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Taylor series and polynomials are mathematical tools used to approximate complex functions through derivatives at a point. They transform functions into infinite sums or finite partial sums, enhancing accuracy with higher degrees. Maclaurin polynomials, a subset centered at zero, simplify calculations near the origin. These methods are crucial in fields like physics and engineering for approximating values and understanding function behavior near specific points.

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## Introduction to Taylor Series

### Definition of Taylor series

Taylor series are infinite sums of terms calculated from the derivatives of a function at a single point

### Taylor polynomials

Definition of Taylor polynomials

Taylor polynomials are finite sums that approximate a function around a specific point

Accuracy of Taylor polynomials

The accuracy of a Taylor polynomial increases with its degree, which corresponds to the number of terms and derivatives used

### Construction of Taylor polynomials

Taylor polynomials are constructed to match the values and derivatives of the original function at the point of expansion

## Linear Approximation

### Definition of linear approximation

The linear approximation is the simplest form of Taylor polynomial, using the function's value and first derivative at a specific point

### Equation of the tangent line

The linear approximation represents the equation of the tangent line to the function at the point of expansion

### Application in differential calculus

The linear approximation is a foundational concept in differential calculus

## Higher-Degree Taylor Polynomials

### Definition of higher-degree Taylor polynomials

Higher-degree Taylor polynomials use more terms and derivatives to provide a more precise approximation of a function

### Maclaurin polynomials

Maclaurin polynomials are a type of Taylor polynomial centered at x=0

### Application in approximating functions

Higher-degree Taylor polynomials are useful for approximating functions, especially further from the point of expansion

## Applications and Limitations of Taylor Polynomials

### Examples of Taylor polynomial applications

Taylor polynomials can be used to approximate functions and compute difficult values, such as estimating square roots

### Precision and proximity to the center point

The accuracy of a Taylor polynomial depends on its degree and the proximity to the center point of expansion

### Local approximations

Taylor polynomials are powerful tools for function approximation, but their accuracy diminishes as the point of interest moves further from the center point