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Average Value of a Function Calculator + Online Solver with Free Steps
The Average Value of a Function Calculator is an online tool that is used to calculate the average value or the mean height of the graph of a function over a specified interval $[a, b]$. This calculator provides accurate results and presents the solutions in a matter of a few seconds.
The Average Value of a Function Calculator is an excellent tool that provides the average value of any type of function $f(x)$ over any given interval $[a,b]$. This tool makes use of the integral formula for determining the average value of the function $f(x)$.
What Is the Average Value of a Function Calculator?
The Average Value of a Function Calculator is a free tool available online that is used to determine the average value for all types of functions $f(x)$, over any specific interval between the points $a$ and $b$.
The Average Value of a Function Calculator is a very efficient tool that provides a detailed step-by-step solution. It simply takes the input from the user and with one click of the button, it presents the desired answer.
The Average Value of a Function Calculator makes use of the following formula for determining the average value for any function $f(x)$ in the interval $[a,b]$:
\[ f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx \]
The best feature of this calculator is its simple yet efficient user interface. This calculator only consists of 3 input boxes with designated titles to help the user in inserting the values. It also consists of a prominent button that says “Submit” which upon clicking presents the solution.
The Average Value of a Function Calculator is not only quick and efficient but it also always provides accurate results. Moreover, this speedy calculator only takes a few seconds to load the solution.
How To Use the Average Value of a Function Calculator?
You can use the Average Value of a Function calculator by entering the value of the function and specifying its limits. The Average Value of a Function Calculator is quite simple to use due to its extremely user-friendly interface. The calculator consists of a simple interface that allows the user to easily navigate through it without any confusion and obtain the desired results.
The interface of the Average Value of a Function Calculator consists of three input boxes. The first input box is titled “y” and it allows the user to enter the value of the function $f(x)$. For this input box, you can take help from the following interpretation:
\[ y = f(x) \]
The second and third input box corresponds to the limits of the integral, or in other words, the starting and ending point of the interval $[a,b]$ in which the function exists. The first input box is labeled with “Lower Limit” and it prompts the user to enter the starting value of the interval, i.e, $a$.
Similarly, the third and last input box is labeled with “Upper Limit” and it allows the user the enter the final or ending value of the interval, which is $b$.
Apart from the three input boxes, the interface of the Average Value of a Function Calculator consists of a “Submit” button which begins the solution.
For a better understanding of using the Average Value of a Function Calculator, a step-by-step guide is given below:
Step 1
Analyze the given function $f(x)$ and also the specified interval $[a.b]$ for the given function. There is no restriction on the type of function being used in the calculator.
Step 2
Now that you have analyzed your function and the interval, the next step is to fill in the input boxes. Enter the given function $f(x)$ in the first input box and then move on to the rest.
Step 3
After entering the value of the function $f(x)$ in the first input box, move on to the second and third input box and enter the lower limit and upper limit of the function respectively. Note that the lower limit corresponds to the starting point of the interval $a$ and the upper limit corresponds to the ending point of the interval $b$.
Step 4
Once all your input values have been added, simply click on the button that says “Submit.” Your solution will begin processing and within a few seconds, the Average Value of a Function Calculator will present the solution.
How Does the Average Value Of a Function Calculator Work?
The Average Value of a Function Calculator works by finding the area under the curve of the function. This is a very handy tool that works on the principle of integrals. This calculator makes use of the following formula for determining the average value of the function:
\[ f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx \]
The Average Value of a Function Calculator works on one of the most fundamental principles of calculus. To fully understand the working of this calculator, let’s revise the average value of a function concept.
What Is the Average Value of a Function?
The Average Value of a Function is the average value or the mean value of the height of the function $f(x)$ in any interval. For understanding this statement, let’s consider a function $f(x)$ specified over two points $a$ and $b$.
These two points $a$ and $b$ mark the starting and ending point of the interval for the function $f(x)$. Now imagine splitting the function $f(x)$ into multiple smaller intervals, each constituting a different height.
The average or the mean of these heights is termed as the Average Value for any Function $f(x)$. This can also be computed with the aid of the following formula:
\[ f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx \]
In this formula, $a$ refers to the starting point of the interval and similarly, $b$ refers to the ending point, where $f(x)$ is the given function.
Solved Example
Now that we have developed an understanding of the working of the Average Value of a Function Calculator, let’s take a look at an example.
Example 1
Consider a function specified over the interval $[1, 5]$. Find the average value of this function. The function is given below:
\[ y = x^{2} + 4\]
Solution
Before using the Average Value of a Function Calculator for determining the average value of this function $f(x)$, let’s first analyze the function. The function $f(x)$ is given below:
\[ y = x^2 + 4 \]
We also know the interval in which the function is specified which is:
\[ [1, 5] \]
Now, simply insert all the desired values into the designated input boxes. Insert the value of the function in the first input box and the values of $a$ and $b$ in the second and third input box respectively.
Once all these input values have been inserted, click on “Submit” to begin the solution. The calculator will take a few seconds for the solution to load. The calculator makes use of the following formula for determining the average value of the function $f(x)$:
\[ f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx \]
The calculator instantly provides a detailed solution for this function and interval. Firstly, the calculator substitutes the values in the formula, and then it begins the solution. The substitution of input values in the formula is shown below:
\[ f_{avg} = \frac{1}{4} \int_{1}^{5} (x^{2} + 4) dx \]
The average value of the function obtained is:
\[ f_{avg} = \frac {43}{3} \approx 14.33\]