Concept: Average value
By this point, you have learned quite a bit about functions in your mathematical career. In fact, given a function, \(f(x)\), you can evaluate it, graph it, find the exact/approximate values of the roots, discuss inflection points, investigate the extrema, and a whole lot more. The list of what you can do with a function at this point is pretty long (trust me, this page originally started with me listing out all your function skills... that sucked); however, one topic that you have missed out on is finding the average value of a function over a given interval.
Recall that the average value of a list of numbers is just the sum of that list of numbers divided by the number of elements in that list of numbers. This same idea is going to be used to create an average value for a continuous function over a given interval.
Let's consider the function \[f(x) = \cos{(x)} + 1\] on \([0,\pi/2]\). This function is displayed in the Wolfram Demonstrations applet below.
If we wanted to compute the average function value for \(f\), then we could imagine taking a finite number of sample points between \(0\) and \(\pi/2\). We would add up the function values at these points and divide by the number of points.
You can change the number of sample points in the app and you will see that the approximation of the average value becomes increasingly more accurate as the number of intervals increases. This should not be a surprise at this point in calculus, right?
By this point, you have learned quite a bit about functions in your mathematical career. In fact, given a function, \(f(x)\), you can evaluate it, graph it, find the exact/approximate values of the roots, discuss inflection points, investigate the extrema, and a whole lot more. The list of what you can do with a function at this point is pretty long (trust me, this page originally started with me listing out all your function skills... that sucked); however, one topic that you have missed out on is finding the average value of a function over a given interval.
Recall that the average value of a list of numbers is just the sum of that list of numbers divided by the number of elements in that list of numbers. This same idea is going to be used to create an average value for a continuous function over a given interval.
Let's consider the function \[f(x) = \cos{(x)} + 1\] on \([0,\pi/2]\). This function is displayed in the Wolfram Demonstrations applet below.
If we wanted to compute the average function value for \(f\), then we could imagine taking a finite number of sample points between \(0\) and \(\pi/2\). We would add up the function values at these points and divide by the number of points.
You can change the number of sample points in the app and you will see that the approximation of the average value becomes increasingly more accurate as the number of intervals increases. This should not be a surprise at this point in calculus, right?
Average Value of a Function from the Wolfram Demonstrations Project by Michael Largey
You should also notice that there is a computed value for the actual average.
"How did they get this actual value?"
Follow me on a journey into the next theorem and we will see... (cue creepy music)
"How did they get this actual value?"
Follow me on a journey into the next theorem and we will see... (cue creepy music)

