Calculus 1 - Calculus Section 4.1 Page1 4.1 Extreme Values...

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Unformatted text preview: Calculus Section 4.1 Page1 4.1 Extreme Values of Functions Absolute (Global) Extreme Values Example 1 Exploring Extreme Values Determine the maximum value and minimum value on - 2 , 2 of : a) y = cos x b) y = sin x Example 2 Exploring Absolute Extrema Determine the absolute extreme of the function y = x 2 on each domain Domain D Graph Absolute Extrema on D - , ( ) 0, 2 [ ] 0, 2 ( ] 0, 2 ( ) Extreme Value Theorem If f is continuous on a closed interval a , b [ ] , then f has both a maximum value and a minimum value on the interval. Calculus Section 4.1 Page 2 Local (Relative) Extreme Values Definition Critical Point Example 3 Finding Absolute Extrema Find the absolute maximum and minimum values of f x ( ) = x 2 3 on the interval - 2, 3 [ ] Solve Graphically Confirm Analytically Example 4 Finding Extreme Values Find the extreme values of f x ( ) = 1 4- x 2 Solve Graphically Confirm Analytically Calculus Section 4.1 Page3 While a functions extrema can occur only at critical points and endpoints, not every critical point or endpoint signals the presence of an extreme value: y = x 3 y = x 1 3 Calculus Section 4.1 Page 4 Example 5 Finding Extreme Values Find the extreme values of f x ( ) = 5- 2 x 2 , x 1 x + 2, x > 1 Solve Graphically Confirm Analytically Example 6 Using Graphical Methods Find the extreme values of f x ( ) = ln x 1 + x 2 Solve Graphically Confirm Analytically Assignment III-1 Page 184 185 #1 18 all, 19 29 odd. 35 43 odd, 45 49 all Calculus Section 4.1 Page1 4.1 Extreme Values of Functions Absolute (Global) Extreme Values Example 1 Exploring Extreme Values Determine the maximum value and minimum value on - 2 , 2 of : a) y = cos x b) y = sin x Example 2 Exploring Absolute Extrema Determine the absolute extreme of the function y = x 2 on each domain Domain D Graph Absolute Extrema on D - , ( ) 0, 2 [ ] 0, 2 ( ] 0, 2 ( ) Extreme Value Theorem If f is continuous on a closed interval a , b [ ] , then f has both a maximum value and a minimum value on the interval. Calculus Section 4.1 Page 2 Local (Relative) Extreme Values Definition Critical Point Example 3 Finding Absolute Extrema Find the absolute maximum and minimum values of f x ( ) = x 2 3 on the interval - 2, 3 [ ] Solve Graphically Confirm Analytically Example 4 Finding Extreme Values Find the extreme values of f x ( ) = 1 4- x 2 Solve Graphically Confirm Analytically Calculus Section 4.1 Page3 While a functions extrema can occur only at critical points and endpoints, not every critical point or endpoint signals the presence of an extreme value: y = x 3 y = x 1 3 Calculus Section 4.1 Page 4 Example 5 Finding Extreme Values Find the extreme values of f x ( ) = 5- 2 x 2 , x 1 x + 2, x > 1 Solve Graphically Confirm Analytically Example 6 Using Graphical Methods Find the extreme values of f x ( ) = ln x 1 + x 2 Solve Graphically Confirm Analytically...
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Calculus 1 - Calculus Section 4.1 Page1 4.1 Extreme Values...

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