Understanding Linear Approximation (LA) is one of the easiest concepts in mathematics. Basically, the Linear Approximation of a function is a concept based on the ability of a linear function formed from a point in a curve to be a good representation of the general formula that represents the entire curve.
The Linear Approximation formula is based on a curve. It is actually an equation formed from a point in the curve. It, however, follows a tangent line that, as it continues to move towards infinity, will eventually reach the same space as the curve. So due to this possibility, the equation (point of the curve that follows a tangent line) is considered a good approximation (Linear approximation) of the entire curve equation.
Understanding the Linear Approximation formula
Getting the Linear approximation of the function is easy. Let y = F(x) be the general function of the curve and let L(x) be the linear approximation formula, function, or equation at the point ‘a’ of the curve. Then the approximation can be represented in the form (a, f(a)) since it is based on the point ‘a’ of the curve. Also, since the equation follows the tangent lie, then its slope would be its derivative
Also, the point-slope has a formula
which in the case of this equation, will be
A diagrammatic presentation of the formula is shown below
The straight line is the tangent line from the above diagram, while the black dot is the point where the linear approximation formula is formed. The linear approximation evaluation basically gives an approximated solution to the main function F(x).
Below is the Linear approximation formula.
Linear Approximation Formula
Let a General function that is continuously differentiable at least twice from a curve be represented as
Then its linear approximation function based on a tangent line very close to its curve is
Where;
F(x) is the general formula
L(x) is the LA formula
f(a) is the formula at the point a
The R in the general formula (fml 1) is the remainder that the Linear approximation formula (fml 2) will have to cut off as it is only an approximation and not the formula itself.
It is very important to understand that the remainder is very small and the Linear approximation formula, as long as it is based on a tangent line, is reliable enough to be a good approximation.
How to Calculate the Linear Approximation?
While there is a Linear Approximation Calculator, manually calculating it is very easy. Below are the steps on how to calculate LA
Determine the point a for which the LA is to be calculated.
Calculate the slope of the tangent line at the point. (calculate the derivate, F(a))
Solve the entire equation at the point a
Solving the Linear Approximation
Example 1
Find and Calculate the linear approximation of
Find the approximate value of 8 with the equation
Solution
From the example, the main function is
With the above solving;
With the above, the Linear Approximation is
L(x) = 1.14 + 0.11x
This means that the approximated equation for the formula
(b) Fine the approximate value of 8 with the equation
We can prove by application that L(x) is the LA of F(x) by finding 8
For F(x) at a=8,
now for the LA,
From the main and approximated equations, we can see that 2 and 2.052 are very similar. In fact, if the latter is approximated to its nearest possible whole, it would become 2
Therefore, it can be proven by application that our determined L(x) is a linear approximation of the F(x).
Example 2 Find the linear approximation of the function
Based on the above solutions,
The Linear Approximation Calculator
As can be seen from the two examples, the basic idea of the LA is pretty straightforward and does not require any tough computation. You may need basic knowledge of differentiation and the different methods for obtaining derivatives of the first orders.
All examples you will get to handle are similar to the two discussed in this article, and you only need to follow them to get your result.
For non-class purposes, online calculators for determining LAs can be found in various trustworthy resources. They can be very helpful in situations when calculations are not very convenient.
Conclusion
The Linear Approximation of a function is very easy to calculate. The process simply requires knowing when the approximation is to be made and how to find the first-order derivative using different differentiation methods. One of the major reasons for calculating the LA is very important is its simplicity compared to that of the main function. It saves time and does not require any technicality whatsoever.
Two examples were considered in this article, and they followed the same method of calculation. This means that you can easily get the calculation of any LA as long as there is a point on which a tangent line is based.
