The relation between marginals and averages

Let x be a variable (the amount of an input, the amount of output, ...) and let f (x) be a function of x (total product, total cost, ...). The marginal of f is its slope:

M(x) = f '(x).

The average of f is defined by

A(x) = f (x)/x.

A useful relationship between the marginal M and the average A is obtained by differentiating A:

A'(x) = f '(x)/x f (x)/x² = [M(x) A(x)]/x.

Thus

if M(x) > A(x) then A is increasing
if M(x) = A(x) then A is constant
if M(x) < A(x) then A is decreasing

That is: whenever M exceeds A, A is increasing, and whenever M is less than A, A is decreasing. If A is U-shaped for example, we get the picture

while if A has the shape of an inverted U then we get the picture

Average curves and marginal curves: interactive example

Instructions

Click once in the gray area below to fix one end of an average curve.
Click again to fix the other end.
Hold the mouse button down and drag the mouse. As you do so, two curves appear. The green one, which connects the first two points you clicked, is an average curve; the red curve is the corresponding marginal curve.
To change the colors of the curves, click on the small colored boxes at the bottom of the screen. (The boxes to the right of the A control the color of the average curve, while those to the right of the M control the color of the marginal curve.)
To draw a new curve, press the New button. (The currently displayed image will disappear when you press the mouse to fix the start of the next average curve.)
The Add button currently doesn't do anything.

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