SUNY Geneseo Department of Mathematics

Rates of Change

Friday, August 29

Math 221 02
Fall 2014
Prof. Doug Baldwin

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Misc

Syllabus now has link to my other section’s lecture notes

The Basics of Limits

Making sense of examples

Predict what you expect from problem statement, especially re past material
Check expectations
- Fulfilled: good, helps track example
- Violated: ask “why?”

Sections 2.1 - 2.2 “Limits of Function Values,” especially example 2

Examples

Ball dropping off table has height y = 34 - 192t² inches after t seconds

Table with XY axes beside it

When does it land?
- Ball lands when y = 34 - 192t² = 0
- i.e., at t = √(-34/-192) = 0.420813 sec
- Demo of how to use R as a calculator for things such as this (and the next calculation too)

            > -34 / -192
            [1] 0.1770833
            > sqrt( 0.1770833)
            [1] 0.4208127
            > sqrt( -34 / -192 )
            [1] 0.4208127
            > 34/ 0.42
            [1] 80.95238

What is its average speed during fall?
- Δy / Δt = 34 / 0.42 = 81 in/sec
How fast is it moving when it lands?
- Graph of “instantaneous speed” issue

Upside-down parabola with small section between t0 and t0+h

Instantaneous speed at t = 0.42 is average speed between 0.42 and 0.42 + h for small h
= ((34 - 192(0.42+h)²) - (34 - 192(0.42)²)) / h
= ((34 - 192(0.42² + 0.84h +h²)) - (34 - 192(0.42)²)) / h
= -192(0.84h + h²) / h
= ( -161.3h - 192h²) / h
= -161.3 - 192h
= -161.3, setting h = 0

Some other questions some but not all of us worked on
- How fast is it moving at half the time to landing?
- Where is it then?

Limits per se

“Limits of Function Values” part of reading from section 2.2

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