In Exercises 15-18, use a finite sum to estimate the average...

In Exercises $15-18,$ use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$f(x)=1 / x \text { on } [1,9]$$

Key Concepts

Average Value of a Function

The average value of a function over an interval is a concept that involves integrating the function over that interval and then dividing by the length of the interval. This gives a single number that represents the central or 'average' output of the function on that interval, which is useful for summarizing the function's behavior over a range.

Partitioning an Interval

Partitioning an interval involves dividing a continuous interval into a number of subintervals. This division is a fundamental step in many numerical methods, as it allows the continuous problem to be approximated by discrete calculations over each smaller segment, facilitating approximation of integrals or averages.

Midpoint Rule

The midpoint rule is a numerical integration technique where the function is evaluated at the midpoint of each subinterval. By taking the value at the center of each subinterval, one can construct an estimate that tends to give a better approximation of the average behavior of the function over that subinterval compared to simply using an endpoint, particularly when the function is smooth.

Finite Sum Approximation

Finite sum approximation refers to the process of replacing an integral or continuous sum with a discrete sum over a finite number of terms. This is particularly useful when exact integration is difficult or impossible, allowing for practical computation of quantities like integrals or average values using sums over partitioned subintervals.

Transcript

00:01 So this question is asking us to estimate the average value of f on the given interval by partitioning the interval into four subintervals of equal length and evaluating f at the sub interval of midpoints.

00:14 So we want to partition this interval into four sub intervals.

00:17 So first we want to find our delta x, how much is just going to be changing.

00:26 So to find that, we're going to take our end points and subtract them.

00:29 So 9 minus 1, divided by a number, sub -intervals 4, and so we get 2.

00:38 So this is going to be the distance in between the sub -intervals.

00:43 So i'm going to start at my end point 1.

00:46 I'm going to add 2 to that and get 3.

00:50 And then keep continuing in that same manner until i hit 9, my other endpoints.

01:06 Okay, so the next part says we want to evaluate f at the sub -interval midpoints.

01:11 So for each of these, i'm going to want to find my midpoint.

01:20 So i'm going to just find the middle number between, which this is simple because there's only one number there.

01:28 So between one and three, it's just two.