everything is in motion … is changing the-universe.jpg
TRANSCRIPT
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Everything is in motion … is changing
http://www.pspgolf.org/wp-content/uploads/2010/08/constant-change-of-the-Universe.jpg
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The Earth spins on its axis and revolves around the sun. The Earth’s position is changing.
http://www.nationsonline.org/bilder/earth_rotation_axis.jpg
http://www.physicalgeography.net/fundamentals/images/earth_rotation.jpg
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Ice cubes melt …
http://dclips.fundraw.com/zobo500dir/ice_cube_jarno_vasamaa_.jpg
We age …
http://www.stylishandtrendy.com/wp-content/uploads/2008/12/aging_aeging-process.jpg
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Political views change …
http://images.worldgallery.co.uk/i/prints/rw/lg/7/3/Maxi-Posters-Barack-Obama---Change-73618.jpg
The climate changes …
http://www.climatescience.gov/Library/stratplan2003/vision/VisionFig1.jpg
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Your Turn ….
What else changes?
What doesn’t change?
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Understanding Change
https://www.teachengineering.org/collection/usf_/lessons/usf_maxwell/usf_maxwell_lesson01_image3web.jpg
http://www.clker.com/cliparts/e/7/5/e/11949849161257289955radio_wireless_tower_cor_.svg.hi.png
http://www.acer.europa.eu/PublishingImages/electricity.jpg
http://grabbaggraphics.com/wp-content/uploads/2014/03/Electricity-Stock-03.jpg
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/magcur2.gif
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Understanding Change
http://2n3904.com/transistor%20(500%20x%20375).jpg
http://teelengineering.com/wp-content/uploads/2014/09/EndlessElectronics.jpg
http://www.goodhousekeeping.com/cm/goodhousekeeping/images/consumer-electronics.jpg
http://www.abc.net.au/reslib/201307/r1144180_14281902.jpg
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Understanding Change
http://1.bp.blogspot.com/-xxq7R9sVKmY/TvHQc6j-r9I/AAAAAAAAAME/qpZWtSmVQLs/s1600/Frame-dragging.jpg
http://media-cache-ec0.pinimg.com/736x/d8/24/a7/d824a7a7ea9883ef7a27b0a1fbc7d23c.jpg
http://undsci.berkeley.edu/images/us101/relativity_lightdemo.gif
https://www.sciencenews.org/sites/default/files/images/blackhole_gravity.png
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Change is everywhere.
To understand the world, we need to understand change.
A type of mathematics that helps us understand change is called calculus.
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http://www.coolmath.com/reference/images/dictionary-slope-of-a-line.gif
Linear mathematical models measure change
We have seen this rate of change (slope) before.
ym
x
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y
x
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Most Mathematical Models are Not Linear
How do we measure all types of change?http://aiche.confex.com/aiche/2006/techprogram/images/68335-0.jpg
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Let’s look at how what we know about linear
functions can apply to nonlinear ones.
Open calculus in motion file “f tan der int.gsp”
point out average rate of change and instantaneous rate
Click on “Show x,P, tan seg”
Move x around
Change f(x)
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The Concept of Instantaneous Rate
http://www.1080p-wallpapers.com/images/exotic-car-wallpapers/lamborghini-murcielago-green-race-car.jpg
http://www.photosot.com/wp-content/uploads/2011/07/Skiing-in-Kufri-Himachal-Pradesh.jpg
http://www.savique.com/system/photos/3/original/skydiving.jpg?1307403043
In order to describe how things change, we need to understand two different types of rates of change.
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Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:mi
200 mi 4 hr 50 hr
total distanceaverage speed
elapsed time
x
t
If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.
Greg Kelly, Hanford High School, Richland, Washington
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Average and Instantaneous Speed
If a rock falls from a high cliff …
The distance the rock is from where it was dropped after t seconds is
216y thttp://fromconcepttocontract.com/wp-content/uploads/2011/04/wile-e-coyote.jpg
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Average and Instantaneous Speed
The distance the rock is from where it was dropped after t seconds is
216y t
http://fromconcepttocontract.com/wp-content/uploads/2011/04/wile-e-coyote.jpg
How far did it fall after 2 seconds?:
216 2 64 feety
What is the rock’s average speed after the first 2 seconds? total distance 64 ft
32 ft/selapsed time 2 s
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Average and Instantaneous Speed
What is the rock’s instantaneous speed at that moment (2 seconds after the drop)?
instantaneous
yV
t
for some very small change in t (remember the speedometer?)
2 216 2 16 2h
h
where h = some very small change in t
average speed216y t
slope
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instantaneous
yV
t
2 2
16 2 16 2h
h
hy
t
1 80
0.1 65.6
.01 64.16
.001 64.016
.0001 64.0016
.00001 64.0002
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.
(Note that h never actually becomes zero.)
Greg Kelly, Hanford High School, Richland, Washington
try different values of h
216y t
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2 2
0
16 2 16 2limh
h
h
The limit as h approaches zero:
2
0
4 4 416 lim
h
h h
h
2
0
4 4 416 lim
h
h h
h
016 lim 4
hh
0
64
Since the 16 is unchanged as h approaches zero, we can factor 16 out.
0
(4 )16 lim
h
h h
h
216y t
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Rates of Change by Graph and equation
open “Define Derivative & NDER.gsp”
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The slope of a line is given by:y
mx
x
y
The slope at (1,1) can be approximated by the slope of the secant through (4,16).
y
x
16 1
4 1
15
3 5
We could get a better approximation if we move the point closer to (1,1). ie: (3,9)
y
x
9 1
3 1
8
2 4
Even better would be the point (2,4).
y
x
4 1
2 1
3
1 3
2f x x
Greg Kelly, Hanford High School, Richland, Washington
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The slope of a line is given by:y
mx
x
y
If we got really close to (1,1), say (1.1,1.21), the approximation would get better still
y
x
1.21 1
1.1 1
.21
.1 2.1
How small should we make the denominator?
2f x x
Greg Kelly, Hanford High School, Richland, Washington
Keep making the denominator smaller until you see a pattern … the limiting value.
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1f
1 1 h
1f h
h
slopey
x
1 1 1 1
( 1) 1
f h f f h f
h h
slope at 1,1 2
0
1 1limh
h
h
2
0
1 2 1limh
h h
h
0
2limh
h h
h
2
The slope of the curve at the point is: y f x ,P a f a
0
lim h
f a h f am
h
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Summing Up:
Meaning of a derivative: The derivative of function f(x) at x = c is the instantaneous rate of change of f(x) with respect to x at x = c.
•Numerically, by taking the limit of the average rate over the interval from c to x as x approaches c
•Graphically, by finding the slope of the line tangent to the graph at x = c
•By table, see slide 17
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Calculus
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NUMBER
FUNCTION
LIMIT
DERIVATIVE ANTI-DIFFERENTIATION
TECHNIQUES AND APPLICATIONS
TECHNIQUES AND APPLICATIONS
FUNDAMENTAL THEOREM OF CALCULUS(INTEGRAL)
CALCULUS
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Knowledge of Calculus opens many doors
understanding t
he universe
jobs
predicting t
he future
Peace an
d happiness