Continuity and One-Sided Limits：连续性和片面的限制.docVIP

Objectives: 2.4.1 Describe continuity and be able to distinguish a continuous function from one with discontinuities. Materials: Exploration 2-4a (IRB) Lesson 1 Warm Up p. 49-50: Quick Review: Q1-Q10 1. What is meant by the derivative of a function? 2. What is meant by the definite integral of a function? 3. Draw a pair of alternate interior angles. 4. What type of function has a graph like the following: 5. Sketch the graph of 6. Factor: 7. Evaluate: 8. Evaluate: 5! 9. No calculator! Divide 50 by ? and add 3. Solutions: Q1. Instantaneous rate of change Q2. Product of x and y, where x varies and y can vary 3. 4. Exponential function 5. 6. 7. 53 8. 120 9. 103 I. Continuity at a Point and on an Open Interval A function is continuous if there are no holes, steps, or asymptotes. Basically, this means that you can draw the function without lifting your pencil. Some of the most interesting functions with discontinuities are “piecewise” functions. Graphing a Piecewise Function Manually Piecewise functions have multiple branches at different sections of the domain of the function. To manually graph a piecewise function, graph each piece separately and then erase the sections that are beyond the domain of that section. Example 1 Manually graph the following piecewise function: Query: Is the function in Example 1 continuous? Solution: No, there is a step discontinuity. Example 2 Manually graph the following piecewise function: Query: Is the function in Example 2 continuous? Solution: No, the vertical asymptote is a discontinuity. Graphing a Piecewise Function with a Calculator Graph the function from Example 1, , by entering the following into Y1: Example Plot the following function on your calculator: Does f(x) have a limit as x approaches 2? No, the function has different values if you approach from the left or from the right. Is f(x) continuous at x = 2? No, there is a gap. Does f(x) have a limit as x approaches 5? Is f(x) continuous at x = 5? “Types” of Discon