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**I can sketch the graph of f given the graph of f’**

Section 2.5 Day 2 Critical Numbers – Relative Maximum and Minimum Points I can determine the key components of a function given the it’s derivative both graphically and numerically. I can sketch the graph of f given the graph of f’

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Vocabulary Critical Number: a number c in the interior of the domain of a function is called this if either f ‘ (c) = 0 or f ‘ (c) does not exist Critical Point: the point (c, f(c)) of the graph f. Local (Relative) Maximum: occurs at the highest point (a, f(a)) if f (your y-value) is the largest value. Local (Relative) Minimum: occurs at the lowest point (a, f(a)) if f (your y-value) is the smallest value.

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More Vocabulary Local Extreme Values: Collectively, local maximum and minimum values Local Extreme Points: local maximum and minimum points Points of Inflection: Where function changes concavity

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**This is the graph of f(x) on the interval [-1, 5].**

Where are the relative extrema of f(x)? x = -1, x = 1, x = 3, x = 5 For what value(s) of x is f ‘ (x) < 0? (1, 3) For what value(s) of x is f ‘ (x) > 0? (-1, 1) and (3, 5) D. Where are the zero(s) of f(x)? x = 0 This is the graph of f(x) on the interval [-1, 5].

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**This is the graph of f ‘ (x) on the interval [-1, 5]. [-1, 1), (3, 5]**

Where are the relative extrema of f(x)? x = -1, x = 0, x = 5 B. For what values of x is f ‘ (x) < 0? [-1, 0) C. For what values of x is f ‘ (x) > 0? (0, 5] For what values of x is f “ (x) > 0? This is the graph of f ‘ (x) on the interval [-1, 5]. [-1, 1), (3, 5]

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**This is the graph of f(x) on [-10, 3].**

Where are the relative extrema of f(x)? x = -10, x = 3 On what interval(s) of x is f ‘ (x) constant? (-10, 0) On what interval(s) is f ‘ (x) > 0? For what value(s) of x is f ‘ (x) undefined? x = -10, x = 0, x = 3 This is the graph of f(x) on [-10, 3].

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**This is the graph of f ‘ (x) on [-10, 3].**

Where are the relative extrema of f(x)? x = -10, x = -1, x = 3 On what interval(s) of x is f ‘ (x) constant? none On what interval(s) is f ‘ (x) > 0? For what value(s) of x is f ‘ (x) undefined? none This is the graph of f ‘ (x) on [-10, 3].

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CALCULATOR REQUIRED Based upon the graph of f ‘ (x) given on the interval [0, 2pi], answer the following: Where does f achieve a minimum value? Round your answer to three decimal places. 3.665, 6.283 Where does f achieve a maximum value? Round your answer to three decimal places. 0, 5.760

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Given the graph of f(x) on to the right, answer the two questions below. Estimate to one decimal place the critical numbers of f(x). -1.4, -0.4, 0.4, 1.6 Estimate to one decimal place the value(s) of x at which there is a relative maximum. -1.4, 0.4

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**Given the graph of f ‘ (x)**

on to the right, answer the three questions below. Estimate to one decimal place the critical numbers of f(x). -1.9, 1.1, 1.8 Estimate to one decimal place the value(s) of x at which there is a relative maximum. 1.1 Estimate to one decimal place the value(s) of x at which there is a relative minimum. -1.9, 1.8

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CALCULATOR REQUIRED a) For what value(s) of x will there be a horizontal tangent? 1 b) For what value(s) of x will the graph be increasing? c) For what value(s) of x will there be a relative minimum? 1 d) For what value(s) of x will there be a relative maximum? none

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**This is the graph of f(x) on**

For what value(s) of x is f ‘ (x) = 0? On what interval(s) is f(x) increasing? . Where are the relative maxima of f(x)? -1 and 2 -1, 4 (-3, -1), (2, 4) This is the graph of f(x) on [-3, 4].

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**This is the graph of f ‘ (x)**

For what value(s) of x if f ‘ (x) = 0? For what value(s) of x does a relative maximum of f(x) exist? For what value(s) of x is the graph of f(x) increasing? For what value(s) of x is the graph of f(x) concave up? -2, 1 and 3 -3, 1, 4 (-2, 1), (3, 4] This is the graph of f ‘ (x) [-3, 4] [-3, -1) U (2, 4]

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**This is the graph of f(x) on**

For what values of x if f ‘(x) undefined? For what values of x is f(x) increasing? For what values of x is f ‘ (x) < 0? Find the maximum value of f(x). 6 (-5, 1) (1, 3) -5, 1, 3 This is the graph of f(x) on [-5, 3]

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**This is the graph of f ‘ (x)**

For what value(s) of x is f ‘ (x) undefined? For what values of x is f ‘ (x) > 0? On what interval(s) is the graph of f(x) decreasing? On what interval(s) is the graph of f(x) concave up? (0, 7) (-7, 0) (0, 7] none This is the graph of f ‘ (x) on [-7, 7].

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**This is the graph of f(x) on**

For what value(s) of x is f ‘ (x) = 0? For what value(s) of x does a relative minimum exist? On what intervals is f ‘ (x) > 0? f “ (x) > 0? (-1, 1), (1, 2) (-2, -1.5), (-0.5, 0.5), (1.5, 2) -2, -0.5, 1.5 -1.5, -0.5, 0.5, 1.5 This is the graph of f(x) on [-2, 2].

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**This is the graph of f ‘ (x) on**

For what value(s) of x is f ‘ (x) = 0? For what value(s) of x is there a local minimum? f ‘ (x) > 0? f “ (x) > 0? (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-2, -1), (0, 1) -2, 0, 2 -2, -1, 0, 1, 2 This is the graph of f ‘ (x) on [-2, 2]

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Finding the Absolute Extreme Values of Functions

Finding the Absolute Extreme Values of Functions

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