Everything is in motion … is changing the-Universe.jpg

The Earth spins on its axis and revolves around the sun. The Earth’s position is changing.

Ice cubes melt … We age … process.jpg

Political views change … Change jpg The climate changes …

Your Turn …. What else changes? What doesn’t change?

Understanding Change

Understanding Change

Understanding Change

Change is everywhere. To understand the world, we need to understand change. A type of mathematics that helps us understand change is called calculus.

dictionary-slope-of-a-line.gif Linear mathematical models measure change We have seen this rate of change (slope) before.

Most Mathematical Models are Not Linear How do we measure all types of change?

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 wallpapers/lamborghini-murcielago-green-race-car.jpg content/uploads/2011/07/Skiing-in-Kufri-Himachal- Pradesh.jpg kydiving.jpg? 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: 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 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 content/uploads/2011/04/wile-e-coyote.jpg How far did it fall after 2 seconds?: What is the rock’s average speed after the first 2 seconds?

Average and Instantaneous Speed What is the rock’s instantaneous speed at that moment (2 seconds after the drop)? for some very small change in t (remember the speedometer?) where h = some very small change in t average speed slope

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

The limit as h approaches zero: 0 Since the 16 is unchanged as h approaches zero, we can factor 16 out.

Rates of Change by Graph and equation open “Define Derivative & NDER.gsp”

The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4). Greg Kelly, Hanford High School, Richland, Washington

The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How small should we make the denominator? Greg Kelly, Hanford High School, Richland, Washington Keep making the denominator smaller until you see a pattern … the limiting value.

slope slope at The slope of the curve at the point is:

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 FUNDAMENTAL THEOREM OF CALCULUS (INTEGRAL) CALCULUS

Knowledge of Calculus opens many doors understanding the universe jobs predicting the future Peace and happiness