SUNY Geneseo Department of Mathematics

Average Rates of Change

Thursday, August 28

Math 221 10
Fall 2014
Prof. Doug Baldwin

Misc

URL for syllabus is http://www.geneseo.edu/~baldwin/math221/fall2014/10/syllabus.php

Lecture notes and handouts online
Syllabus now has link to my other section’s lecture notes

To make appointments via Google calendar you have to log in to it with your Geneseo ID, enter my name in “add a calendar” box if you want to see my calendar, then make an appointment and invite me to it

Access to these notes (also need to be logged in to Google w/ Geneseo ID): https://docs.google.com/a/geneseo.edu/document/d/1RYckbLjsH7A_5DkElS0dUE96D0pN9yHs67Qf2hr74Cc/edit

The Basics of Limits

Sections 2.1 - 2.2 “Limits of Function Values”

Chapter 1?
- Background, review
How to read this? Lots of dense info
- Lots of teaching in examples
- Connect to past material
- Translate to words, math uses lots of symbols that stand for entire words or ideas
  - “+” = “plus”
  - “Δ” = “change in”
- Read very closely. Every word matters
Limits of function values?
- What is a limit?
- Slope at a point?

Examples

Ball dropping off table has height y = 28.75 - 192 t² inches after t seconds

Table with XY axes at bottom

When does it land?
- Lands when y = 0 = 28.75 - 192 t²
- -28.75 = -192 t²
- -28.75 / -192 = t²
- t = √(-28.75 / -192)
- t = 0.387 seconds
- These calculations are best done with a calculator. Demo how R can be used as one

            > -28.75 / -192
            [1] 0.1497396
            > sqrt(0.1497396)
            [1] 0.386962
            > sqrt( -28.75 / -192 )
            [1] 0.386962
            > ratio = -28.75 / -192
            > sqrt(ratio)
            [1] 0.386962
            > ratio
            [1] 0.1497396

How fast is it moving when it lands?
- Average speed = Δy/Δt = 28.75 / 0.387 = 74.289
  - But this is the average speed over the whole roughly 1/3 second the ball takes to fall, not the speed at the moment the ball reaches the ground.
- speed = acceleration × time
  - = 384 in/sec² × 0.387 sec
  - = 148.56 in/sec
  - This solution works if you understand the physics behind it, but can we solve the problem via the math in the reading?
- Δy/Δt = ((28.75 - 192(0.387+h)²) - (28.75 - 192 (0.387)²) / h
  - = ((28.75 - 192(t+h)²) - (28.75 - 192 t²) / h (temporarily replacing 0.387 with t to reduce typing)
  - = ((28.75 - 192(t²+2th+h²)) - (28.75 - 192 t²) / h
  - = (28.75 - 192(t²+2th+h²) - 28.75 + 192 t²) / h
  - = (-192(t²+2th+h²) + 192 t²) / h
  - = (-384th-192h²) / h
  - = (-384t-192h)
  - = -384t (setting h = 0)
  - = -148.56 in/sec

Questions and examples related to the “Limits of Function Values” part of the reading

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