Thursday,
April 20, 2006

More on Areas, Rates, and Slopes

More on Areas, Rates, and Slopes

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

Examples?

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

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.

Note: The simplest non-linear algebraic curves are the conics: especially- the parabolas.

slope of straight line:

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

Problem: Find the Area "under a straight line":

Motion: time: t ; position:s

constant and average velocity:

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

Problem: Given a constant velocity or a constant acceleration for a moving object, find the distance travelled.

Note: The motion of a falling object is perhaps the simplest physical example of motion with a variable velocity. Such an object has a constant acceleration.

Questions:

What is the "slope" on a curved line?

What is the instaneous velocity of a falling object?

What is the area of a region in the plane bounded by a curved line?

Given an object moving at a variable velocity, how far will it travel in a fixed interval of time?

Activity: Estimations of slopes of tangent lines and area for a parabola given by a quadratic relation between coordinates y and x.

"zooming"

What if motion does not proceed at constant velocity?

Example: What line does the graph of y = x

Related measurement concepts:

- Instantaneous Rate of growth or motion.
- Slope (gradient) of a curved line, tangent line to a curve.
- Net Change in measured variable based on rate of growth.
- Area of a region in the plane.

What is the calculus?

A systematic method for answering these questions of velocity, tangents, change in position and area that uses the algebraic representation of a relation between y and x .

Examples: if y = .... then the slope of the tangent to the curve described by this relationship at the point (a,b) is....

If the position s = ... t.... then the instantaneous velocity of the object moving to corresponfd to this relation at time t is .................

If the velocity of a moving object is given by v= .... t..... , then the object has a net change during the interval from t = a to t = b of .....

the area of the region in the plane bounded by the Xis, the lines X = a and X= b and the graph of y= ...x... is....