# 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) **

Here, f(a) represents the value of the function f(x) after substituting the value of a in it.

The function f´(x) is obtained by taking the first derivative of the function f(x). The value of f´(a) comes by putting the value of a in the derivative of the function f’(x).

The point a lies on the function f(x). The function f(x) is a non-linear function. It is a function with a degree greater than 1.

The calculator gives 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 **

It lies on the function $f(x) = x^4 + 6 x^{2}$. The calculator computes the linearization equation of the function f(x) at the point a.

### 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) **

The calculator also provides all the **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 **

The value for f(a) is obtained by putting the value of a in the non-linear function f(x) as follows:

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

**f(a) = 7 **

For f´(a), the first derivative of the function f(x) is given as follows:

\[ f´(x) = \frac{ d ( x^4 + 6 x^{2} ) }{ dx } = 4 x^{3} + 6 ( 2x) \]

\[ f´(x) = 4 x^{3} + 12x \]

Th value of a = -1 is placed into the function f´(x) to obtain f´(a) as follows:

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

**f´(- 1) = – 16 **

Putting the value of f(a), f´(a), and a in the equation of L(x) gives the linearization approximation at the point a on the curve.

**L(x) = f(a) + f’(a) (x – a) **

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

**L(x) = – 16x – 9 **

The calculator shows the **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 **

Also plot the curve f(x) and the linearization function L(x) in a 2-D plane.

### 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**

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

** y = 6x – 4 **

The calculator also displays the **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 **

Also plot the graph for f(x) and the linearization equation L(x).

### 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 **

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

**y = 16x – 15 **

The **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.*