Problem 12 - Calculus: Early Transcendentals

Question

Use a computer algebra system to find the Taylor polynomials \( T_n \) centered at \( a \) for \( n \) = 2, 3, 4, 5. Then Graph these polynomials and \( f \) on the same screen. \( f (x) = \sqrt [3]{1 + x^2,} \) \( a = 0 \)

Step-by-Step Solution

Answer

The Taylor polynomials are \( T_2(x) = 1 + \frac{1}{3} x^2 \), \( T_3(x) = 1 + \frac{1}{3} x^2 \), \( T_4(x) = 1 + \frac{1}{3} x^2 - \frac{1}{54} x^4 \), \( T_5(x) = 1 + \frac{1}{3} x^2 - \frac{1}{54} x^4 \).

Step 1: Find the first few derivatives of the function

The function given is \( f(x) = \sqrt[3]{1 + x^2} \). We begin by finding the derivatives of \( f \) at \( x = 0 \). The first derivative is found using the chain rule. The second derivative, third derivative, and fourth derivative are necessary for our polynomials.

Step 2: Compute the first derivative

The first derivative of \( f(x) = (1 + x^2)^{1/3} \) is \( f'(x) = \frac{2x}{3(1 + x^2)^{2/3}} \). Evaluating at \( x = 0 \), we find \( f'(0) = 0 \).

Step 3: Compute the second derivative

The second derivative is \( f''(x) = \frac{2}{3(1 + x^2)^{5/3}} - \frac{8x^2}{9(1 + x^2)^{5/3}} \). Evaluating at \( x = 0 \), \( f''(0) = \frac{2}{3} \).

Step 4: Compute the third derivative

The third derivative is found via differentiation of \( f''(x) \). This is complicated, and hence a computer algebra system is often used. However, the evaluation at \( x = 0 \) gives \( f'''(0) = 0 \).

Step 5: Compute the fourth derivative

The fourth derivative is similarly complex and requires careful application of differentiation rules. The evaluation at \( x = 0 \) produces \( f''''(0) = -\frac{16}{9} \).

Step 6: Determine the Taylor Polynomials

Using the derivatives and the Taylor series formula \( T_n(x) = f(a) + f'(a)x + \frac{f''(a)}{2}x^2 + \frac{f'''(a)}{6}x^3 + \frac{f''''(a)}{24}x^4 \), calculate the Taylor polynomials for \( n = 2, 3, 4, 5 \).

Step 7: Construct and Simplify Taylor Polynomials

For \( T_2(x) \), use the terms up to \( x^2 \): \[ T_2(x) = 1 + \frac{1}{3} x^2 \]For \( T_3(x) \), it's the same as \( T_2(x) \) because \( f'''(0) = 0 \).For \( T_4(x) \), include up to the \( x^4 \) term: \[ T_4(x) = 1 + \frac{1}{3} x^2 - \frac{1}{54} x^4 \]For \( T_5(x) \), it's the same as \( T_4(x) \) because there is no \( x^5 \) term in the expansion.

Step 8: Graph the Taylor Polynomials and the Function

Graph each Taylor polynomial alongside the original function \( f(x) = (1 + x^2)^{1/3} \) using a graphing tool. Visualize how each polynomial approximates \( f(x) \) at and around \( x = 0 \).

Key Concepts

Taylor Series

A Taylor series is a way to approximate a complex function with an infinite sum of terms, calculated from the function's derivatives at a single point. This method is highly useful in both mathematics and computer science for simplifying functions that are otherwise difficult to work with.

• The essence of the Taylor series is to use polynomial expressions to approximate a function.
• It uses derivatives of the function evaluated at a specific point to construct this series.
• The terms in a Taylor series follow the formula: \[ T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \frac{f'''(a)}{6}(x-a)^3 + \ldots \]

The Taylor polynomials are truncated versions of these infinite series, providing practical approximations with only a finite number of terms.

Derivative Calculation

Derivative calculation is at the heart of creating Taylor polynomials. It involves differentiating the function multiple times and evaluating these derivatives at a specific point, usually where the function is centered in the Taylor series.

• Start by finding the first derivative of the function. For example, using the chain rule on our function, we got:\[ f'(x) = \frac{2x}{3(1 + x^2)^{2/3}} \]
• Continue this process for higher derivatives. It can become complex, especially for functions like \[ f(x) = (1 + x^2)^{1/3} \]
• Once you have computed the derivatives, evaluate them at the center of the Taylor series, in our example, at \( x = 0 \).This step is crucial for forming the coefficients of the polynomial.

While some derivatives can be simple to compute by hand, others may require a more systematic approach with computational help.

Computer Algebra System

A computer algebra system (CAS) is an invaluable tool for finding derivatives and constructing polynomials when the manual calculations become intimidating or cumbersome.

• CAS software like Mathematica, Maple, or even Python's SymPy can handle complex symbolic calculations effectively.
• They automate the derivative process, providing accurate results ready for use in polynomial approximation.This is particularly useful when dealing with functions that produce complex derivatives, as seen in our example.
• Additionally, CAS can graph the Taylor polynomials and the original function simultaneously. This visual aspect helps in understanding how well the polynomials approximate the function near the center point \( a \).

Leveraging technology like CAS allows students to focus more on understanding concepts rather than getting stuck in lengthy manual computations.

Polynomial Approximation

Polynomial approximation is a core aspect of mathematical analysis and helps make complicated functions easy to analyze and compute.

• The idea is to approximate a function like \( f(x) = (1 + x^2)^{1/3} \) using its Taylor polynomials.
• The Taylors up to the nth degree provide increasingly better approximations near the point of expansion (in our case, \( a = 0 \)).For instance, for \( T_2(x) \), we found:\[ T_2(x) = 1 + \frac{1}{3} x^2 \]
• Each additional term in a higher-degree polynomial makes the approximation more accurate, as it accounts for additional curvature of the function.However, this also adds computation complexity.

Importantly, not every function has a Taylor polynomial that converges everywhere. It's an approximation valid typically only within a certain range or interval around the center \( a \).