Problem 22 - Essential Calculus

Question

In Exercises \(19-24,\) find the \(n\) th Taylor polynomial centered at \(c\). $$ f(x)=\sqrt[3]{x}, \quad n=3, \quad c=8 $$

Step-by-Step Solution

Answer

The third degree Taylor polynomial for the function \(f(x)=\sqrt[3]{x}\) centered at \(c=8\) is \(P_3(x) = 2 + \frac{1}{12} (x - 8) - \frac{1}{288} (x - 8)^2 + \frac{1}{5832}(x - 8)^3\)

Step 1: Calculation of the Value of Function at \(c\)

Firstly, calculate the value of the function \(f(x)=\sqrt[3]{x}\) at \(x=8\): \(f(8) = \sqrt[3]{8} = 2\). This gives the first term of the Taylor polynomial.

Step 2: Calculation of the First Derivative and Its Value at \(c\)

Next, calculate the first derivative of the function: \(f'(x) = \frac{1}{3\sqrt[3]{x^2}}\). Then find its value at \(x=8\): \(f'(8) = \frac{1}{3\sqrt[3]{64}} = \frac{1}{3*4} = \frac{1}{12}\). This provides the second term of the polynomial.

Step 3: Calculation of the Second Derivative and Its Value at \(c\)

Now calculate the second derivative: \(f''(x) = -\frac{2}{9\sqrt[3]{x^4}}\). Find the value at \(x=8\): \(f''(8) = -\frac{2}{9\sqrt[3]{4096}} = -\frac{1}{144}\). This gives the third term of the polynomial.

Step 4: Calculation of the Third Derivative and Its Value at \(c\)

Proceed with calculation of the third derivative: \(f'''(x)=\frac{8}{27\sqrt[3]{x^5}}\). Then evaluate this at \(x=8\): \(f'''(8)=\frac{8}{27\sqrt[3]{32768}}=\frac{1}{648}\). This gives the fourth term of the polynomial.

Step 5: Construction of the Third Degree Taylor Polynomial

Finally, collect all terms to form the third degree Taylor polynomial using the formula: \(P_3(x) = f(8) + f'(8)(x - 8) + \frac{f''(8)(x - 8)^2}{2!} + \frac{f'''(8)(x - 8)^3}{3!}=2 + \frac{1}{12} (x - 8) - \frac{1}{288} (x - 8)^2 + \frac{1}{5832}(x - 8)^3\)

Key Concepts

Taylor Polynomial

The Taylor polynomial allows us to express functions in simpler forms by approximating them around a specific point. A Taylor polynomial is particularly helpful because it provides a polynomial approximation of a function near the given center point, denoted as \(c\). The general formula for the Taylor polynomial of degree \(n\) centered at \(c\) is: \[P_n(x) = f(c) + f'(c)(x - c) + \frac{f''(c)(x - c)^2}{2!} + \cdots + \frac{f^n(c)(x - c)^n}{n!}\] The terms \(f'(c)\), \(f''(c)\), \(f^n(c)\) are the derivatives of the function evaluated at \(c\). Each derivative plays a critical role in defining the coefficients of the polynomial. The degree of the polynomial \(n\) determines how many terms the polynomial will have. More terms usually result in a better approximation of the function near the point \(c\). For functions with derivatives of higher order, Taylor polynomials become more precise, providing a greater understanding of the behavior of the function around the center.

Derivatives

The concept of derivatives is foundational to calculus and particularly to constructing Taylor polynomials. A derivative indicates how a function changes at any point. It provides the slope of the tangent line to the function at a specific point. For a Taylor polynomial, each derivative contributes a different degree of approximation to the function:

• First Derivative (\(f'(x)\)): Provides linear approximation, giving the slope at point \(c\).
• Second Derivative (\(f''(x)\)): Adds curvature to the approximation, adjusting the polynomial to reflect the concavity.
• Third Derivative (\(f'''(x)\)): Makes further refinements to curvature, improving the fit of the polynomial to the function.

Each derivative is calculated at the center \(c\), forming the coefficients that define the polynomial's degree. Understanding these derivatives involves calculating the slope, acceleration, and higher-order rates of change of the function, enabling the creation of a polynomial that closely resembles the function around the point of interest. This step-by-step derivation enables us to craft detailed and accurate approximations for complex functions.

Third Degree Polynomial

A third degree polynomial is a polynomial of degree 3, which means it includes terms up to \(x^3\). This type of polynomial is used in Taylor polynomial approximations to carefully and efficiently model functions around a specific point. A third degree Taylor polynomial includes not just linear and quadratic approximations, but also cubic: \[P_3(x) = f(c) + f'(c)(x - c) + \frac{f''(c)(x - c)^2}{2!} + \frac{f'''(c)(x - c)^3}{3!}\] This formula demonstrates how various derivatives contribute to the terms of a third degree polynomial:

• Constant term: \(f(c)\) reflects the initial value of the function at \(c\).
• Linear term: \(f'(c)(x-c)\) shows how the function initially changes around \(c\).
• Quadratic term: \(\frac{f''(c)(x-c)^2}{2!}\) provides adjustment for curvature.
• Cubic term: \(\frac{f'''(c)(x-c)^3}{3!}\) improves modeling by accounting for further changes in curvature over small intervals.

The third degree polynomial thus offers a balance between accuracy and simplicity, making it a valuable tool for approximating functions within a localized region.