Thursday,
April 13, 2006

More on Infinity, Rates, and Slopes

More on Infinity, Rates, and Slopes

Arithmetic growth: 2, 7, 12, 17, 22, ...

Examples?

Formula: A (x) = 5x + 2; Y(x) = mx + b, m>0

Geometric growth: 2, 10, 50 , 250, 1250, 6250,

Examples?

Formula: P(n) = 2*5

For each type of growth, when x > > 0.

Infinite curves: The snowflake.

Activity on snowflakes.

WinFeed (old version)

Space filling curves.

Other examples using approximations- measurements of curves and surfaces:

Archimedes: The area of the circle- the area of a triangle.

Kepler: The volume of a torus- the volume of a cylinder.

Growth Rates: m - linear, r - exponential.

Graphing and rates: slopes of lines, approximating curves with lines.

slope of straight line:

m = (change in y)/(change in x)

Motion: time: t ; position:s

constant and average velocity:

v = (change in s)/(change in t)

Questions: What if line is curved?

"zooming"

What if motion does not proceed at constant velocity?

Example: What line does the graph of y = x

Two related measurement concepts:

- Instantaneous
Rate of growth or motion.

- Slope (gradient) of a curved line, tangent line to a curve.