Linearizing

TL;DR

• Approximates a non-linear function by a linear expression, typically via truncating a series expansion.
• Commonly implemented using the Taylor series and is useful when the non-linear form is hard to work with.
• The approximation is valid only near the expansion point (small values of the variable), where higher-order terms are negligible.

Definition

Linearizing is a mathematical method used to approximate a non-linear function with a linear one. This is useful when a non-linear function is difficult to work with, or when a linear approximation is desired for simplicity or ease of calculation.

Explanation

Linearizing typically uses a series expansion (such as the Taylor series) of the non-linear function and retains only the lower-order terms to form a linear approximation. The resulting linear expression is an approximation that is most accurate for small deviations around the expansion point, where higher-order terms in the series can be ignored.

Examples

Exponential function

The exponential function is defined as

$$y = a^x$$

A Taylor series expansion of this function is given in the source as

$$y = 1 + ax + (1/2!)ax^2 + ...$$

Keeping only the first two terms yields the linear approximation

$$y = 1 + ax$$

Example given in the source:

$$y = 2^x$$

Linearized (first two terms):

$$y = 1 + 2x$$

This linear approximation is stated to be valid for x values between -1 and 1.

Logarithmic function

The logarithmic function is defined as

$$y = \log(x)$$

A Taylor series expansion of this function is given in the source as

$$y = x - (1/2)x^2 + (1/3)x^3 + ...$$

Keeping only the first two terms (as presented in the source) yields

$$y = x - (1/2)x^2$$

This linear approximation is stated to be valid for x values between 0 and 2.

Use cases

• When a non-linear function is difficult to work with.
• When a linear approximation is desired for simplicity or ease of calculation.

Notes or pitfalls

• The linear approximation is valid only for small values of the variable (near the expansion point) where higher-order terms are small compared to the retained terms.
• Ignoring higher-order terms reduces accuracy as the variable moves away from the region where the approximation is valid.

Related terms

• Taylor series
• Exponential function
• Logarithmic function