Warmup – No calculator 1) is? 2) Sketch a function f(x) that has all of the following properties: could you write a function that would have this?

2.3: Continuity

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

Continuity: You can draw a function without lifting your pencil off the paper

Ex. ALL 3 of these conditions need to be true for the function to be continuous at x = a. Definition of continuity at a point: A function f(x) is said to be continuous at a point x = a if 1) f(a) exists 2) 3)

Free response type of question: Find a and b such that the function is continuous. Justify your answer using the definition of continuity

Find the values of the constants a and b such that the function is continuous on the entire real line

Removing a discontinuity: has a discontinuity at. Write an extended function that is continuous at. Note: There is another discontinuity at that can not be removed. and -1

Removing a discontinuity: Note: There is another discontinuity at that can not be removed. (VA)

Find the x-values (if any) at which f is not continuous. continuous on entire domain

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. Also: Composites of continuous functions are continuous.

Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and. Because the function is continuous, it must take on every y value between and.

My son was born on February 10 th, He weighed 6 lbs, 15 ounces and was 20.5 inches in length At his 14 months he weighed 23 lbs, 12 ounces and was 33 inches in length Using the intermediate value theorem, what kind of statements can I make that are known to be true?

Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists.) Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2.

Explain why the functions have a zero in the domain indicated No Calculator

Recap: Definition of continuity at a point: A function f(x) is said to be continuous at a point x = a if 1) f(a) exists 2) 3) ALL 3 of these conditions need to be true for the function to be continuous at x = a.

Free response type of question: Find a and b such that the function is continuous. Justify your answer using the definition of continuity

What value of k makes the function continuous at x=0 ?

the end Day 1: p. 80 ( 1-26, 35, 36, 39, 41) Day 2: p. 80 ( 27-30, state the domain only w/out calc 37,38,40)