Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Compute the third degree Taylor polynomial for \(\tan x\) about \(x=0\). (b) Why is it reasonable to expect the coef cient of the \(x^{2}\) term to be zero?

Step-by-Step Solution

Answer

The third-degree Taylor polynomial for \( \tan x \) about \( x=0 \) is \( P_{3}(x) = x + x^3 / 3 \). The coefficient of the \( x^{2} \) term is zero in the Taylor polynomial because the function \( \tan x \) is odd, and thus its Taylor series around \( x = 0 \) only contains odd-degree terms.

Step 1: Compute the Successive Derivatives

Firstly, compute the successive derivatives for \( \tan x \). The first derivative of \( \tan x \) is \( \sec^{2} x \), the second derivative is \(2\sec^{2} x \cdot \tan x\), and the third derivative is \(2\sec^{2} x + 4\sec^{4} x \cdot \tan x\). Evaluate the derivatives at \( x = 0 \), they result in 1, 0 and 2 respectively.

Step 2: Apply the Taylor Series Formula

The Taylor series formula plans out as \( f(a) + f'(a)(x - a) / 1! + f''(a)(x - a)^2 / 2! + f'''(a)(x - a)^3 / 3! \), substituting \( a = 0 \) and the earlier calculated derivatives at \( x = 0 \), we get \( P_{3}(x) = x + 0 + 0 + (x^3 / 3) \), which simplifies to \( P_{3}(x) = x + x^3 / 3 \)

Step 3: Analyze the Coefficient of the \(x^{2}\) term

Looking at the Taylor polynomial calculated, the \( x^{2} \) term is missing, implying that the coefficient is zero. This is reasonable because the tangent function is an odd function, and the Taylor series developed around \( x = 0 \) only has odd-degree terms. Thus, the coefficient of the \( x^{2} \) term is indeed zero.

Key Concepts

Derivative Calculation

Calculating derivatives is an essential part of finding a Taylor Polynomial, as it helps understand how a function behaves at a certain point. When we compute derivatives, we are essentially looking at the rate at which the function changes. For the tangent function, which is slightly more complex due to its periodic nature, derivatives help to simplify understanding its variations.
The first derivative of the tangent function is particularly unique as it turns into a positive secant squared function:

• First Derivative, \( \frac{d}{dx}(\tan x) = \sec^2 x \)
• Second Derivative, \( \frac{d^2}{dx^2}(\tan x) = 2\sec^2 x \cdot \tan x \)
• Third Derivative, \( \frac{d^3}{dx^3}(\tan x) = 2\sec^2 x + 4\sec^4 x \cdot \tan x \)

Evaluating these derivatives at \( x = 0 \) helps in constructing the Taylor polynomial, providing values like 1, 0, and 2 for the derivatives, making it simpler to plug into the formula to find the polynomial terms.
With derivative calculation, understanding function behavior at specific points becomes more accessible, enabling the expansion into Taylor Series effectively.

Tangent Function

The tangent function is a fundamental trigonometric function with unique properties, especially when dealing with its behavior around certain points, such as zero. Unlike many other functions, \( \tan x \) is periodic and undefined at odd multiples of \( \frac{\pi}{2} \), making it interesting to analyze around those points for a Taylor Polynomial.

• The tangent function is an odd function, evident from its symmetry about the origin.
• Symmetry plays a crucial role when developing Taylor series because it dictates which terms have non-zero coefficients. This is why only odd-degree terms appear when the Taylor polynomial is derived at \( x = 0 \).

Understanding the tangent function's nature helps explain the absence of even power terms in its Taylor series around \( x = 0 \). This fundamental recognition is key when predicting which terms will emerge in its polynomial representation.

Taylor Series Formula

The Taylor series formula is a powerful tool in calculus for approximating functions as polynomials. The formula represents a function as an infinite sum of terms calculated from the function's derivatives at a single point.
The Taylor series is defined as: \\[ f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots \]

• Each term in the series expands the original function into a polynomial.
• The coefficients are determined by the function's derivatives at point \(a\).

When using the Taylor series for the tangent function centered about zero, it simplifies greatly due to zero values in the second derivative and positive values for odd powers. This particular simplicity in the tangent function's derivatives explains the non-appearance of even-degree terms.
By applying the formula, you can convert complicated trigonometric behavior into manageable polynomial segments, making complex functions easier to analyze under standard polynomial operations.

Polynomial Coefficient Analysis

Polynomial coefficient analysis in a Taylor series is critical for understanding how each term contributes to approximating the original function. Every derivative calculated plays a role in forming these coefficients, influenced by the function's nature.
In the Taylor polynomial of \( \tan x \), calculated about \( x = 0 \): \\[ P_{3}(x) = x + \frac{x^3}{3} \]

• Notice the absence of the \(x^2\) term as its coefficient is zero.
• This absence is because the second derivative of \(\tan x\) evaluated at zero equals zero.

This zero-value stems from the tangent function being an odd function. Understanding why some coefficients are zero helps predict the polynomial shape and improve accuracy. By focusing on odd-degree terms, particularly those derived from derivatives that are non-zero, you can accurately capture the essence and behavior of the original function.