2.1 Rates of Change and Limits

A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous speed at 2 seconds?

for some very small change in t where h = some very small change in t We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.)

The limit as h approaches zero: 0

Secant Line y = f(x)

Solution:

Example: The function gives the height (in feet) of a ball thrown straight up as a function of time, t (in seconds). (a) Find the average rate of change of the height of the ball between 1 and t seconds. Solution: (b) Use the result found in part (a) to find the average rate of change of the height of the ball between 1 and 2 seconds. (a)

If t = 2, the average rate of change between 1 second and 2 seconds is: - 4(4(2) - 21) = 52 ft/second. The average rate of change between 1 second and t seconds is: -4(4t - 21) (b)

Example: How does the the function f(x)= x 2 -x+2 behave near x=2. x 2 f(x) Before we give a definition of limit, let us look at the following examples. Solution:

We see that when x is close to 2 (x>2 or x<2), f(x) is close to 4. Then we can say that the limit of the function f(x) = x 2 -x+2 as x approaches 2 is equal to 4 And we write this as the notation :

Value of x f(x)=(x 2 –1)/(x – 1) y 12x Solution: The function f(x) is defined for all x except x=1

For any x=1 we simplify the formula of the function by So the graph of the function f is the line y=x+1 with the removable point (1,2) From the table we say that f(x) approaches the limit 2 as x approaches 1 And we write this as the notation

Definition: Informal Definition of limit Let f(x) be defined on an open interval about a, except possibly at a itself. If we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a. And we say “the limit of f(x) equals L” as x approaches a. And we write

The limit of a function refers to the value that the function approaches, not the actual value (if any). But not 1

(3, 7) y x Solution: From graph we get

Try This Find: f(0)is undefined; and 2 Solution:

Try This 2 1 Solution: f(0)is undefined; and Find:

Try This Find the limit of f(x) as x approaches 3 where f is defined by: Solution:

(2, 7) (2, -1) does not exist, there is a jump at x =2. Solution:

Example: Consider Solution: The limit is 3

Example: Discuss the behavior of the following functions as x  0 The unit step function, it values jump at x=0. For negative values of x close to 0, U(x)=0. For positive values of x close to 0, U(x)=1. jump infinite oscillating

The values of g grow too large in the absolute values As x  0 and do not stay close to any real number The function’s values oscillate between +1 and -1 in every interval containing 0.

oscillating The function’s values oscillate between +1 and -1 in every interval containing 0. Example: Discuss the behavior of the following functions as x  0 Solution: