Everything is in motion … is changing

http://www.pspgolf.org/wp-content/uploads/2010/08/constant-change-of-the-Universe.jpg

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

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

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

Your Turn ….

What else changes?

What doesn’t change?

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

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

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

Change is everywhere.

To understand the world, we need to understand change.

A type of mathematics that helps us understand change is called calculus.

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

y

x

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

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)

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.

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

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

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

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

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

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

Rates of Change by Graph and equation

open “Define Derivative & NDER.gsp”

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

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.

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

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

Calculus

NUMBER

FUNCTION

LIMIT

DERIVATIVE ANTI-DIFFERENTIATION

TECHNIQUES AND APPLICATIONS

TECHNIQUES AND APPLICATIONS

FUNDAMENTAL THEOREM OF CALCULUS(INTEGRAL)

CALCULUS

Knowledge of Calculus opens many doors

understanding t

he universe

jobs

predicting t

he future

Peace an

d happiness