Thursday,  April  13, 2006
More on Infinity, Rates, and Slopes

Recall from last class:

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*5n   ; P(x) = A r
x ,   r>1.
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 = x2 look like near (3,9).

Two related measurement concepts:
• Instantaneous Rate of growth or motion.
• Slope (gradient) of a curved line, tangent line to a curve.