Continuity Discontinuity Increasing Decreasing Of Functions Objective SWBAT

Continuity & Discontinuity Increasing & Decreasing Of Functions

Objective • SWBAT: – Identify whether a function is continuous or discontinuous – Identify the types of discontinuity – Identify when a function is increasing, decreasing, or constant with the intervals respectively.

Definition of Continuity A function is continuous on an open interval (a, b) if it is continuous on each point in the interval. A function that is continuous on the entire real line is everywhere continuous. f(x) is continuous on (-3, 2)

A function is continuous if you can draw it in one motion without picking up your pencil.

Removable Discontinuities: (You can fill the hole. ) Nonremovable Discontinuities: jump infinite

“Discussing Continuity” • Continuous or discontinuous? • If discontinuous – Removable or nonremovable discontinuity? – At what x-value is the discontinuity? 6

Continuity by Function Type • Polynomials are everywhere continuous • Sine and Cosine are everywhere continuous • Rational functions and other trig functions are continuous except at xvalues where their denominators equal zero. – “Removable” discontinuity if factoring and canceling “removes” the zero in the denominator – “Non-removable” otherwise. (Recall that vertical asymptotes occur where numerator is nonzero and the denominator is zero. ) • Root functions are continuous, except at x-values that would result in a negative value under an even root • For piecewise functions, find the f(x) values at the x-value separating the regions of the function. – If the f(x) values are equal, the function is continuous. – Otherwise, there is a (non-removable) discontinuity at this point. 7

Increasing and Decreasing Functions

Definitions • Given function f defined on an interval – For any two numbers x 1 and x 2 on the interval • Increasing function – f(x 1) < f(x 2) when x 1 < x 2 • Decreasing function – f(x 1) > f(x 2) when x 1< x 2 X 1 X 2 f(x) • Constant Function – f(x 1) = f(x 2) when x 1< x 2 9

Check These Functions • By graphing on calculator, determine the intervals where these functions are – Increasing – Decreasing 10

Notes Over 2. 3 Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is decreasing on the interval increasing on the interval

Decreasing on(-∞, -1) U (0, 1) Increasing on (-1, 0) U (1, ∞) Using compound Interval Notation is More Effective

Notes Over 2. 3 Increasing and Decreasing Functions Describe the increasing and decreasing behavior. The function is increasing on the interval constant on the interval decreasing on the interval

Applications • Digitari, the great video game manufacturer determines its cost and revenue functions – C(x) = 4. 8 x -. 0004 x 2 0 ≤ x ≤ 2250 – R(x) = 8. 4 x -. 002 x 2 0 ≤ x ≤ 2250 • Determine the interval(s) on which the profit function is increasing 14