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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 20 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 x0 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) x3 2 lim x 2 x 4 x2 x 2 lim tan x x0 x