Summary of First

Summary of First-Week Ideas

Rate of Change

Rate of Change and Graphs

Interpreting Graphs

Precision in Experiment

Rate of Change

The rate of change of a quantity A with respect to a quantity B is

rate of change = change in A / change in B.

This concept is usually applied when quantity A is dependent on quantity B. Examples include:

The period of a pendulum is dependent on its length.
The tension force exerted by a given rubber band is dependent on its length.
The rate at which the volume of a filling container increases at a given depth is dependent on its cross-sectional area at that depth.
Position of an object rolling down an incline is dependent on clock time. Rate of change of position with respect to clock time is velocity.
Velocity of an object rolling down an incline is dependent on clock time. Rate of change of velocity with respect to clock time is acceleration.
Rate of change of velocity on a straight incline tends to be constant. If an incline is not straight, or if the object is rolling fast enough that air resistance is a factor, acceleration is a nonconstant function of clock time.

If the rate of change of A with respect to B is constant then a graph of A vs. B is linear. If the rate of change is zero then the graph is horizontal.

Rate of Change and Graphs

Suppose you have a graph of quantity A vs. quantity B. This means that quantity A is on the vertical axis and quantity B is on the horizontal axis.

The rise of a graph represents the change in the quantity A and the run represents the change in quantity B.

The slope, which is rise / run, therefore represents change in quantity A / change in quantity B, which is the average rate at which quantity A changes with respect to quantity B.

Interpreting Trapezoidal Approximation Graphs

To intepret a graph we first interpret the following quantities:

rise of each trapezoid (defined as change in altitude from left to right side)
run of each trapezoid (defined as change in the horizontal coordinate from left to right side)
average altitude of each trapezoid (defined as the average of the two altitudes)

If the graph is of quantity A vs. quantity B then

rise represents change in A
run represents change in B
average altitude represents the average value of A

and

slope = rise / run represents [change in A / change in B ] = average rate of change of A with respect to B
area = average value of A * change in B

If quantity A is a the rate of change r with respect to clock time then we think of the interpretation as follows:

rise represents change in rate r
run represents change in B
average altitude represents the average value of the rate r

and

slope = rise / run represents [change in rA / change in B ] = average rate of change of r with respect to B = average rate at which the rate r changes with respect to B
area = average value of rate r * change in B = change in A.

Precision in Experiment

In any experiment the precision of our observations affects the precision of our conclusions.

If we have insufficient precision we might find the following:

the period of a pendulum doesn't change with length
the rate at which the depth of water in a leaking cylinder changes is constant
the time required for a ball to travel from rest down a ramp doesn't depend on the slope of the ramp
the temperature of an object cooling in a constant-temperature environment changes at a constant rate
the tension force exerted by a rubber band is a linear function of its length

Given sufficient precision we can find the following:

the period of a pendulum in seconds is .2 sqrt(L), where L is length in cm
a graph of the rate at which the depth of water in a leaking cylinder changes vs. clock time is linear, meaning that the rate at which depth changes changes at a constant rate
the rate at which the velocity of a ball traveling from rest down a ramp changes is linearly dependent on ramp slope, for ramps with sufficiently small slopes
the rate at which the temperature of an object cooling in a constant-temperature environment changes is directly proportional to the difference between object temperature and the temperature of the environment
the tension force exerted by a rubber band within its elastic limits first increases at a decreasing rate, then increases at an increasing rate

Period of a Pendulum

The period of a pendulum is proportional to the square root of its length. Near the surface of the Earth the period of a pendulum is given by the equation T = .20 sqrt(L), with T being period in seconds and L being length in cm.

Describing Graphs

We describe graphs as increasing or decreasing, at an increasing or decreasing rate; and we indicate key points, zeros, asymptotes, relative maxima and minima and other characteristics relevant to the meaning of the graph and the behavior of the system it represents. We also indicate the meaning of the slopes and areas defined by the graph.

Marble on Incline

Assuming a constant rate of velocity change we can determine for a marble starting from rest and coasting to the end of a ramp the following quantities:

the average rate at which position changes
a graph of rate of position change vs. clock time, which is linear, which starts from the origin and which has average vertical coordinate equal to the average rate at which position changes
the rate at which velocity changes, noting that velocity is rate of change of position

Rubber Bands

We can observe the force exerted by a rubber band vs. the length of the rubber band.

The slopes of a graph of force vs. length tell us the rate at which force changes with respect to length.

The areas under a graph of force vs. length tell us the energy stored in the rubber band (caveat: thermal energy is also involved)