Problem 30 - Essential Calculus

Question

Use the series for \(f(x)=\arctan x\) to approximate the value, using \(R_{N} \leq 0.001\). \(\int_{0}^{1 / 2} x^{2} \arctan x d x\)

Step-by-Step Solution

Answer

The approximated value can be easily calculated by substituting 1/2 and 0 into the series, and adding up the terms until the difference between two successive terms is less than 0.001.

Step 1 - Express \(\arctan x\) using its Taylor series

The Taylor series for \(\arctan x\) is \(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\). This forms an infinite series and provides a mathematical representation of \(\arctan x\) as a sum of terms containing different powers of x.

Step 2 - Multiply by \(x^{2}\) and express the function as a series

Now, multiply each term in the Taylor series by \(x^{2}\). This results in the new infinite series: \(x^{3} - \frac{x^{5}}{3} + \frac{x^{7}}{5} - \frac{x^{9}}{7} + \cdots\). This series represents the function \(f(x) = x^{2} \arctan x\).

Step 3 - Integrate the function

Now, integrate the function from 0 to 1/2. As this is an infinite series, we need to integrate term by term. This results in: \(\frac{x^{4}}{4} - \frac{x^{6}}{18} + \frac{x^{8}}{40} - \frac{x^{10}}{70} + \cdots\) evaluated from 0 to 1/2.

Step 4 - Evaluate the integral

Substitute 1/2 and 0 into the series to determine the approximate value. Continue approximating with each term of the series until the desired degree of accuracy (within 0.001) is reached. This provides an approximate value for the integral.

Key Concepts

Arctan

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function. It is a fundamental function in trigonometry used to find angles whose tangent is the given number. This function is particularly useful in scenarios involving right triangles and circular measurements.

In calculus, the arctangent function can be expressed using a Taylor Series, which is an infinite series that represents functions as sums of power series. The Taylor series for \( \arctan(x) \) is:

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

The convergence of this series refers to how accurately the series represents the \( \arctan(x) \) within a certain interval, typically for values of \( x \) between -1 and 1.

Infinite Series

An infinite series is a sum of infinitely many terms. These series are foundational in mathematics because they help to express functions precisely or to approximate them closely.

In the context of the arctangent function, each term in its Taylor series is derived from the powers of \( x \) and involves alternating signs. The significance of an infinite series lies in its potential to approximate complex functions using just a few initial terms for calculation purposes. However, the precision depends on how many terms we include.

In the problem at hand, the infinite series:

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

is transformed by multiplying each term by \( x^2 \), resulting in a new series that can be integrated to find areas under curves and approximate definite integrals.

Integration

Integration is a fundamental concept in calculus concerning the accumulation of quantities, such as areas under curves. It is the inverse process of differentiation.

To integrate a function expressed as a series, such as our modified Taylor series for \( x^2 \arctan(x) \), each term must be integrated separately. This approach is called term-by-term integration:

• \( \int x^2 \arctan(x) \, dx = \int (x^3 - \frac{x^5}{3} + \cdots) \, dx \)

This operation involves integrating each power of \( x \), yielding terms like:

• \( \frac{x^4}{4} - \frac{x^6}{18} + \cdots \)

Once integrated, the limits from 0 to 1/2 are applied to evaluate the definite integral.

Approximation

Approximation is the process of finding a value that is close to, but not exactly, the true value. It is especially useful when dealing with complex mathematical expressions or when solving problems precisely is not feasible.

To approximate the value of \( \int_{0}^{1/2} x^2 \arctan(x) \, dx \), we compute the terms of the series until the remainder (the sum of the remaining terms) is less than or equal to 0.001, as specified in the exercise.

This method ensures that our result is not only accurate but also efficient in terms of computational effort. By calculating the first few terms of our integrated series, we can add them together to estimate the integral's value with high precision, lower the error margin, and meet the required accuracy level.