Marquette University High Schoolfaculty.muhs.edu/hardtke/ABCalcNotes_2011_12/ABNotes_2.2... · 2011. 9. 12. · Created Date: 9/12/2011 11:59:24 AM

Given, f(x) = 2x - 7, what is

Recall: we don't really care what happens when x is exactly 3, but do the walkers from each of the road get close enough to shake hands when they approach 3. The left-hand limit and the right-hand limit must agree for the overall limit to exist.

IF you know that a function is continuous and well-behaved at "a", you find the limit as x →a by simple SUBSTITUTION.

Given

, what is

Simply SUBSTITUTING 3 for x might work, but how much do we recognize about this graph? Let's use an "ASK" table on the TI-89 to see if the limit exists. Looking at the graph may be helpful, but recall that the calculator graph doesn't show all

AB Calc Assignment Page

Upcoming Assignments & Opportunities:

This Week's Quiz Retake Options: Mon (5th), Mon(2:55 pm), Tues (7:20am), Fri(7:20am) Fri (5th), You may retake any quiz of your choice. Just show up and I'll have copies of all three types. Times are also posted on website assignment page.

Start retaking today - don’t procrastinate.

Return Exit Slip and discuss writing mathematical arguments. Except for multiple choice or T/F question, your answers must be JUSTIFIED. Get used to adding words, phrases, sentences, diagrams and graphs. How and why did you chose your response?

Notes on LimitsIf time allows, return to Friday's TI-89 handout.

Today's Topics & Class Plan:

A Math Phobic’s Nightmare

AB Calculus Section 2.2 NotesMonday, September 12, 2011

Chapter 2 Page 1

http://faculty.muhs.edu/hardtke/ABCalc_Assignments.htm

at the graph may be helpful, but recall that the calculator graph doesn't show all discontinuities and weird behaviors correctly.

Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m

1. The ave velocity over interval from 5 to 8 second.2. The (instantaneous) velocity at 5 seconds.

above the ground. Find

We need Galileo's formula for free fall (discounting airresistance) to find the distance traveled, s(t), in meters,at t seconds.s(t) = 4.9t 2

1. Recall we already know how to find the AVERAGE VELOCITY for a falling body overan interval of time. Use your TI-89 to find theaverage velocity over t (0,8):

1. Now use your TI-89 to find the velocity at 5 seconds. Be prepared to explain your method.

Vav=

Chapter 2 Page 2

Find the limits for the graph of f(x) shown.

C. F.

Use your calculator to estimate

Be prepared to explain.

Tomorrow we'll find an EXACT way to do this without a calculator.

Some functions behave so wildly that your TI-89's graph may be completely

misleading. Try to estimate

with your calculator.

Chapter 2 Page 3

misleading. Try to estimate

with your calculator.

So the limit d.n.e. since this graph oscillates up and down more and more times as x gets close to 0, we can't say whether y is getting closer to 0 or not.

QUICK SUMMARY:

If and then we say the limit of f(x) as x approaches a is equal to Lor in symbols

Always remember that we do not care about what happens exactly at the x-value, simply whether the left-hand limit and the right-hand limit are equal so the two walkers get close enough to shake hands.

• Recall the left and right hand limits must agree so for graphs with weird "corners" or discontinuities, your answer for the limit might be d.n.e

• You may ESTIMATE an instantaneous velocity by finding the average velocity over a very small time interval. Perhaps we will find some shortcuts for this later.

• If you don't recognize the function or its graph, you may be forced to rely on your calculator to estimate a limit (even though it isn't always as reliable as you thought)!

Definition of Limit: L is a real number

Chapter 2 Page 4

Chapter 2 Page 5