedure to follow in determining the intervals where f increases or decreases is the following: 1. Find all values of x for which f ’(x)= 0 of f’(x) DNE. 2. Identify the open intervals determined by these x values. 3. Choose a test value, c, in each of these intervals. 4. If f ’(c) > 0, f is increasing in the interval. 5. If f ’(c) < 0, f in decreasing in the interval.

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Let f(t) = 2t / (t 2 +1) Find the largest open intervals where f is increasing Find the largest open intervals where f is decreasing

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21
A critical number is an x value in the domain of f such that:

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Unformatted text preview: f ’(x) = 0 or f ’ (x) DNE • Relative Extrema: A function f has a relative maximum at x = c if there is an open interval (a,b) containing c such that f(x) < f(c) for all x in (a,b) • A function f has a relative minimum at x = c if there is an open interval (a,b) containing c such that f(x) > f(c) for all x in (a,b) To find Relative Extrema: 1 st derivative test • 1. Determine the Critical Numbers of f(x) • 2. Determine the sign of f ’(x) to the left and to the right of each critical number. • 3. If f ’(x) changes sign from positive to negative across the critical number, f(c) is a relative maximum. • 4. If f ’(x) changes sign from negative to positive across the critical number, f(c) is a relative minimum. • 5.If f ‘(x) does not change sign across the critical number, then f(c) is not a local extremum. • Let f(t) = 2t / (t 2 +1) . Determine the t value of any relative extrema....
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4.1_ - f ’(x = 0 or f ’(x DNE • Relative Extrema A...

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