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 x values 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 Constant Function – f(x 1 ) = f(x 2 ) when x 1 < x 2 X 1 X 2 X 1 X 2 f(x) 9

Check These Functions • By graphing on calculator, determine the intervals where these functions are – Increasing – Decreasing  2 3

x

3 

x

2  4

x

 2

y x

5 

x

 3

x

 4 10

Notes Over 2.3

Increasing and Decreasing Functions Describe the increasing and decreasing behavior. 8 .

The function is decreasing on the interval    ,  1  increasing on the interval   1 , 0  decreasing on the interval  0 , 1  increasing on the interval  1 ,  

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. 9 .

The function is increasing on the interval   4 ,  1  constant on the interval   1 , 2  decreasing on the interval  2 , 5 

Applications • • Digitari, the great video game manufacturer determines its cost and revenue functions – C(x) = 4.8x - .0004x

2 – R(x) = 8.4x - .002x

2 0 ≤ x ≤ 2250 0 ≤ x ≤ 2250 Determine the interval(s) on which the profit function is increasing 14