Average of a Function

The concept of average is used daily in our lives whether we calculate our test marks or go to buy something in bulk. Let’s pretend you got the following test results in algebra class: 92, 81, 85, 94, 75 ,89and 80 The average of your exam scores determines your semester grade, and you want to know what to expect. By summing all of the scores and dividing by the number of scores, we can calculate the average. There are seven test scores in this case. Thus,

92+81+85+94+75+89+80  / 7 = 5967 ≈ 85.14

As a result, your average test grade is around 85.14.

This concept of average can also be used in other ways like ; suppose we have a function v(t) that tells us an object’s speed at every time t, and we wish to find the object’s average speed. We can’t apply the technique outlined above since the function v(t) can take on an endless number of values. Fortunately, we can find the average value of a function like this using a definite integral.

Let f(x) be a continuous function across the interval [a,b], and divide [a,b] into n subintervals of width  x=(b-a)/n. 

Calculate f(xi) for i=1,2,…,n for each subinterval by selecting a representative xi*. To put it another way, think of each f(xi*) as a sample of the function over each subinterval. The function’s average value can therefore be estimated as

f(x1*)+f(x2*)+f(x3*)+…………f(xn*) / n

This is the same formula that is used to calculate the average of discrete values.

Definition of Average of the Functions :

Average Value of a Function is a term used to describe the average value of a function.

If there is a continuous and positive function F(x) that is closed on the interval [a, b], then there must be a number x = c in that interval such that:

The integral of a function over a certain interval is equal to the average value of the function multiplied by the length of the interval.

The formula may be rewritten as: 

Geometrical Representation :

This means that there is a rectangle whose area exactly equals the area of the region under the curve y = f (x) The height of the rectangle is represented by the value of f(c), and the width is represented by the difference (b-a).

Guidelines of finding the Mean Value Theorem to a Function:

Find the function’s definite integral over the given interval:

Determine the interval’s length: a – b

To get f, use the formula (c)

Set the value of f(c) equal to the original function and solve for the variable to find the value(s) of c.

Note: If you get more than one value of c, use the one that falls within the interval.

An Example :

Q. Let’s find the average of the given cubic function.

f(x) = 2×2  on the interval [0,2].

Sol.

We use the formula 

favg= 1b-aabf(x) dx

In this case, the value of b is 2 and the value of an is 0. Substitute these values in the equation above as follows:

favg = 12-0022×2 d(x)

favg = 12022×2 d(x)

To integrate the function f(x) = 2×2, we must first get the function’s antiderivative. 2×33+c is the antiderivative of this function. Assume C = 0. The calculus fundamental theorem states:

In the antiderivative of the function, substitute 2 and 0 as follows:

2(23)3 – 2(03)3 = 163

As a result, our definite integral is 163. Replacing it,

As a result, the function’s average value in the interval [0,2] is 83

Conclusion :

The Mean Value Theorem for Integrals says that there exists a rectangle with the same area and width (w = b-a) for any definite integral  abf(x) dx. The average value of the function is the top of the rectangle, which intersects the curve, f(c). The definite integral, the width of the interval, and the average value of the function are all related by the formula:

Which states that the function’s average value is equal to the definite integral divided by the interval’s length.