# Linearization Calculator + Online Solver With Free Steps

The**Linearization Calculator**is used to compute the linearization of a function at a given point. The point a lies on the curve of the function f(x). The calculator provides a

**tangent line**at the given point a on the input curve. Linearization is an essential tool in

**approximating**the curved function into a linear function at a given point on the curve. It computes the

**Linearization function,**which is a tangent line drawn at the point a on the function f(x). The Linearization function L(x) of a function f(x) at a given point a is obtained by using the

**formula**as follows:

**L(x) = f(a) + f´(a) (x – a) **

**slope-intercept form**of the linearization function L(x) and also provides a plot for the function f(x) and L(x) in the x-y plane.

## What Is a Linearization Calculator?

**The Linearization Calculator is an online tool that is used to calculate the equation of a linearization function L(x) of a single-variable non-linear function f(x) at a point a on the function f(x).**The calculator also plots the

**graph**of the non-linear function f(x) and the linearization function L(x) in a 2-D plane. The linearization function is a tangent line drawn at the point a on the curve f(x). The Linearization formula used by the calculator is the

**Taylor series**expansion of

**first**order. The

**Linearization Calculator**has a wide range of usage when dealing with non-linear functions. It is used to approximate the

**non-linear**functions into

**linear**functions that change the shape of the graph.

## How To Use the Linearization Calculator

The user can follow the steps given below to use the Linearization Calculator.### Step 1

The user must first enter the function f(x) for which the linearization approximation is required. The function f(x) should be a**non-linear function**with a degree greater than one. It is entered in the block titled, “

**linear approximation of**” in the calculator’s input window. The calculator takes the function as a

**one-variable**function of x by default. The user should not use another variable in the non-linear function. The calculator uses the function as given below by

**default**for which the linearization approximation is calculated: \[ f(x) = x^4 + 6 x^{2} \] It is a non-linear function with a

**degree**of 4.

### Step 2

The user must now enter the**point**at which the linearization approximation is needed. This point lies on the curve or the non-linear function f(x). The point is named as a by the calculator. It is entered in the block labeled ”

**when a=**” in the calculator’s input window. This is the point at which the

**tangent line**is drawn on the input curve which gives the linear approximation. The calculator sets the value of a by

**default**as:

**a = – 1 **

### Step 3

The user must now enter the “**Submit**” button for the calculator to compute the output. If a

**two-variable**function f(x,y) is entered in the block “linear approximation of”, the calculator gives the signal “Not a valid input; please try again”. If the value of a entered by the user is

**incorrect**or not an integer, the calculator again gives the signal that the input is not valid.

### Output

The calculator processes the input data and computes the output in the**three**windows given below.

#### Input Interpretation

The calculator interprets the input and displays it in this window. For the**default**example, it displays the input as follows: \[ tangent \ line \ \ to \ y = x^4 + 6 x^{2} \ \ at \ a = – \ 1 \] It shows that the calculator will compute the

**equation**for the

**tangent**line on the non-linear function at the point a on the curve. The user can

**verify**the entered input from the input interpretation window whether the calculator has taken the input according to the user’s requirements.

#### Result

The Result’s window shows the**linear approximation**of the function f(x) at the point a on the curve. The calculator computes an equation which is the “slope-intercept form” of the linearization function L(x). This

**equation**is obtained by using the Linearization formula for the linearization function L(x), that is:

**L(x) = f(a) + f´(a) (x – a) **

**mathematical steps**required for the particular problem by clicking on “Need a step-by-step solution for this problem?” For the default example, the mathematical steps are given as follows. For the

**default example**, the function f(x) and the point a is given as: \[ f(x) = x^4 + 6 x^{2} \]

** a = – 1 **

**f(a) = f(- \ 1) = $(- \ 1)^{4}$ + 6 $(- 1)^{2}$ = 1 + 6 **

**f(a) = 7 **

** f´(- 1) = 4 $(- 1)^{3}$ + 12(- 1) = 4(- 1) – 12 = – 4 – 12 **

**f´(- 1) = – 16 **

**L(x) = f(a) + f’(a) (x – a) **

**L(x) = 7 + (- 16) ( x – (- 1) ) = 7 – 16x – 16 **

**L(x) = – 16x – 9 **

**Result**for the linear approximation as follows:

** y = – 16x – 9**

#### Plot

The Linearization Calculator also provides a**graph**plot for the linearization approximation of f(x) at the point a in a x-y plane. The plot shows the non-linear

**curve**of the function f(x). It also displays the linear approximation at the

**point**a, which is a

**tangent line**drawn at the point a on the curve.

## Solved Examples

Here are some of the examples solved through the Linearization Calculator.### Example 1

For the non-linear function: \[ f(x) = 2 x^{3} \] Calculate the linear approximation of the function f(x) at the point a on the curve given as:**a = 1 **

### Solution

The user must first enter the non-linear function f(x) and the point a in the input window of the Linearization Calculator. After pressing “**Submit**”, the calculator opens the output window which shows the three windows as given below. The

**Input Interpretation**window shows the entered input by the user. For this example, it displays the input as follows:

**tangent line to y = 2 $x^{3}$ at a = 1**

**Result’s**window displays the equation for the linear approximation L(x) of the function at the given point as follows:

** y = 6x – 4 **

**plot**for the function f(x) and the linearization equation L(x) as shown in figure 1. The tangent line represents the linear approximation shown in figure 1.

### Example 2

Compute the linearization equation for the function: \[ f(x) = 4x^{2} + 1 \] At the point:**a = 2 **

### Solution

The function f(x) and the point a is entered in the input window of the Linearization Calculator. The user submits the input data and the calculator first shows the**Input Interpretation**as follows:

**tangent line to y = 4 $x^{2}$ + 1 at a = 2 **

**Result’s**window displays the linearization equation as follows:

**y = 16x – 15 **

**Plot**for the non-linear function f(x) and the linearization equation L(x), which is a tangent line drawn at the point a on the curve is shown in figure 2 given below.

*All the images are created using Geogebra.*

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