Problem 39 - Calculus Early Transcendentals

Question

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{-2} ; \text {approximate } 1 / 1.21=1 / 1.1^{2}$$

Step-by-Step Solution

Answer

Question: Find the Taylor series expansion centered at 0 for the function \(f(x) = (1+x)^{-2}\), and use it to approximate the value of \(1/1.21\). Answer: Using the first four terms of the Taylor series centered at 0 for the given function, we approximate the desired quantity \(1/1.21\) as approximately 0.826.

Step 1: Find the derivatives of the given function

We need to find the first four derivative terms, as our Taylor expansion will require them: 1. \(f(x) = (1+x)^{-2}\) 2. \(f'(x) = -2(1+x)^{-3}\) 3. \(f''(x) = 6(1+x)^{-4}\) 4. \(f'''(x) = -24(1+x)^{-5}\)

Step 2: Evaluate the derivatives at the center x = 0

Now we need to substitute the center into the derivatives to get the corresponding coefficients for our Taylor expansion: 1. \(f(0) = (1+0)^{-2} = 1\) 2. \(f'(0) = -2(1+0)^{-3} = -2\) 3. \(f''(0) = 6(1+0)^{-4} = 6\) 4. \(f'''(0) = -24(1+0)^{-5} = -24\)

Step 3: Construct the first four terms of the Taylor series

Using the center and the function evaluations at the center, we can construct the Taylor series expansion: \(f(x) \approx 1 - 2x + 3x^2 - 4x^3\)

Step 4: Approximate the given quantity using Taylor series

Now we will use the Taylor series expansion to approximate the desired quantity: \(1/1.1^2 = (1+x)^{-2}\) with \(x = 0.1\) Using the approximation, we get: $$\frac{1}{1.1^2} \approx 1 - 2(0.1) + 3(0.1)^2 - 4(0.1)^3 = 1 - 0.2 + 0.03 - 0.004 = 0.826$$ Therefore, using the first four terms of the Taylor series centered at 0 for the given function, we approximate the desired quantity as approximately 0.826.

Key Concepts

Polynomial Approximation

Polynomial approximation is a powerful mathematical technique used to estimate complex functions with simpler polynomial expressions. This makes challenging calculations more manageable. The Taylor series is a frequently used method for polynomial approximation.

For any given function, the Taylor series approximates it as a polynomial. This is particularly useful when you need an approximation at a specific point. For instance, we have a function like \(f(x) = (1+x)^{-2}\). Using polynomial approximation, we can estimate its value using a polynomial - a sum of powers of \(x\) multiplied by determined coefficients.

• This makes it easier to perform calculations, especially for functions that involve powers, roots, or transcendental functions.
• Polynomial approximations become more accurate as more terms are added to the series.

Thus, polynomial approximation offers a practical solution to simplify complex expressions in various fields, from physics to computer science.

Derivatives

Derivatives play a crucial role in the process of creating a Taylor series for polynomial approximation. They measure how a function changes as its input changes, providing the necessary coefficients for each term in the series.

In the Taylor series expansion of a function at a specific point (like 0 in our case), the derivatives of the function are evaluated at that point. These values inform us of the behavior and shape of the function around the approximation center.

• First-Order Derivative: Represents the rate of change of the function and provides the coefficient for the first-order term in the polynomial.
• Higher-Order Derivatives: Indicate how the rate of change itself is changing, informing subsequent terms in the series.

For example, for \(f(x) = (1+x)^{-2}\), calculating the first few derivatives gives us the rates at which the function's slope changes, crucial for constructing a robust approximation.

Series Expansion

The concept of series expansion involves expressing a function as an infinite sum of terms calculated from its derivatives. Taylor series is a specific form of series expansion focused on producing polynomial approximations.

A Taylor series expands a function based on the function’s behavior near a specific point. The expansion uses a sum of terms, each derived from the function's derivatives at that point, multiplied by powers of \(x\). In practical use, only a finite number of terms are taken to approximate the function adequately.

• Taylor Series Expansion: Each term in this expansion is calculated as \(\frac{f^{(n)}(0)}{n!}x^n\), where \(f^{(n)}(0)\) is the \(n\)-th derivative evaluated at the center of expansion.
• Accuracy of Approximation: More terms in the series typically increase the accuracy of the approximation.

By creating a series expansion for \(f(x) = (1+x)^{-2}\), we strategically use calculated derivatives at \(x = 0\) to build a polynomial with terms like \(1 - 2x + 3x^2 - 4x^3\).

Mathematical Approximation

Mathematical approximation is the practice of estimating a value or function by another simpler representation that's close to the actual value. The aim is to make complex calculations feasible without a significant sacrifice in accuracy.

One of the key areas where approximation comes into play is when calculating values for functions that are difficult to compute directly. For example, using a Taylor series to approximate the value of \(1/1.1^2\) by substituting \(x = 0.1\) into our constructed polynomial.

• Estimate Complex Expressions: Approximations help simplify the computation of functions like logarithms, exponentials, and roots.
• Reduce Computational Cost: They reduce the time and resources spent on producing precise results.

In our example, by truncating the series after four terms, we can quickly approximate \(1/1.1^2\) as 0.826, combining speed with an acceptable level of accuracy.