Problem 44 - Essential Calculus

Question

In Exercises \(43-46,\) determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of \(x\) to be less than \(0.0001 .\) Use a computer algebra system to obtain and evaluate the required derivatives. $$ f(x)=\cos \left(\pi x^{2}\right), \text { approximate } f(0.6) $$

Step-by-Step Solution

Answer

The solution requires determining the appropriate degree of the Maclaurin polynomial by evaluating its derivatives, and checking the absolute error of the approximation, until it becomes less than 0.0001. The exact degree can only be determined using a computer algebra system to compute the derivatives and check the approximation error as it involves extensive mathematical computations.

Step 1: Understand the Maclaurin Series Expansion

The Maclaurin series for a function \(f(x)\) is a power series expansion of the function around the point \(x=0\). It is a specific case of the Taylor series: \(f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! +...\) The error in the approximation by a polynomial of degree \(n\) is given by the remainder \(R_n = |f^n(c)x^n/n!|\) where \(c\) is between 0 and \(x\). The given function is \(f(x) = \cos(\pi x^2)\), and we are required to find the degree of polynomial for which \(R_n < 0.0001\) at \(x = 0.6\).

Step 2: Computation of Derivatives

Taking derivatives of the function \(f(x) = \cos(\pi x^2)\) and evaluating them at \(x = 0\) can best be done using a computer algebra system due to their complexity, and the fact that the derivatives need to be evaluated until the error becomes less than 0.0001.

Step 3: Evaluation

With the derivatives computed, they are plugged into the Maclaurin expansion and evaluated at \(x = 0.6\). This should be done with the polynomial of degree 'n' where \(n\) increases with each iteration until the absolute error between the actual and approximate values of \(f(0.6)\) is less than 0.0001

Key Concepts

Taylor Series

The Taylor Series is an essential mathematical tool used to represent a function as an infinite sum of terms. These terms are calculated from the values of the function's derivatives at a single point. Specifically, the Taylor series of a function f around a point a is given by:

\[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots \]
The Maclaurin series is a special case of the Taylor series where the point a is zero, simplifying the series to only depend on the function's derivatives at zero. A Maclaurin series takes the form:

\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]
In practical situations, it's often necessary to truncate the series after a certain number of terms to create a polynomial approximation of the function. The challenge in applications is determining the degree of the polynomial required to approximate the function accurately within a specified tolerance.

Polynomial Approximation

Polynomial approximation is the process of estimating a function with a polynomial of finite degree. The goal is to find a simpler function that closely resembles the original function within a certain range. The Maclaurin series provides a straightforward way to create such an approximation by truncating the series after a few terms.

The closer the approximation needs to be to the actual function, the higher the degree of the polynomial will generally need to be. For the exercise mentioned, f(x) = cos(πx^2) is approximated near the point x = 0 with a Maclaurin polynomial. To determine the appropriate degree for the polynomial, it's necessary to consider both the function's behavior and the acceptable error margin. The result is a finite series deeming a 'close enough' representation that maintains a balance between accuracy and simplicity.

Derivatives Computation

The computation of derivatives is fundamental in formulating a Taylor or Maclaurin series, as the series consists of terms involving derivatives of various orders evaluated at a specific point.

For the given function f(x) = cos(πx^2), computing these derivatives is not straightforward due to the complexity introduced by the power of x within the cosine function. Hence, using a computer algebra system to obtain these derivatives is recommended to ensure accuracy and efficiency. Derivative computation plays a crucial role in determining the coefficients of the terms in the polynomial approximation, and must be precise to minimize error margins in the final approximation.

Remainder Term Error

The remainder term error quantifies the difference between the true value of a function and its approximation by a Taylor or Maclaurin series. Represented by R_n, it's the bound on the error when approximating a function with an nth degree polynomial. The formula for the remainder term in terms of the Maclaurin series is:

\[ R_n = \left| \frac{f^{(n+1)}(c)}{(n+1)!} \right| x^{n+1} \]
where c is some point between 0 and x. Choosing the degree n depends on the acceptable error; the series should be truncated just enough to ensure that R_n is less than the tolerance, here 0.0001. As such, calculating the remainder term for successive values of n allows one to determine the minimum required degree of the polynomial for an acceptable approximation accuracy.