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Derivatives

Derivatives

Average Rate of Change

A rate is a value that expresses how one quantity changes with respect to another quantity. For example, a rate in "miles per hour" expresses the increase in distance with respect to the number of hours we've been driving.

If we drive at a constant rate, the distance we travel is equal to the rate at which we travel multiplied by time:

Dividing both sides by time, we have

Sample Problem

If we drive at 50 mph for two hours, the distance we'll travel is 50 mph × 2 hrs = 100 miles.

Sample Problem

 If we drive at a constant speed for 3 hours and travel 180 miles, we must have been driving

.

In real life, though, we don't drive at a constant rate. When we start our trip through Shmoopville, we first climb into the car, traveling at a whopping 0 miles per hour. We speed up gradually (hopefully), maybe need to slow down and speed up again for traffic lights, and finally slow down back to a speed of 0 when reaching our destination, The Candy Stand. We can still divide the distance we travel by the time it takes for the trip, but now we'll find our average rate:

To calculate the average rate of change of a dependent variable y with respect to the independent variable x on a particular interval, we need to know

  • the size of the interval for the independent variable, and
      
  • the change in the dependent variable from the beginning to the end of the interval.

Depending on the problem, we may also need to know

  • the units of the independent and dependent variables.

Then we can find 

The average rate of change of y with respect to x is the slope of the secant line between the starting and ending points of the interval:

Relating this to the more math-y approach, think of the dependent variable as a function f of the independent variable x. Let h be the size of the interval for x:

and let a be one endpoint of the interval, so the endpoints are a and a + h, with corresponding y-values f(a) and f(a + h):

Then the slope of the secant line is

We also write this as

.

This is the definition of the slope of the secant line from (a, f(a)) to (a + h, f(a + h)).

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