Taylor and Maclaurin Polynomials

Linear Approximation and Tangent Line

The linear approximation states that if a function f(x) is differentiable at a point x = a, then near x = a it can be approximated by its tangent line:

• Tangent line equation:

• This is a first-order approximation, using only the value and first derivative at a.

Key Point: The tangent line provides a simple local approximation, but higher accuracy can be achieved using polynomials of higher degree.

Taylor Polynomial: Definition and Construction

A Taylor polynomial of degree n for f(x) at x = a is a polynomial p_n(x) that matches f(x) and its first n derivatives at x = a:

• ... up to

The general form is:

• In sigma notation:

Key Point: The Taylor polynomial provides a polynomial approximation to f(x) near x = a, matching derivatives up to order n.

Maclaurin Polynomial: Special Case

The Maclaurin polynomial is a Taylor polynomial centered at a = 0:

Key Point: Maclaurin polynomials are useful for approximating functions near the origin.

Examples of Taylor and Maclaurin Polynomials

• Example 1: For , the Maclaurin polynomial of degree n is

• Example 2: For , the Maclaurin polynomial of degree n is

• Example 3: For , the Maclaurin polynomial contains only even powers of x.

• Example 4: For , the Maclaurin polynomial contains only odd powers of x.

• Example 5: For , the Taylor polynomial of degree n at x = 2 is

Key Point: The form of the polynomial depends on the function and the center a.

Even and Odd Functions in Maclaurin Series

• If f(x) is even (), its Maclaurin polynomial contains only even powers of x.

• If f(x) is odd (), its Maclaurin polynomial contains only odd powers of x.

• Examples: is even, is odd.

Maclaurin Polynomials for Sine and Cosine

By repeated differentiation, we find the derivatives cycle every four steps:

Thus, the Maclaurin polynomial for (degree n even) is:

For :

Taylor's Theorem

Statement and Remainder Term

Taylor's Theorem provides a formula for the error (remainder) when approximating f(x) by its Taylor polynomial:

• Let be the Taylor polynomial of degree n for f(x) at x = a. Then:

Key Point: If is small, the Taylor polynomial is a good approximation to f(x).

Applications and Approximations

• Taylor and Maclaurin polynomials are used to approximate function values, especially when direct computation is difficult.

• Example: Approximating using the Maclaurin polynomial for is not accurate, since is far from 0. Better results are obtained when x is close to the center a.

• For , the Taylor polynomial centered at gives a good approximation.

The Taylor polynomial of degree 4 for at is:

Degree of Approximation and Error Analysis

• The degree n required for a good approximation depends on how close x is to a and the behavior of higher derivatives.

• Higher-degree polynomials provide better approximations, but may require more computation.

Example: Limit Calculation Using Taylor Polynomials

• Limits involving complicated functions can be simplified using Taylor expansions.

• Example: can be evaluated by expanding and in Taylor series.

Summary Table: Taylor and Maclaurin Polynomials for Common Functions

Additional info:

• Taylor and Maclaurin polynomials are foundational tools in calculus for approximating functions, analyzing limits, and understanding the behavior of functions near a point.

• The error term in Taylor's theorem provides a way to estimate the accuracy of the polynomial approximation.

• These polynomials are also used in numerical methods, such as Newton's method for finding roots.