Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The Maclaurin polynomial approximation of \( f(0.12) \) is 0.120576.

Step 1: Understanding the Maclaurin Series

The Maclaurin series is a special case of the Taylor series for a function expanded around zero. It is given by \[ f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \frac{f^{(4)}(0)x^4}{4!} + \cdots \] Here, we need to find the Maclaurin polynomial of order 4 for the function \( f(x) = \tan x \). This means we will calculate up to the fourth derivative at \( x = 0 \).

Step 2: Calculate the Derivatives

First, calculate the required derivatives of \( f(x) = \tan x \). 1. \( f(0) = \tan(0) = 0 \) 2. \( f'(x) = \sec^2 x \Rightarrow f'(0) = \sec^2(0) = 1 \)3. \( f''(x) = 2\sec^2 x \cdot \tan x \Rightarrow f''(0) = 2 \times \sec^2(0) \times \tan(0) = 0 \)4. \( f'''(x) = 2 \sec^2 x (\sec^2 x + 2 \tan^2 x) \Rightarrow f'''(0) = 2 \times 1 = 2 \)5. \( f^{(4)}(x) = 8 \sec^2 x \tan x \cdot (\sec^2 x + \tan^2 x) \Rightarrow f^{(4)}(0) = 0 \)

Step 3: Construct the Maclaurin Polynomial

Using the derivatives calculated, the Maclaurin polynomial of order 4 for \( f(x) = \tan x \) is:\[ P_4(x) = 0 + 1 \cdot x + 0 \cdot x^2 + \frac{2 \cdot x^3}{3!} + 0 \cdot x^4 = x + \frac{x^3}{3} \]

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

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

Key Concepts

Taylor series

A Taylor series is a way to represent a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. This series is highly valued for approximating functions that may be difficult to compute otherwise.
The Taylor series takes the form:

• The value of the function at the point of expansion.
• The first derivative times the input offset, divided by 1!.
• Plus the second derivative times the input offset squared, divided by 2!.
• And so on for as many terms as needed.

For a function centered at zero, this becomes what is known as the Maclaurin series.
The key in using Taylor series is recognizing how derivatives can tell us about the shape and behavior of a function near the expansion point, providing a polynomial that serves as a piecewise approximation.

polynomial approximation

Polynomial approximation employs polynomials to approximate functions. This tactic allows a simpler way of representing complex functions, especially when functions do not have straightforward closed forms.
Polynomials, being sums of power terms, are simply one of the most straightforward forms of mathematical expressions to work with. They are often used because:

• Polynomials are easy to differentiate and integrate.
• They can provide close approximations over a certain interval.
• Low-degree polynomials enable quick computations.

In the realm of calculus, the Maclaurin polynomial is a polynomial approximation around zero. This allows one to approximate trigonometric, exponential, and logarithmic functions efficiently, as we did with \( \tan x \), capturing essential behavior with fewer terms.

trigonometric functions

Trigonometric functions like sine, cosine, and tangent have unique properties that make them crucial in many mathematics applications, such as engineering, physics, and computer graphics. These functions describe angles and can model periodic phenomena.
When Taylor or Maclaurin series are used with trigonometric functions, it can simplify complex computations by providing a polynomial representation, especially useful for functions like tangent, which can become challenging near specific points.
For example, with \( \tan x \), calculating and using derivatives in a Maclaurin series helps approximate the function values in scenarios where a calculator cannot specify precise outputs due to their transcendental nature.

mathematical derivatives

Mathematical derivatives tell us how a function changes as its input changes. They're at the heart of calculus and analysis. The derivative of a function at a point measures the sensitivity to change of the function value with respect to a change in its input.
For calculating derivatives, several basic rules are essential:

• The Power Rule for functions such as \( x^n \), where it provides straightforward calculations for many polynomial terms.
• The Chain Rule, which allows us to compute the derivative of composite functions.
• Derivatives of trigonometric functions like \( \sin x \), \( \cos x \), and \( \tan x \), using fundamental identities like the fact \( \frac{d}{dx}(\tan x) = \sec^2 x \), are essential for expanding these functions into series.

Understanding derivatives lets us construct series such as the Maclaurin series by evaluating each successive derivative at a certain point to form terms for building polynomial approximations. This mathematical tool lets us gain insights into function’s behavior near specific points, essential for the simplified analysis in various disciplines.