Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Maclaurin polynomial of order 4 for \(f(x)\) and use it to approximate \(f(0.12) .\) $$ f(x)=\tan ^{-1} x $$

Step-by-Step Solution

Answer

The Maclaurin polynomial is \(x + \frac{1}{3}x^3\), and \(f(0.12) \approx 0.120576\).

Step 1: Understand the Maclaurin Series

The Maclaurin series is a Taylor series expansion of a function at zero. The formula for the Maclaurin series is: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f''''(0)}{4!}x^4 + \cdots \] We need to find these derivatives for the function \(f(x) = \tan^{-1}x\).

Step 2: Calculate Derivatives of \(f(x)\)

Calculate the first few derivatives of \(f(x) = \tan^{-1}x\) at \(x = 0\):1. \(f(x) = \tan^{-1}x\) implies \(f(0) = \tan^{-1}(0) = 0\).2. \(f'(x) = \frac{1}{1+x^2}\) implies \(f'(0) = 1\).3. \(f''(x) = \frac{-2x}{(1+x^2)^2}\) implies \(f''(0) = 0\).4. \(f'''(x) = \frac{-6x^2+2}{(1+x^2)^3}\) implies \(f'''(0) = 2\).5. \(f''''(x) = \frac{24x(1+x^2) - 16x^3}{(1+x^2)^4}\) implies \(f''''(0) = 0\).

Step 3: Construct the Maclaurin Polynomial of Order 4

Using the calculated derivatives, construct the Maclaurin polynomial:\[ P_4(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f''''(0)}{4!}x^4 \]Substituting the values:\[ P_4(x) = 0 + 1x + \frac{0}{2}x^2 + \frac{2}{6}x^3 + \frac{0}{24}x^4 \]Thus,\[ P_4(x) = x + \frac{1}{3}x^3 \]

Step 4: Approximate \(f(0.12)\)

Substitute \(x = 0.12\) into the polynomial:\[ P_4(0.12) = 0.12 + \frac{1}{3}(0.12)^3 \]Calculate \((0.12)^3 = 0.001728\), so:\[ P_4(0.12) = 0.12 + \frac{1}{3} \times 0.001728 = 0.12 + 0.000576 = 0.120576 \]

Key Concepts

Taylor Series

The Taylor series is a powerful mathematical tool that allows us to express complex functions as infinite sums of polynomial terms. It provides a way to approximate the value of functions that are otherwise difficult to compute. The Taylor series is centered at a specific point, often zero, making it particularly useful in calculus and analysis for understanding function behavior around that point.

• The general form of a Taylor series centered at zero (also called the Maclaurin series) is: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]
• Each term in the series depends on the derivatives of the function at the center point.
• The more terms included in the series, the closer the approximation is to the actual function.

This process is invaluable for problems where computing the function directly is complex. By using Taylor series, one can derive simpler polynomial expressions to approximate values of functions like \(\tan^{-1}(x)\).

Derivatives

Derivatives are fundamental to the construction of Taylor series and are central in calculus for analyzing function behavior and change. They measure how a function's value changes as its input changes.

• The first derivative \(f'(x)\) represents the slope of the tangent at a point and tells us how the function grows or shrinks.
• Higher-order derivatives, like \(f''(x)\) and \(f'''(x)\), provide deeper insights into curvature and the nature of extrema.

In the context of the Maclaurin series for \(f(x) = \tan^{-1}(x)\), we calculated these derivatives up to the fourth order to construct the polynomial approximation:
- \(f(x) = \tan^{-1}(x)\) implies \(f(0) = 0\) - \(f'(x) = \frac{1}{1+x^2}\) gives \(f'(0) = 1\) - \(f''(x) = \frac{-2x}{(1+x^2)^2}\) gives \(f''(0) = 0\) - \(f'''(x) = \frac{-6x^2+2}{(1+x^2)^3}\) gives \(f'''(0) = 2\)

Approximation

Approximation is a mathematical strategy used to find a simpler representation or estimate of a more complex value. In many cases, exact values might be challenging to find, making approximations crucial, especially for transcendental functions like arctan.

• In our exercise, we used the Maclaurin polynomial, which is a finite Taylor series, to approximate \(f(0.12)\), the value of \(\tan^{-1}(0.12)\).
• By substituting values into our polynomial up to the fourth term, we found an approximate value without calculating the exact arctan value.

The resulting approximation was: \[ P_4(0.12) = 0.120576 \]
This process highlights how polynomial approximations can provide practical solutions for otherwise complex computations.

Polynomial Expansion

Polynomial expansion is the process of expressing a function as a sum of polynomial terms. This approach is central to methods like the Taylor and Maclaurin series.

• It helps simplify complex functions into a combination of terms that are easier to work with.
• In the case of \(f(x) = \tan^{-1}(x)\), the Maclaurin series expansion allowed us to express it as: \[ P_4(x) = x + \frac{1}{3}x^3 \]
• This is known as a polynomial expansion because it breaks down the function into components that can be easily calculated.

This breakdown turns an intricate function into something that can be evaluated with basic arithmetic, providing a practical tool for both theoretical and applied mathematics.