At the very front of your binder. Write a message to yourself At the very front of your binder. Write a message to yourself. Tell yourself what you will need to hear in the difficult times. Tell your self what you need to do to get through this course with the grade you want. Then write: “I make the following 3 promises to myself…”

I firmly believe that any man's finest hour, the greatest fulfillment of all that he holds dear, is that moment when he has worked his heart out in a good cause and lies exhausted on the field of battle – victorious.

I firmly believe that any man's finest hour, the greatest fulfillment of all that he holds dear, is that moment when he has worked his heart out in a good cause and lies exhausted on the field of battle – victorious.

Rates of Change and Limits 2.1 Rates of Change and Limits “Once we accept our limits, we go beyond them.” Albert Einstein

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-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. 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: Y= WINDOW GRAPH

Looks like y=1

Numerically: TblSet TABLE You can scroll down to see more values.

It appears that the limit of as x approaches zero is 1 TABLE You can scroll down to see more values.

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.) 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: WINDOW

p

Its your Fun and Happy Joy Joy Pleasure Time #3-30 (multiples of 3), 32, 35, 39, 42, 44, 45-52, 55, 58 Optional over 2 weeks: Overview Chapter 1.4 - 3 to 30 (multiples of 3) Chapter 2.1 Self Assessment: Write a paragraph describing any difficulties you encounter while finding limits.