Miss Battaglia AB/BC Calculus
What does it mean to be continuous? Below are three values of x at which the graph of f is NOT continuous At all other points in the interval (a,b), the graph of f is uninterrupted and continuous f(c) is not defined does not exist
Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. f(c) is defined 2. exists 3. Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞,∞) is everywhere continuous.
Removable (f can be made continuous by appropriately defining f(c)) & nonremovable. Removable Discontinuity Nonremovable Discontinuity
Discuss the continuity of each function
Limit from the right Limit from the left One-sided limits are useful in taking limits of functions involving radicals (Ex: if n is an even integer)
Find the limit of as x approaches -2 from the right.
Definition of Continuity on a Closed Interval A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and and The function f is continuous from the right at a and continuous from the left at b. Theorem 1.10: The Existence of a Limit Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iff and
Discuss the continuity of
By Thm 1.11, it follows that each of the following functions is continuous at every point in its domain.
Consider a person’s height. Suppose a girl is 5ft tall on her thirteenth bday and 5ft 7in tall on her fourteenth bday. For any height, h, between 5ft and 5ft 7in, there must have been a time, t, when her height was exactly h. The IVT guarantees the existence of at least one number c in the closed interval [a,b]
Use the IVT to show that the polynomial function f(x)=x 3 + 2x – 1 has a zero in the interval [0,1]
AB: Page 78 #27-30, odd, odd, 78, 79, 83, 91, BC: Page 78 #3-25 every other odd, 31, 33, 34, every other odd, 61, 63, 69, 78, 91,99-103