The elementary method of tracing (or plotting) a curve whose equation is given in rectangular coordinates, and one with which the student is already familiar, is to solve its equation for (or ), assume arbitrary values of (or ), calculate the corresponding values of (or ), plot the respective points, and draw a smooth curve through them, the result being an approximation to the required curve. This process is laborious at best, and in case the equation of the curve is of a degree higher than the second, the solved form of such an equation may be unsuitable for the purpose of computation, or else it may fail altogether, since it is not always possible to solve the equation for or .
The general form of a curve is usually all that is desired, and the Calculus furnishes us with powerful methods for determining the shape of a curve with very little computation.
The first derivative gives us the slope of the curve at any point; the second derivative determines the intervals within which the curve is concave upward or concave downward, and the points of inflection separate these intervals; the maximum points are the high points and the minimum points are the low points on the curve. As a guide in his work the student may follow the
Rule for tracing curves. Rectangular coordinates.
If the calculated values of the ordinates are large, it is best to reduce the scale on the -axis so that the general behavior of the curve will be shown within the limits of the paper used. Coordinate plotting (graph) paper should be employed.
david joyner 2008-11-22