Rates of Change and Limits 2.1 Rates of Change and Limits Grand Teton National Park, Wyoming Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2007

Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

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-84 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. 1 80 0.1 65.6 .01 64.16 .001 64.016 .0001 64.0016 .00001 64.0002 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: Since the 16 is unchanged as h approaches zero, we can factor 16 out.

Consider: What happens as x approaches zero? Graphically:

It appears that the limit of as x approaches zero is 1

Limit notation: “The limit of f of x as x approaches c is L.” So:

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

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See your book for details. Page 99) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

does not exist because the left and right hand limits do not match! 2 1 1 2 3 4 At x=1: left hand limit right hand limit value of the function

because the left and right hand limits match. 2 1 1 2 3 4 At x=2: left hand limit right hand limit value of the function

because the left and right hand limits match. 2 1 1 2 3 4 At x=3: left hand limit right hand limit value of the function

The Sandwich Theorem: Show that: The maximum value of sine is 1, so The minimum value of sine is -1, so So:

By the sandwich theorem:

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