Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt[3]{8.3} $$

Step-by-Step Solution

Answer

A second degree Taylor polynomial can approximate \( \sqrt[3]{8.3} \) as \( 2 + (1/12)*0.3 + (1/2)*(-1/288)*0.3^2 \).

Step 1: Identify the function

First, identify the function that needs to be approximated. Here, \(f(x) = \sqrt[3]{x}\) along with the value to approximate, \(x=8.3\). The base point which we centre our approximation around is \(x=8\) to reduce error.

Step 2: Calculate Taylor polynomial

Next, we calculate the terms for the second degree Taylor series centred around 2. The formula for the Taylor series is \(f(a) + f'(a)(x-a) + (f''(a)(x-a)^2)/2\). We will need to calculate the derivative \(f'(x)\) and the second derivative \(f''(x)\) of the function \(f(x)\). With \(f'(x) = \frac{1}{3\sqrt[3]{x^2}}\), \(f''(x) = \frac{-2}{9\sqrt[3]{x^4}}\), we find \(f(8)=2\), \(f'(8) = 1/12\), and \(f''(8)=-1/288\). Plugging these values into the Taylor series formula gives us the second degree Taylor polynomial.

Step 3: Substitute for required value

Now substitute \(x-8 = 0.3\) into the second degree Taylor Polynomial. The result will give an approximation of the value of \( \sqrt[3]{8.3}\). This approximation is accurate to the range around x=8 that we have selected.

Step 4: Verify validity

Verify if the approximation is reasonable according to the values of \(f(8)\), \(f'(8)\), \(f''(8)\), and \(x-8 = 0.3\). This is done by checking whether the result is close to the actual value \( \sqrt[3]{8.3}\).

Key Concepts

Taylor Series

The Taylor Series is a mathematical tool used to approximate functions by expanding them around a specific point, known as the base point. It provides a polynomial that approximates the function. For example, if you know the function and its derivatives at a specific point, you can write an approximating polynomial.

• The general form of a Taylor series for a function \(f(x)\) around a point \(a\) is \(f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2} + \ldots\)
• This expansion allows us to approximate complex functions with simpler polynomials.
• The degree of the polynomial determines the number of terms and the accuracy of the approximation.

In this exercise, we use a second-degree Taylor polynomial to approximate the cube root of \(8.3\) centering at \(x=8\). This involves using up to the second derivative of the function.

Approximation

Approximation in mathematics involves finding a value that is close to but not exactly equal to a desired quantity. Taylor polynomials are particularly useful for approximations near the chosen base point.

• The accuracy of the approximation depends on how many terms of the series you include and how close the value of \(x\) is to the base point \(a\).
• In this example, we approximate the cube root \(\sqrt[3]{8.3}\) using a second-degree polynomial centered at \(x=8\).
• This approximation simplifies complex calculations, providing an insight into the behavior of functions with simpler expressions.

By substituting \(x = 8.3\) into the polynomial, we find an efficient way to compute approximately \(\sqrt[3]{8.3}\).

Second Derivative

Understanding derivatives is vital for creating Taylor polynomials. The first derivative \(f'(x)\) describes the rate of change of the function, while the second derivative \(f''(x)\) indicates how the rate of change itself is changing.

• The first derivative of \(f(x) = \sqrt[3]{x}\) is \(f'(x) = \frac{1}{3\sqrt[3]{x^2}}\).
• The second derivative is \(f''(x) = \frac{-2}{9\sqrt[3]{x^4}}\).

These derivatives help form the Taylor polynomial by providing necessary coefficients. Substituting \(f(8)\), \(f'(8)\), and \(f''(8)\) into the polynomial helps estimate the function near \(x = 8\). Understanding how these derivatives work gives insight into the shape and behavior of the function around the point of interest.

Cube Root Approximation

Cube root approximation involves using methods like Taylor polynomials to find an approximate value for the cube root of a number. For \(\sqrt[3]{8.3}\), the function \(f(x) = \sqrt[3]{x}\) is expanded around \(x=8\) using a Taylor polynomial.

• Choosing \(x=8\) as the center helps use nearby known values, minimizing errors in approximation.
• Using the Taylor polynomial: \(f(8) = 2\), \(f'(8) = \frac{1}{12}\), and \(f''(8) = -\frac{1}{288}\), provide coefficients for the second-degree approximation.
• Finally, substituting \(x-8 = 0.3\) into this polynomial gives us a close approximation to \(\sqrt[3]{8.3}\).

This method is practical for calculating roots when exact computation may be complex or time-consuming, giving a broader understanding of how mathematical approximations can simplify intricate calculations.