AP Calculus September 6, 2016 Mrs. Agnew

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Presentation transcript:

AP Calculus September 6, 2016 Mrs. Agnew Continuity AP Calculus September 6, 2016 Mrs. Agnew

Essential Question Essential Vocabulary What does it mean for a function to be continuous? Essential Vocabulary Continuity Discontinuity (jump, hole, break, gap)

What is Continuity? A function that is continuous at x = c has no jumps, holes, or gaps at c. Let’s examine some functions that are not continuous due to the following: Hole Jump Gap

Continuity at a Point A function f is continuous at x = c if the following conditions are met: Continuous when limit can be found using method of direct substitution. Let’s examine a graph…

Continuity on an Interval A function is continuous on the open interval (a,b) if it is continuous at every point in the interval. A function that is continuous for all real values of x is said to be everywhere continuous. Give some examples of functions that are everywhere continuous.

Discontinuities There are two types of discontinuities: Removable (holes) Nonremovable (breaks, gaps) A function that can be factored and the discontinuity canceled has a removable discontinuity. Examples

Continuous Functions Polynomials: continuous everywhere Rationals: continuous over domain Abs Value: continuous everywhere Trig Functions Sine and Cosine: continuous everywhere Others: continuous over domain Radicals: continuous over domain Exponential: continuous everywhere Logarithmic: continuous over domain

Intermediate Value Theorem Suppose f is continuous on the open interval (a,b) and suppose that N is a number between f(a) and f(b). Then there exists a number x between a and b such that f(x) = N.

Notes on the IVT The IVT is an existence theorem… tells us a value exists, not what the value is. Usefulness: Proving that roots of a polynomial exist How a calculator graphs… Examples

Homework: 9/6/16 Page 79 – 82 #10, 12, 15, 33, 43 – 49, 53, 54, 61 – 67 (O) 68, 83, 85, 93, 99 – 103, 108