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Understanding Continuity and Critical Points in Functions: Key Concepts and Examples

In this comprehensive guide, you will explore essential concepts related to continuity, end behavior, critical points, and extrema of functions. By the end of the session, you'll have clear diagrams and examples to illustrate: types of discontinuities, methods for testing continuity, behaviors of functions at critical points, and techniques for identifying maximums, minimums, and inflection points. Engage in practical exercises to apply these concepts and develop a strong understanding of how to analyze and classify functions in calculus.

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Understanding Continuity and Critical Points in Functions: Key Concepts and Examples

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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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