The average of a function

If you read the newspaper, you will frequently see averages used to convey quantitative information in a concise way. Remember how an average is formed: if n students write an exam and their scores are denoted by $ s_i $ , then the average score on the exam is

\[ \bar{s} = \frac 1n \sum_{i=1}^{n} s_i \]


Average temperature

Now an important average we hear discussed within the context of global warming is Earth's average temperature or perhaps the average temperature in Vancouver. While this kind of an average still makes sense to us, it is something quite different. There are not a discrete set of temperatures which we can add but rather, the temperature varies continuously from one time to the next.

One way we could try to compute the average temperature in Vancouver for one day is simply to average the temperature at midnight and the temperature at noon. If we call $ T(t) $ the temperature t hours after midnight, our average temperature would be

\[ \frac{T(0) + T(12)}{2} \]

While this is easy to do, it somehow doesn't give us an accurate feel for what we expect the average temperature to be. For instance, if the temperature is 5o C. at midnight and at noon, we would find an average temperature of 5o C. However, if a cold front comes in that afternoon and lowers the temperature to 0o, its effect would not be seen in the average temperature we compute.

So we could try to find a better average by averaging the hourly temperatures. This would be

\[ \frac{T(0) + T(1) + T(2) + \ldots + T(23)}{24} = \sum_{t=0}^{23} \frac{T(t)}{24} \]

Of course, now that you see this idea, you might think that an even more accurate average could be found by reading the temperature every minute and averaging these readings. If there are n minutes in a day and $ t_i $ represents the time after i minutes have passed, we would have

\[ \sum_{i=0}^{n-1} \frac{T(t_i)}{n} \]

Clearly, we could continue in this way by breaking the day into smaller and smaller pieces--that is, using a larger value for n and computing the average. We will call this the average value of the function $ T(t) $ :

\[ \bar{T} = \lim_{n\to\infty} \sum_{i=0}^{n-1} \frac{T(t_i)}{n} \]


More generally

For a function $ f(x) $ , we can define the average of $ f $ over an interval $ [a,b] $ by breaking the interval into n pieces and calling $ x_i $ the resulting points. That is, if $ \Delta x = \frac{b-a}{n} $ measures the width of each sub-interval, then $ x_i = a + i\Delta x $ . The average value of $ f $ is then

\[ \bar{f} = \lim_{n\to\infty} \sum_{i=0}^{n-1} \frac{f(x_i)}{n} \]

This should seem somewhat similar to how we computed the area under the graph of a function. In fact, we can rearrange the definition of the average above to reflect this similarity more clearly.

Since we have $ \Delta x = \frac{b-a}{n} $ , this means that $ \frac 1n = \frac{\Delta x}{b-a} $ and hence the average value above is

\[ \bar{f} = \frac 1{b-a} \lim_{n\to\infty} \sum_{i=0}^{n-1} f(x_i)\Delta x \]

This is precisely the expression we found for the area under graph of a function divided by the width of the interval $ b-a $ . This gives us a geometric interpretation of the average value of the function: if we write

\[ (b-a)\bar{f} = \lim_{n\to\infty} \sum_{i=0}^{n-1} f(x_i)\Delta x \]

we see that $ \bar{f} $ is the height of the rectangle whose area is the same as the area under the graph of the function. The following demonstrations will show you this for two of the functions whose area we computed.


Average velocity

Let's suppose that we know $ v(t) $ , the velocity of an object as a function of time t. From what we have seen above, we know that the average velocity over the time interval from, say, t = 0 to t = T is

\[ \bar{v} = \frac 1T \lim_{n\to\infty} \sum_{i=0}^{n-1}v(t_i)\Delta t \]

Remember that if we multiply the average velocity over a time interval by the length of the time interval, we have the distance travelled during that time interval. This tells us that

\[ \bar{v}T = d = \lim_{n\to\infty} \sum_{i=0}^{n-1}v(t_i)\Delta t \]

In other words, the distance travelled is obtained through this process of summing. Of course, the velocity is the derivative of the distance travelled so we may think that we have reversed the differentiation process through this process. This should remind you of our discussion of area since there we found that the area function, obtained through a summing process, had as its derivative the function we were summing.

We are being led to an inescapable conclusion: that the process of summing somehow undoes the process of differentiation. We will now attempt to make this more clear.