Download section 2.1- rates of change and limits

SECTION 2.1- RATES OF CHANGE AND LIMITS
Average speed = distance divided by elapsed time
Example 1. A rock breaks loose from the top of a tall cliff. What is
its average speed during the first 2 seconds of fall?
Solution:
It has been show experimentally that a dense solid object
dropped from rest will fall y=16t2 feet in the first t sec.
The average speed over any given time is the change
in the distance traveled, y ,divided by the change
in time, t
y 16(2)2 16(0)2

 32 ft / sec.
t
20
Instantaneous Speed – speed at a particular instant in time
Example 2. Find the speed of the rock in example 1 at the instant
that t=2.
Solution:
We can calculate the avg. speed from t=2 to any slightly
later time t=2+h as:
y 16(2  h)2 16(2)2

t
h
We cannot use this formula to find the exact speed at t=2
because that would require h to equal 0, which would
result in an undefined value. However, we can look at
values as h gets closer and closer to 0.
y 16(2  h)2 16(2)2

t
h
LENGTH OF TIME
INTERVAL (h)
y
( ft / sec)
t
LIMIT – allow us to describe how outputs of a function behave as
the inputs approach a particular value.
Some functions can be simplified to make finding the limit easy. But
some cannot. That is where graphing utilities come in.
Example. Find the limit of the function below as x approaches 0.
f ( x) 
sin x
x
Solution:
We cannot evaluate the limit at 0. Take a look at the graph of
the function. Also, look at the table of values. What appears
to be the limit as x approaches 0?
DEFINITION OF A LIMIT
Let c and L be real numbers. The function f has limit L
as x approaches c if, given any positive number  , there is
a positive number  such that for all x,
0 < x - c   whenever f ( x )  L   .
We write:
lim f ( x )  L .
x c
Example
lim sin x  1
x0 x
*The value of the limit does not depend on whether the function is
defined at c.
Examples (Examine the graph on p.57 of the following functions
and find the limits)
x 2 1
lim
x 1
x 1
 x2 1
,x 1

lim  x  1
x 1
1, x  1

lim x  1
x 1
**(Examine the properties of limits beginning on p. 58)**
EXAMPLES:
Find the limits of the following:
lim  x 2 (2  x)

x3
2
lim x  2 x  4
x2 x  2
lim tan x
x0 x