STANDARD GRAPHING APPLICATIONS

A plot of p vs. q where p and q are any two physical variables can yield up to 5 pieces of information that may be useful.  If the plot is a straight line, then the slope or the first derivative, dp/dq, is constant and the second derivative or curvature dp/dq is zero.  If the plot’s line is curved, then the first derivative is variable and the second derivative is non-zero.  We can infer whether the curvature (second derivative) is positive or negative by seeing if the curve is concave up or down.  The area under the curve or plot is the integral,  evaluated at the two end points of the curve.  For a straight line, the vertical intercept is useful for writing the equation of the graph in the form of y = mx + b as well as for general information.  The horizontal intercept may have some efficacy.  

Slope				dp/dq

Curvature			d/dq

Vertical Intercept		p

Horizontal Intercept		q

Area under Curve		dq

KINEMATIC APPLICATIONS

v = dx/dt,  a = dv/dt,  j = da/dt;	a =  + a,  v =  + v,  x =  + x

 For a plot of position (x) vs. time (t) , the slope is the velocity (v).  The curvature is the acceleration (a), the vertical intercept is the initial position (x), and the horizontal intercept is the first time when the position was zero (at the origin).  The area under the curve is not used.

For a plot of velocity (v) vs. time (t), the slope is the acceleration (a).  The curvature is the jerk (j = da/dt), the area under the curve is the displacement (), and the vertical intercept is the initial velocity (v).  The horizontal intercept is the first time the velocity is zero (not moving.)