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What you will learn

What you will learn. Oh Boy! A smorgasbord! By the end of today, you should have examples or diagrams of: 1. Diagrams of discontinuities 2. Example of testing continuity 3. Examples/diagrams of end behaviors 4. Examples/diagrams of increasing and decreasing “portions” of functions

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What you will learn

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  1. What you will learn • Oh Boy! A smorgasbord! • By the end of today, you should have examples or diagrams of: • 1. Diagrams of discontinuities • 2. Example of testing continuity • 3. Examples/diagrams of end behaviors • 4. Examples/diagrams of increasing and decreasing “portions” of functions • 5. Diagrams of and ways to find maximums, minimums, and points of inflection • 6. Examples of method to find whether critical points are maximums, minimums, or points of inflection

  2. What Do These Graphs Have in Common? Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  3. Infinite Discontinuity • |f(x)| becomes greater and greater as the graph approaches a given x-value. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  4. Jump Discontinuity The graph stops at a given value of the domain (x) and then begins again at a different range value for the same value of the domain (example: piecewise functions) Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  5. Point Discontinuity When there is a value of the domain for which the function is undefined, you have point discontinuity. Example: Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  6. Continuous Functions • If there is no discontinuity, the function is continuous. • If you can trace the entire function without lifting your pencil, it is continuous. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  7. Testing for Continuity at a Point • A function is continuous at a given value of x (in this case, we will say, x = c) if it satisfies the following conditions: 1. The function is defined at c (there is a y-value for that x-value). 2. The function approaches the same y-value on the left and right sides of x = c. 3. The y-value that the function approaches from each side is the y for the x we are testing (duh). Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  8. Example of Continuity Test • Determine whether the function is continuous at x = 1: f(x) = 3x2 + 7 • Test 1: Defined at x = 1? • Test 2: Approaches same y from just above x and just below x. • Test 3: Since test 2 resulted in a number close to 10 (approaches 10) we make sure that f(1) = 10 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  9. Another Example Determine whether the function is continuous at x = 2: Test 1: Test 2: Test 3: Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  10. Yet Another example • Determine whether the function is continuous at x = 1; Test 1: Test 2: Test 3: If x > 1 If x < = 1 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  11. Using the Calculator Examples Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  12. You Try • Perform all three tests to determine whether the following functions are continuous at the indicated value: 1. f(x) = 3x2 + x + 7; x = 1 2. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  13. Continuity on an Interval • A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval. • Example: Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  14. End Behavior – Notes from Page 163 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  15. Increasing/Decreasing on Intervals • Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. • F(x) = 3 – (x – 5)2 • F(x) = 2x3 + 3x2 – 12x + 3 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  16. You Try • Determine the interval(s) on which the functions are increasing or decreasing. 1. f(x) = ½ |x + 3| - 5 2. f(x) = 5x3 + x2 – x + 4 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  17. Critical Points and Extrema • Lots of definitions! • Critical points: points on the graph at which a line drawn tangent to the curve is horizontal or vertical. • Maximum, minimum, point of inflection. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  18. Yet More Definitions • Absolute maximum – the greatest value a function assumes over its domain. • Absolute minimum – the least value a function assumes over its domain. • Extremum – general term for a maximum and minimum Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  19. And Even More • Relative maximum: the greatest y-value on an interval. • Relative minimum: the least y-value on an interval. • Relative extrema: the general term for the above. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  20. Example • Locate and classify the extrema for the function: f(x) = 5x3 – 10x2 – 20x + 7 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  21. You Try • Determine and classify the extrema for the following: f(x) = x3 – 8x + 3 Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  22. Another Little Trick • You can determine whether an extrema is a relative minimum, relative maximum, or a point of inflection. • How? We test points on either side of the identified extrema and check the values. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  23. Example • The function f(x) = 2x5 – 5x4 – 10x3 has critical points at x = -1, x = 0, and x = 3. Determine whether each of these critical points is the location of a maximum, minimum, or a point of inflection. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  24. Summary of Tests Results Relative Maximum: Relative Minimum: Point of Inflection: Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  25. You Try • The function f(x) = 3x4 – 4x3 has critical points at x = 0 and x = 11. Determine whether each point is the location of a maximum, a minimum, or a point of inflection. Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

  26. Homework • page 166, 12, 16, 18, 20, 24-30 evenpage 177, 14, 15, 20-24 even, 26-28 all Objective: 3-5 Continuity and End Behavior, 3-6 Critical Points and Extrema

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