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4-4 Average Value of a Function

A Mathematics. Academy. Production. 4-4 Average Value of a Function. Let’s get started: . You already know about functions and how to take the average of some finite set.

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4-4 Average Value of a Function

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  1. A Mathematics Academy Production 4-4 Average Value of a Function

  2. Let’s get started: • You already know about functions and how to take the average of some finite set. • Today we’re going to take the average over infinitely many values (those that the function takes on over some interval)…which means CALCULUS! • Before I show how to do this…let’s talk about WHY we might want to do this?

  3. Consider the following picture: • How high would the water level be if the waves all settled?

  4. Okay! So, now that you have seen that this an interesting question... Let’s forget about real life, and... Do Some Math

  5. Suppose we have a “nice”function and we need to find its average value over the interval [a,b].

  6. Let’s apply our knowledge of how to find the average over a finite set of values to this problem: First, we partition the interval [a,b] into n subintervals of equal length to get back to the finite situation: In the above graph, we have n=8

  7. Let us set up our notation: comes from the i-th interval Now we can get an estimate for the average value:

  8. Let’s try to clean this up a little: Since

  9. In a more condensed form, we now get: But we want to get out of the finite, and into the infinite! How do we do this? Take Limits!!!

  10. In this way, we get the average value of f(x) over the interval [a,b]: So, if f is a “nice” function (i.e. we can compute its integral) then we have a precise solution to our problem.

  11. Let’s look back at our graph:

  12. Remember the original MVT? If f is continuous on then at some point c in (a, b), When looking at anti-derivatives and definite integrals, we write it another way: Average Value Theorem (for definite integrals) So we just say that: Average Value of f (x)

  13. Average value of a function The average value of function f on the interval [a, b] is defined as Note: For a positive function, we can think of this definition as saying area/width = average height Example: Find the average value of f(x)=x3 on [0,2].

  14. So we’ve solved our problem! If I give you the equation and ask you to find it’s average value over the interval [0,2], you’ll all say

  15. Now we can answer our fish tank question!(That is, if the waves were described by an integrable function)

  16. The Mean Value Theorem for Integrals Theorem: If f is continuous on [a, b], then there exists a number c in [a, b] such that

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