Question

Approximate the function value with the indicated Taylor polynomial and give approximate bounds on the error. Approximate \(\sin 0.1\) with the Maclaurin polynomial of degree 3 .

Short answer

Approximate value: 0.0998333; Error bound: < 0.0000008333.

Step-by-step solution

Understand the Maclaurin Series for Sine

The Maclaurin series for the sine function is given by \( \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \). This is an infinite series that approximates \( \sin(x) \) around 0. For a degree 3 polynomial, we will use the first two non-zero terms.

Write the Degree 3 Maclaurin Polynomial

Considering the Maclaurin series, the degree 3 polynomial for \( \sin(x) \) is \( P_3(x) = x - \frac{x^3}{3!} \). This uses terms up to \( x^3 \).

Substitute the Value into the Polynomial

Now, substitute \( x = 0.1 \) into the polynomial: \( P_3(0.1) = 0.1 - \frac{0.1^3}{6} \). Simplifying this gives \( P_3(0.1) = 0.1 - \frac{0.001}{6} = 0.1 - 0.0001667 \approx 0.0998333 \).

Approximate Error using Taylor's Theorem

Taylor's theorem tells us that the error for a polynomial of degree 3 is proportional to the next term, which involves \( x^5 \). For sine, this is \( \frac{x^5}{5!} \). Evaluate this for \( x = 0.1 \): \( \frac{0.1^5}{120} \approx 0.0000008333 \). This gives an approximate error bound.

Interpret the Results

The approximation for \( \sin(0.1) \) using the degree 3 Maclaurin polynomial is 0.0998333 with an error less than 0.0000008333. This means our polynomial is a very close approximation.

Key concepts

Maclaurin series

The Maclaurin series is a special type of Taylor series that expands a function around 0. It is named after the Scottish mathematician Colin Maclaurin. This series is an infinite sum of terms calculated from the derivatives of the function at a single point. For many common functions, the Maclaurin series provides a simple and elegant way to approximate function values.

The general formula for the Maclaurin series of a function \( f(x) \) is:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \)

The sine function, \( \sin(x) \, \), has a specific Maclaurin series. This series can be represented as:

• \( \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \)

This series alternates between subtracting and adding terms as the powers of \( \( x \) \) increase by two in each subsequent term. Due to the fast decreasing nature of the factorial in the denominators, this series converges quickly, making it easy to approximate \( \sin(x) \, \) around 0.

Error approximation

When using polynomial approximations like the Maclaurin series, it's important to evaluate how closely the polynomial approximates the true function, which is where error approximation comes in. Taylor's theorem gives us a way to determine this error. The error approximation is expressed in terms of the next term in the series that we have not used.

For our degree 3 approximation of the sine function, this error is given by:\

• The next term in the series, which would incorporate \( x^5 \), specifically: \( \frac{x^5}{5!} \)

In Taylor's theorem, the error bound for a function \( f(x) \) approximated by a polynomial of degree \( n \) involves the \( (n+1) \)th derivative of \( f \) evaluated at some point \( c \) within the range of approximation.

For \( \sin(x) \), since the series used for approximation was a degree 3 polynomial, the error approximation would consider\( \frac{x^5}{5!} \) as the term providing the leading contribution to the error. Since \( 0.1^5 \) remains tiny, it shows that even the simplest Maclaurin polynomials can deliver very accurate estimates.

Sine function

The sine function is one of the fundamental trigonometric functions, often abbreviated as \( \sin \). It is a periodic function with a wave-like graph that repeats every \( 2\pi \) units. It is defined for all real numbers and is particularly important in the study of phenomena involving waves and oscillations.

Beyond its role in geometry and physics, the sine function is crucial in mathematical analysis due to its smoothness and periodicity, as well as its beautiful properties when expressed as a series.

Its Maclaurin series expansion is of special interest, particularly:

• \( \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)

This expansion is especially useful for approximating the function near \( x = 0 \). In our case, simply using the first two terms \( x - \frac{x^3}{6} \) delivers a highly accurate approximation to \( \sin(x) \, \) for small values of \( x \).

The simplicity and elegance of the sine function's properties make it a cornerstone function in mathematics.

Degree 3 polynomial

A degree 3 polynomial is one that contains terms up to the third power of a variable. Such polynomials are known for their balanced computational simplicity and ability to provide fairly accurate approximations of more complex functions over an interval.

In the context of the Maclaurin series, a degree 3 polynomial for a function like \( \sin(x) \, \) involves the initial two non-zero terms. For sine, this means:

• \( P_3(x) = x - \frac{x^3}{3!} \)

This forms a simple and concise polynomial that surprisingly closely estimates the sine function for small values of \( x \), such as \( x = 0.1 \). This makes it particularly useful for quick calculations in settings where detailed computation is impractical.

By evaluating this polynomial at \( x = 0.1 \), we got an estimate for \( \sin(0.1) \, \, \) of approximately 0.0998333. The simplicity of the polynomial demonstrates why lower-degree Taylor polynomials often provide sufficient accuracy, reflecting their value in both educational and practical applications.