Linear Approximation Formula | Linear Interpolation & Regression Formula

Linear Approximation Formula

For the function of any given value, we have to determine the closest estimation value of a function and it is given by the Linear approximation Formula. The other name for this mathematical concept is tangent line approximation or approximate tangent value of a function.

This is the process to find the equation of a line or closest approximation of a given function for the given value. It is used in Trigonometry or optics. This is an infinitely small like where the curve is almost straight and imitates the function. For the real-valued function F (x), the linear approximation value is given as below –

\[\LARGE f(x) = f(x_{0}+f'(x_{0}(x-x_{0})+R_2\]

where R2 is the remainder term. The linear approximation, then, is given by

\[\LARGE f(x)\approx f(x_{0}+f'(x_{0}(x-x_{0})\]

f(x0) is the value of f(x) at x = x0.
f'(x0) is the derivative value of f(x) at x = x0.

The approximation value is almost equal to the tangent line at a given point a?You could also opt for Euler’s method to find the solution for linear approximation values. In the end, only the closeness of the tangent line matters and tangent values can be calculated with specific formulas in mathematics. The other popular name for linear approximation is linearization.

So, how can you calculate the tangent at a given point a. There are two main things to remember here. First is the slope of a line and second is the particular point (a, b) line passes through. In this case, the equation of the line could be given as –

\[\LARGE Y – b = m(X-a)\]

Here, values of b and m will not be given automatically but you have to calculate it yourself only with formula derivatives.

Linear Interpolation Formula

Interpolation is a popular statistical tool in mathematics that is used to calculate the estimated values between two points. Here, we will discuss the formula for the concept. Interpolation is also used in science, businesses, or many other fields too. Every time when you have to predict values between data points, linear interpolation formula is helpful.

Take an example of Tomato plant planted by a curious gardener. He wanted to know how much the plant has grown on the third day? If it is 6mm tall on the third day then its height should increase in a linear pattern in coming days too. It means the growth chart should create a straight line when plotting data on the graph. Here, the linear Approximation formula comes in handy to solve the issue.The Linear Interpolation Formula in mathematics is given as below.

\[\large y=y_{1}+\frac{\left(x-x_{1}\right)\left(y-y_{1}\right)}{x_{2}-x_{1}}\]

x1 and y1 are the first coordinates
x2 and y2 are the second coordinates
x is the point to perform the interpolation
y is the interpolated value.

Linear Regression Formula

Linear regression is the highly common and predictive analysis technique used by the mathematicians or scientists. In this case, there are two variables, one is taken as the explanatory variable, and the other is taken as the dependent variable. With the help of a linear regression model, you can always relate the weights of individuals with heights.

There are a variety of linear regression techniques available in the mathematical world. These are -simple linear regression. Multiple linear regression, Logistic regression, Ordinal regression, Multinomial regression, Discriminant analysis etc. The Formula for linear regression equation in mathematics is given by:

\[\large y=a+bx\]

a and b are given by the following formulas:

\[\large b\left(slope\right)=\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{n\sum x^{2}-\left(\sum x\right)^{2}}\]

\[\large a\left(intercept\right)=\frac{n\sum y-b\left(\sum x\right)}{n}\]

x and y are two variables on regression line.
b = Slope of the line.
a = y-intercept of the line.
x = Values of first data set.
y = Values of second data set.