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Mathematical Expression Editor
In this section we interpret the derivative as an instantaneous rate of change.
1 Rates of Change
We begin by comparing the notion instantaneous rate of change to average rate of
change. For two points and with on the graph of a function , the slope of the secant
line connecting them is This slope can be interpreted as the average rate of change
of with respect to over the interval .
The instantaneous rate of change of with respect to at is denoted by and it is
obtained by letting approach in the formula for Graphically, is the slope of the
tangent line to the graph of at . Generally speaking, represents the average rate of
change of relative to whereas represents the instantaneous rate of change of
relative to .
Examples of Rates of Change
example 1
If is the distance traveled by an object as a function of time, then is the average
speed of the object and is the instantaneous speed of the object.
If is the volume of water in a tank as a function of time, then is the average rate
that water is entering (or leaving, if this is negative) the tank and is the
instantaneous rate that water is entering (or leaving) the tank.
example 2 If is the cost of producing units then represents the instantaneous rate of
change of cost with respect to the number of units produced. This is called the
marginal cost of production.
example 3 If represents population at time , then is the average growth rate of the
population and is the instantaneous growth rate of the population. If the
population of a certain bacteria is 50 (thousand) and it doubles every hour, then
what is the population after 3 hours? How fast is the population growing
at hours? First, from the description of the population, we have . Hence
the population after 3 hours is given by (thousand) bacteria. The growth
rate at time is given by Computing the derivative, we have and hence,
example 4 If represents the quantity of a (medical) drug in the bloodstream at time ,
then is the average rate of change of the quantity in the bloodstream and is the
instantaneous rate of change of the quantity in the bloodstream.