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Definitions

The average rate of change of A with respect to B is

average rate = (change in A)/(change in B).

The average velocity of an object over an interval is its average rate of change of position with respect to clock time.

The average acceleration of an object over an interval is its average rate of change of velocity with respect to clock time.

Graphs on Coordinate Systems

A right-handed Cartesian coordinate system consists of two axes at right angles, with the positive axes of one oriented at 90 degrees 'to the right' of the other. 

• The first axis, traditionally the vertical axis, is often labeled the y axis. 

• The second is traditionally oriented horizontally and is often labeled the x axis.

The two axes of the Cartesian coordinate system can represent any two quantities. 

• For example they can represent velocity and clock time. 

• Or they can represent position and clock time. 

• They can represent force and position. 

• They can represent velocity and position.

They can represent any two quantities that help us understand a situation or solve a problem.

The graph is said to be a graph of y vs. x.  In the traditional orientation (vertical-horizontal) the variable in front of 'vs.' is represented along the vertical axis, and the variable following the 'vs.' is represented along the horizontal axis. 

• A graph of v vs. t, for example, represents the velocity v along the vertical axis and the clock time t along the horizontal axis.

Usually when we graph one quantity vs. another, one quantity will be dependent on the other. 

• Traditionally, in cases where we have an independent and the dependent variable, the first quantity (the one in front of the 'vs.') is the dependent variable, and the second quantity (the one following the 'vs.') is the independent variable.

When making a table representing the values of the dependent variable vs. the values of the independent variable, the first column of the table lists the values of the independent variable and the second column lists the values of the dependent variable. 

• Thus, for example, if we have a graph of y vs. x, it is the x variable that is listed in the first column and the y variable in the second. 

• So the columns are listed in the reverse order of the 'y vs. x' reference.

Points, line segments and trapezoids

A point on an x-y coordinate system has two coordinates which are obtained by projecting the point to the x-axis and to the y axis, in a way to which you should be accustomed.

Two points on a Cartesian coordinate system will represent two values of the 'horizontal' coordinate and two values of the 'vertical' coordinate. 

• These two points may be connected by a straight line segment. 

• The midpoint of this segment will represent the 'straight-line average' of the 'horizontal' and 'vertical' coordinates. 

• The 'straight-line average' of two coordinates is easily obtained by adding the two coordinates and dividing by two. 

• As long as the two 'vertical' coordinates are different, the line segment has a slope, equal to the 'rise' between the two points divided by the 'run' between them. 

• Since the 'rise' represents the change in the 'vertical' quantity and the 'run' represents the change in the 'horizontal' quantity, this slope represents the average rate of change of the 'vertical' quantity with respect to the 'horizontal' quantity, over the interval represented by the line segment.

If the two points lie above the 'horizontal' axis, then the line segment and two projection lines from the points to the horizontal axis, along with a corresponding segment of the horizontal axis, form a trapezoid. 

• This trapezoid has two 'altitudes', corresponding to the projection lines, and a 'width', corresponding to the length of the interval along the horizontal axis. 

• The trapezoid can easily be rearranged to form a rectangle by making a horizontal cut starting from the midpoint of the line segment, trimming off a triangle that can be rotated 180 degrees and 'pasted' onto the remaining region to form the rectangle. 

• The dimensions of the rectangle are the width of the original trapezoid and the vertical coordinate of the midpoint (represeting the width and altitude of the newly formed rectangle). 

• The area of the trapezoid therefore represents the product of the 'average altitude' and the 'width' of the trapezoid.

The 'average altitude' of the trapezoid represents the average value of the 'vertical' quantity, and its width represents the change in the value of the 'horizontal' quantity. 

• If the product of these two quantities has a meaningful interpretation, then since this product represents the area of the trapezoid, that interpretation applies to the area.

For example, if the vertical quantity represents the velocity of an object and the horizontal quantity represents clock time, then the altitudes of the trapezoid represents the initial and final velocities over the interval. 

• The average altitude therefore represents the average velocity (again assuming that the graph is in fact a straight line between the points), and the width represents the change in the clock time. 

• Since average velocity is the average rate of change of position with respect to clock time, i.e., (change in position)/(change in clock time), when we multiply the average altitude in the width we find

ave altitude * width = (change in position)/(change in clock time)*(change in clock time) = change in position. 

 

So the area represented by the region lying 'beneath' the graph of velocity versus clock time represents change in position.

• In the same situation, with the vertical quantity representing velocity and the horizontal quantity representing clock time, the rise of the graph between two points represents the change in velocity and the run represents the change in clock time. 

• The slope therefore represents

slope = (rise)/(run) = (change in velocity)/(change in clock time). 

 

By the definition of average rate of change, this is the average rate of change of velocity with respect to clock time, or the average acceleration for this interval.

Question:  If the vertical quantity represents the position of an object and the horizontal quantity represents clock time, then what does each of the following represent:

• The altitudes of the trapezoid defined by two points.

• The average altitude of the trapezoid defined by two points.

• The rise between the two points.

• The run between the two points.

• The area of the trapezoid.

• The slope between the two points.

• The midpoint of the corresponding interval of the horizontal axis.

• The midpoint of the corresponding interval of the vertical axis.

Question:  If the vertical quantity represents the acceleration of an object and the horizontal quantity represents clock time, then what does each of the following represent:

• The altitudes of the trapezoid defined by two points.

• The average altitude of the trapezoid defined by two points.

• The rise between the two points.

• The run between the two points.

• The area of the trapezoid.

• The slope between the two points.

• The midpoint of the corresponding interval of the horizontal axis.

• The midpoint of the corresponding interval of the vertical axis.

Question:  If the vertical quantity represents the force exerted on an object and the horizontal quantity represents the position of the object, then what does each of the following represent:

• The altitudes of the trapezoid defined by two points.

• The average altitude of the trapezoid defined by two points.

• The rise between the two points.

• The run between the two points.

• The area of the trapezoid.

• The slope between the two points.

• The midpoint of the corresponding interval of the horizontal axis.

• The midpoint of the corresponding interval of the vertical axis.

Question:  If the vertical quantity represents the velocity of your car and the horizontal quantity represents clock time, then what does each of the following represent:

• The altitudes of the trapezoid defined by two points.

• The average altitude of the trapezoid defined by two points.

• The rise between the two points.

• The run between the two points.

• The area of the trapezoid.

• The slope between the two points.

• The midpoint of the corresponding interval of the horizontal axis.

• The midpoint of the corresponding interval of the vertical axis.

Question:  If the vertical quantity represents the landing position of a ball rolling off a ramp and the horizontal quantity represents the slope of the ramp, then what does each of the following represent:

• The altitudes of the trapezoid defined by two points.

• The average altitude of the trapezoid defined by two points.

• The rise between the two points.

• The run between the two points.

• The area of the trapezoid.

• The slope between the two points.

• The midpoint of the corresponding interval of the horizontal axis.

• The midpoint of the corresponding interval of the vertical axis.