Inside the context of calculus, the common fee of a function over an c language is given as follows:
$$ {f_{avg}} = \frac{1}{{b - a}}\int_{{\,a}}^{{\,b}}{{f\left( x \right)\,dx}} $$
in which; f(x) is a non-stop function and \(\left[ {a,b} \right]\) is the c program languageperiod wherein its continuity stays.
If you want to calculate average feature’s price manually, you can discover it pretty tough. however don't worry as we will be resolving multiple examples to clarify your concept. For greater example, you may also let this common value of a characteristic calculator to determine common value vital.
Example:
How to discover the average value of a characteristic as underneath:
$$ f(x) = 2x^2 + 5x $$
$$ interval = [1, 3] $$
Solution:
Finding the average value of a function given:
$$ \bar{f}= \frac{1}{b-a} \int_a^b f \left( x \right) dx $$
Substitute the given function and interval:
$$ \bar{f}= \frac{1}{3-1} \int_{1}^{3} \left( 2x^2 + 5x \right) dx $$
Simplify:
$$ \bar{f}= \frac{1}{2} \int_{1}^{3} \left( 2x^2 + 5x \right) dx $$
Evaluate the integral:
$$ \int \left( 2x^2 + 5x \right) dx = \frac{2x^3}{3} + \frac{5x^2}{2} $$
Apply the limits \( [1, 3] \):
At \( x = 3 \):
$$ \frac{2(3)^3}{3} + \frac{5(3)^2}{2} = \frac{54}{3} + \frac{45}{2} = 18 + 22.5 = 40.5 $$
At \( x = 1 \):
$$ \frac{2(1)^3}{3} + \frac{5(1)^2}{2} = \frac{2}{3} + \frac{5}{2} = 0.67 + 2.5 = 3.17 $$
Subtract the results:
$$ 40.5 - 3.17 = 37.33 $$
Find the average value:
$$ \bar{f}= \frac{1}{2} \times 37.33 = 18.665 $$
The average value of the given function is approximately 18.67.
For operating this loose calculator, you're required to offer the following enter values:
Input:
Output:
The free average price of function calculator does the subsequent calculations:
The common cost of the characteristic is used to examine graphical interpretation of the function and test its behaviour. moreover, you could additionally employ every other exponential growth calculator to peer any function’s graph along side all parameters described.
