Page 40 - How To Solve Word Problems In Calculus

Chapter 2

Rates of Change in the

Natural and Social Sciences

The average rate of change of a function f with respect to its in-
f (x) f (x + h) − f (x)
dependent variable x is the quotient = .
x h
f (x + h) − f (x)
The instantaneous rate of change is lim which,
h→0 h

by definition, is f (x).
The instantaneous rate of change of a function with respect
to its independent variable is the derivative of the function
with respect to that variable.

Most problems dealing with rates of change involve instanta-
neous rates of change, and the word “instantaneous” is usually
omitted. In these problems we simply compute the derivative
of the function and evaluate it at the point in question. If the
average rate is required, the word “average” will usually be
mentioned.
Graphically, the (instantaneous) rate of change of a func-
tion is the slope of the tangent line at a point. The average
rate of change over an interval is the slope of the secant line
connecting the points on the curve corresponding to the end-
points of the interval.

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