Warmup: Without a calculator find: 1)

2.1c: Rate of Change, Step Functions (and the proof using sandwich th.) (DAY 3) From:

1) Lets use the calculator to verify this 2) Lets verify it algebraically

The limit as h approaches zero: Using an algebraic approach

Use the position function which gives the height (in meters) of an object that has fallen from a height of 150 meters. Time (seconds) 1) Find the average velocity of the object from 1 second to 4 sec. 2) Find the average speed of the object from 2 second to 3 sec. 3) Find the velocity of the object at 3 seconds. 4) Find the speed of the object when it hits the ground.

“Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function.

This notation was introduced in 1962 by Kenneth E. Iverson. Recent by math standards! Greatest Integer Function: The greatest integer function is also called the floor function. The notation for the floor function is: Some books use or.

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

We are stopping here If you would like to explore the proof of the sandwich function, feel free to look over the rest of the slides at your convenience.

The end… p. 91 (1-5,8,10-12, 15-20)

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) Note: The following proof assumes positive values of. You could do a similar proof for negative values.

(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: 

The end… p. 62 (22-30 even, 35-42, 44, 47, 48,50,52, 57, 59) p. 91 (1-5,8,10-12, 15-20)