Interpreting the Derivative

Now that we know what the derivative is and how to calculate it, let's take a look at how we can interpret it in contexts.

Example 1

Let P(t) be the average monthly Social Security benefit for men, measured in dollars, where t is measured in years since 1940.

Interpret the meaning of P(60) = 959.79.

In the year 2000, the average Social Security benefit for men was $959.79.

Interpret the meaning of P'(60) = 36.33.

In the year 2000, the average Social Security benefit for men was increasing by $36.33 per year.

Example 2

The temperature in degrees Fahrenheit of a cup of coffee placed on a desk is given by f(t) where t is the minutes since the coffee was poured into the cup.  Interpret the following.  Give units for each.

interp_example.png

1. 5 minutes after the coffee was placed on the desk the temperature of the coffee is 90 degrees farenheit.

2. The temperature of the coffee will be 85 degrees farenheit 7 minutes after it was placed on the desk.

 

3. 5 minutes after the coffee was placed on the desk the temperature was decreasing by 3 degrees farenheit per minute.

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