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Linear Approximation Calculator

Choose Type:

Enter a function: \(y = f(x)\) \(x(t)\) \(r(t)\) \(f(x,y)\)

\(y(t) \) = \(g(x,y)\)

Enter a point: \(x_0\) \(t \) =

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An online linear approximation calculator helps you to calculate the linear approximations of either parametric, polar, or explicit curves at any given point. The idea behind linearization or local linear approximation is to find a value of the function at the given point and evaluate the derivative to find the slope of entered points.

Here’s we know all about how to do linear approximation of different types of curves.

What is Linear Approximation?

In mathematics, use a linear approximation to estimate the value of a general function \(f(x)\) by using linear expressions. This is also known as tangent line approximation, which is the method of determining the line equation that is nearer estimation for entered linear functions at any given value of x.

So, the linear approximation calculator approximates the value of the function and finds the derivative of the function to evaluate the derivative to find slope with the help of the linearization formula.

Linear Approximation Formula:

In order to find the linear approximation of the function L at a point \(x = x_0\), where the slope of the line is m and a point that where the lines go through, (a and b).

So, the equation of the line is:

\[y – b = m(x–x_0)\]

In some problems, the values of b or m will not be given. So, use the linear approximation and differentials steps to calculate them.

Firstly, \(m = f ‘(a), \text{ Then, b} = f (b)\), where collect all these to find value of L using multivariable linear approximation calculator, the equation will be as follows:

$$y – b = m(x–x_0)$$

$$y = b + m(x–x_0)$$

$$m(x–x_0)$$

$$L (x) ≈ f (x_0) + f ^`(x_0) (x – x_0)$$

With local linearization formula, we can estimate the value of a function, f(x), near a point, \(x = x_0\).

However, an Online Derivative Calculator helps to find the derivative of the function with respect to a given variable.

How to do Linear Approximation?

A linear approximation equation can simplify the behavior of complex functions. The point x = k is the accurate linear approximation. As we get farther away from a point\( x = k\), the estimation becomes less accurate.

A simple curve linear approximation envies the direction of the curve. But, it does not predict the concavity of any curve.

Example:

Find the value of the \(f (8.3)\) by using the linear approximation \(x_0=2\), whose function is differentiable such as \(f (3)= 12, \text{ and} f’(3) = -2\).

Solution:

By using the linear approximation formula:

\[L (x) ≈ f (x_0) + f ^`(x_0) (x – x_0)\]

By putting the values in the formula, we get

$$L(x) = f (3) + f^’(3)(x – 3) = 18 – 2x$$

$$Hence, f(8.3)= 18-2(3.5)$$

$$f(8.3)= 18 – 7$$

$$f(8.3) = 11$$

Moreover, an Online Integral Calculator helps you to evaluate the integrals of the functions with respect to the variable involved.

How Linear Approximation Calculator Works?

The online linearization calculator will estimate the values of a given function by using linear approximation formula with the following steps:

Input:

  • First, choose the type of linear function for approximation from the drop-down menu.
  • Enter a function that requires finding the linear approximation.
  • Now, enter a point to find the value function at the given point.
  • In some cases, when you select the type parametric from the drop-down then put the value of t in to find x and y coordinates, that corresponds to the t.
  • Hit the calculate button.

Output:

The multivariable linear approximation calculator determines the following results:

  • Displays the linear approximation values with step-by-step calculations.
  • Evaluate the derivatives at the given point to find a slope.

Frequently Asked Questions for Linear Approximation:

How to Calculate the Amount of Error in Approximation?

In many areas, quantities are calculated on the basis of measurements. For instance, the region of a circle is computed by finding the radius of a circle.

When the Value of Linear Approximation Exact?

In a case, when the function \(y = f(x)\) is linear or constant then the value of linear approximation exact.

What is the Difference Between Propagated and Relative Error?

The propagated error occurs when we calculate quantity \(f (x)\) which we getting from the measurement error dx and the relative error given absolute error A for specific quantity, \frac{A}{A} is a relative error.

Last Points:

Well, an online linear approximation calculator allows you to determine the approximations of given functions, also finds the x and y coordinates and their derivative for evaluating the slope of given points with a linearization formula. Therefore manual calculations for estimation are a little tricky so we should use this linearization calculator to get rid of a lengthy and time-consuming process. Other than that, this online calculator equally beneficial for learners and professionals.

Reference:

From the source of Wikipedia: Period of oscillation, Electrical resistivity, Gaussian optics.

From the source of Paul’s Notes: Linear Approximations, Linearization of a function, Find the linearization of L(x) of the function.

From the source of Math24: Polynomial (Given An Initial Value), Cube Root (Given An Initial Value), Linear Approximation of a Function at a Point.