2.1 Rates of Change and 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. 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

Look at this chart that shows “h” getting smaller and smaller… We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. 1 80 0.1 65.6 .01 64.16 .001 64.016 .0001 64.0016 .00001 64.0002 (Note that h never actually becomes zero.) We can determine limits with t-charts, graphs, and/or algebraic methods.

The limit as h approaches zero: Now let’s prove the instantaneous rate algebraically using limits: Since the 16 is unchanged as h approaches zero, we can factor 16 out.

Today we will… 1.) Have an intuitive understanding of the limiting process 2.) Calculate limits using algebra 3.) Estimate limits from graphs or tables of data

What is a limit???? A limit is a statement that tells you what height (y-value) a function is headed for as you get close to a point specific x-value. It does not matter if the function actually gets there (like if there is a hole or asymptote at that point) All that matters is that you can tell where the function INTENDS TO GO!

Sketch: What happens as x approaches zero? Graphically: Looks like y=1 REMEMBER THIS LIMIT, IT COMES UP A LOT IN THE FUTURE!!! So:

Basics 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.

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

because the left and right hand limits do not match! 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

Find the following limits:

Sketch the graph, and find the following. a.) b.) c.) d.) = 1 = 1 = -1 DNE since b ≠ c

How to evaluate a limit algebraically: 1.) Simplify the function if you can 2.) Substitute the x-value into the function. If you get a number back, that is your answer 3.) If the answer is in the form something over zero, the answer is “undefined” 4.) If the answer is in the form 0 over 0, you have to follow an alternative: a.) Try Factoring to see if something cancels out b.) Try Rationalizing if one of the terms is a square root c.) Try Trig Substitution d.) Try Graphing or looking at a Table

Evaluate the following limits: a.) b.) c.) Now plug in: Can’t plug in, we get an undefined answer, so let’s simplify first: Since we still can’t plug in, THE LIMIT DOES NOT EXIST

Evaluate the following limits: a.) b.) Since we still can’t plug in, THE LIMIT DOES NOT EXIST

Evaluate a.) b.) c.) 7/8

Evaluate the Limit: The limit does not exists since the two Since its absolute value, we have the following: The limit does not exists since the two one-sided limits are not equal

Evaluate the limit:

Evaluate the following limits: a.) b.) c.) 1/25

Limits of Trig Functions

Limits of Trig Functions

Limits of Trig Functions

Limits of Trig Functions does not exist!

Determine

Limits of Trig Functions

Limits of Trig Functions

Limits of Trig Functions

Limits of Trig Functions does not exist!

This video will help explain and prove this function using the sandwhich (squeeze) theorem https://www.khanacademy.org/math/differential-calculus/limits_topic/squeeze_theorem/v/proof-lim-sin-x-x

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

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