Download Continuous and Discontinuous Functions

Continuous and Discontinuous Functions
A continuous function can be described as one that can be sketched without lifting the pencil off the paper. A
discontinuous function can only be sketched if the pencil is lifted.
Types of discontinuous functions:
1. removable discontinuities
2. non-removable discontinuities
Removable discontinuity:
Also known as a hole. As x approaches c from the left ( c  ) and from the right ( c  ), f(x) approaches the
same numerical value.
Non-removable discontinuities
Jump: As x approaches c from the left ( c  ) and from the right ( c  ), f (x) approaches a different numerical
value.
Vertical asymptote: As x approaches c from the left ( c  ), f (x) increases/decreases without bound.
As x approaches c from the right ( c  ), f (x) increases/decreases without bound.
Determine from the table the type of discontinuity.
x
f (x)
1.9
-.02
1.99
-.002
1.999
-.0002
2
dne
2.001
.0002
2.01
.002
2.1
.02
x
f (x)
2.9
-1
2.99
-1
2.999
-1
3
dne
3.001
2
3.01
2
3.1
2
x
f (x)
.9
-10
.99
-100
.999
-1000
1
dne
1.001
1000
1.01
100
1.1
10
Average Rate of Change
The average rate of change of the function y = f (x) between x = a and x = b is
∆y = f (b) – f (a)
∆x
b–a
(this is a difference quotient)
The average rate of change is the slope of the secant line between x = a and x = b , passing through the points
( a, f(a)) and ( b, f(b))
If f is an increasing function, then the slope of the secant line is positive and the rate of change is positive.
If f is a decreasing function, then the slope of the secant line is negative and the rate of change is negative.
Example: If an object is dropped from a height of 3000 feet, its distance above the ground (in feet) after t
seconds is given by h (t) = 3000 - 16t2.
a. Find the average rate of change (average velocity) between 2 and 8 seconds.
b. Find the average velocity between 6 and 10 seconds.
c. Write the equation of the secant line for between 6 and 10 seconds.