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1.5 Increasing/Decreasing; Max/min Tues Sept 16

1.5 Increasing/Decreasing; Max/min Tues Sept 16. Do Now Graph f(x) = x^2 - 9. HW Review: p.87 #43-51, 55-61. 43) (-infinity, infinity) 45) (-infinity, 0) U (0, infinity) 47) (-infinity, 2) U (2, infinity) 49) (-infinity, -1) U (-1, 5) U (5, infinity) 51) (-infinity, 8]

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1.5 Increasing/Decreasing; Max/min Tues Sept 16

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  1. 1.5 Increasing/Decreasing; Max/minTues Sept 16 Do Now Graph f(x) = x^2 - 9

  2. HW Review: p.87 #43-51, 55-61 • 43) (-infinity, infinity) • 45) (-infinity, 0) U (0, infinity) • 47) (-infinity, 2) U (2, infinity) • 49) (-infinity, -1) U (-1, 5) U (5, infinity) • 51) (-infinity, 8] • 55) D: [0, 5] R: [0, 3] • 57) D: [-2pi, 2pi] R: [-1, 1] • 59) D: (-infinity, infinity) R: {-3} • 61) D: [-5, 3] R: [-2, 2]

  3. Increasing and Decreasing Functions • A function’s behavior can be described as one of three types: • Increasing • Decreasing • Constant • These behaviors can be described in interval notation as well

  4. Ex 1 • Graph on page 120 • Determine the intervals on which the function is increasing, decreasing, constant

  5. Relative Extrema • Certain functions can have relative extrema, a point on the graph where the function changes from increasing to decreasing, or vice versa • These are called relative maxima or minima • Sometimes called local maxima or minima

  6. Finding relative extrema using a calculator • 1) Type the function in “Y=“ • 2) Graph • 3) “2nd” -> “Calc” • 4) Min or Max • 5) Left bound: Select a point to the left of the max/min • 6) Right bound: Select a point to the right of the max/min • 7) Guess: hit enter

  7. Ex: • Find the relative extrema of and determine when it is increasing or decreasing

  8. You try • Graph each function. Find any relative extrema, and determine when each function is increasing or decreasing • 1) • 2) • 3)

  9. Closure • What are relative extrema? How can we find them? • HW: p.127 #1-21 odds

  10. 1.5 Piecewise FunctionsWed Sept 17 • Do Now • Graph • Find the relative minimum, and determine where the function increases / decreases

  11. HW Review: p.127 #1-21 • 1) a: (-5,1) b: (3, 5) c: (1, 3) • 3) a: (-3, -1), (3, 5) b: (1, 3) c: (-5, -3) • 5) a: (-inf, -8) (-3, -2) b: (-8, -6) c: (-6, -3), (-2, inf) • 7) D: [-5, 5] R: [-3, 3] • 9) D: [-5, -1] U [1, 5] R: [-4, 6] • 11) D: (-inf, inf) R: (-inf, 3] • 13) max: (2.5, 3.25), inc (-inf, 2.5) dec (2.5, inf) • 15) max: (-0.667,2.37), min: (0,2) inc (-inf, -0.667) U (2, inf) dec(-0.667, 2)

  12. 17-21 • 17) min (0,0) inc (0, inf) dec (-inf, 0) • 19) max (0, 5) inc (-inf, 0) dec (0, inf) • 21) min (3, 1) inc (3, inf) dec (-inf, 3)

  13. Piecewise functions • A piecewise function is a function that uses different output formulas for different parts of the domain • Each piece is only considered for the given domain

  14. Graphing Piecewise Functions • It is important to graph the endpoints of each piece, so we know where they fit in

  15. Ex • Graph

  16. Ex 2 • Graph

  17. Ex 3 • Graph

  18. Greatest Integer Functions • The greatest integer function is defined as the greatest integer less than or equal to x This function is also known as a step function - Its graph looks like steps

  19. Closure • Graph • HW: p.131 #39-49 odds, 59-63 odds

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