2.1: Rates of Change & Limits

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. What is this? 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 calculator to evaluate this expression for smaller and smaller values of h.

We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.) Lets go to TABLESET what do we want to do? We want to make the time become smaller and smaller.

The limit as h approaches zero: 0

Consider: What happens as x approaches zero? Graphically? What should our window be?

Looks like y=1

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 page 58 for details.) 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.

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

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

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

The Sandwich Theorem: Show that: The maximum value of sine is 1, soThe minimum value of sine is -1, soSo:

By the sandwich theorem: Lets graph this and see what is happening around 0.

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2.1 Step Functions “Miraculous Staircase” Loretto Chapel, Santa Fe, NM Two 360 o turns without support!

“Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function. The TI-83 contains the command, but it is important that you understand the function rather than just entering it in your calculator.

Greatest Integer Function:

The greatest integer function is also called the floor function. The notation for the floor function is: We will not use these notations. Some books use or.

Least Integer Function:

The least integer function is also called the ceiling function. The notation for the ceiling function is: Least Integer Function:

Using the Sandwich theorem to find

If we graph, it appears that

We might try to prove this using the sandwich theorem as follows: Unfortunately, neither of these new limits are defined, since the left and right hand limits do not match. We will have to be more creative. Just see if you can follow this proof. Don’t worry that you wouldn’t have thought of it. Unfortunately, neither of these new limits are defined, since the left and right hand limits do not match.

(1,0) 1 Unit Circle P(x,y)

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

(1,0) 1 Unit Circle P(x,y) T AO

multiply by two divide by Take the reciprocals, which reverses the inequalities. Switch ends.

By the sandwich theorem: 