Problem 54 - Thomas Calculus

Question

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$

Step-by-Step Solution

Answer

Analyze Taylor polynomials over the range to assess error. Approximate and actual errors guide conclusions.

Step 1: Plot the Original Function

The function given is \( f(x) = (1+x)^{3/2} \). Plot this function over the interval \(-\frac{1}{2} \leq x \leq 2 \). This step ensures we understand the behavior of the function within the specified range.

Step 2: Find Taylor Polynomials

The Taylor polynomials are determined from the derivatives of \( f(x) \) at \( x=0 \).- First derivative: \( f'(x) = \frac{3}{2}(1+x)^{1/2} \)- Second derivative: \( f''(x) = \frac{3}{4}(1+x)^{-1/2} \)- Third derivative: \( f'''(x) = -\frac{3}{8}(1+x)^{-3/2} \)Taylor polynomials:- Linearization: \( P_1(x) = 1 + \frac{3}{2}x \)- Quadratic: \( P_2(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 \)- Cubic: \( P_3(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 - \frac{1}{16}x^3 \)

Step 3: Calculate Next Derivative and Estimate Maximum

The next derivative \( f^{(4)}(x) = \frac{15}{16}(1+x)^{-5/2} \). Plot \( f^{(4)}(c) \) over the interval \(-\frac{1}{2} \leq c \leq 2 \) to estimate its maximum absolute value, \( M \). This estimate will be used in the Taylor remainder term analysis.

Step 4: Compute Remainder Terms

The remainder terms are given by:- \( R_1(x) = M \cdot \frac{|x|^2}{2!} \)- \( R_2(x) = M \cdot \frac{|x|^3}{3!} \)- \( R_3(x) = M \cdot \frac{|x|^4}{4!} \)Plot these remainders over the interval to find the values of \( x \) for which each approximation satisfies the error less than \( 10^{-2} \).

Step 5: Estimate Actual Errors

Plot the actual error \( E_n(x) = |f(x) - P_n(x)| \) for each polynomial over the interval. This provides insight into the actual performance of each approximation and helps answer the error-related question.

Step 6: Graph and Compare Approximations

Graph \( f(x) \) along with \( P_1(x), P_2(x), \) and \( P_3(x) \) over the interval. Discuss how well each polynomial approximates \( f(x) \) and analyze where the approximations deviate significantly based on the remainder calculations and error estimates from Steps 4 and 5.

Key Concepts

Polynomial Approximation

Polynomial approximation is a way to estimate complex functions using simpler polynomial expressions. For this exercise, Taylor Series is used to approximate the function \( f(x) = (1+x)^{3/2} \). The Taylor polynomial provides a sum of terms based on derivatives of the function at a particular point. In this case, we linearize the function by using different polynomial degrees, like linear, quadratic, and cubic approximations:

• Linearization: Represents the function using the first-degree polynomial \( P_1(x) = 1 + \frac{3}{2}x \).
• Quadratic Approximation: Utilizes up to the second-degree term \( P_2(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 \) to capture more of the function's curvature.
• Cubic Approximation: Employs up to the third-degree term \( P_3(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 - \frac{1}{16}x^3 \), providing an even closer fit to the function.

When \( n = 1, 2, \text{ or } 3 \) are applied, these polynomials approximate the function more closely as \( n \) increases, giving a more accurate representation of \( f(x) \) over a specified interval.

Error Analysis

Error analysis is crucial to understanding how well polynomial approximations represent the original function. It involves determining the difference between the true function value and the polynomial approximation. This difference is known as the error.
In this exercise, we aim for an error less than \( 10^{-2} \). A practical approach is to estimate this error using the remainder term \( R_n(x) \) in the Taylor series:

• The remainder term quantifies the deviation of the polynomial from the actual function.
• For linear approximation: \( R_1(x) \) involves second-degree error term \( M \cdot \frac{|x|^2}{2!} \), where \( M \) is the maximum value of the next derivative within the interval.
• For quadratic: \( R_2(x) = M \cdot \frac{|x|^3}{3!} \).
• For cubic: \( R_3(x) = M \cdot \frac{|x|^4}{4!} \).

By plotting and analyzing these remainder terms, it helps determine for what intervals each approximation provides a sufficient level of accuracy.

Calculus

Calculus is the mathematical study of change, utilizing differentiation and integration to solve problems concerning rates of change and accumulation. It plays a pivotal role in approximating functions like \( f(x) = (1+x)^{3/2} \), particularly in deriving derivatives and constructing Taylor Series.
In the context of this problem:

• The first derivative \( f'(x) = \frac{3}{2}(1+x)^{1/2} \) provides the slope of the tangent, which is the basis for linear approximation.
• The second derivative \( f''(x) = \frac{3}{4}(1+x)^{-1/2} \) reveals how the function's rate of change is changing, improving accuracy in the quadratic approximation.
• The third derivative \( f'''(x) = -\frac{3}{8}(1+x)^{-3/2} \) further refines the approximation through the cubic term.

Calculating these derivatives not only aids in forming the Taylor polynomials but also enables error estimation and verification by tracking higher-order trends in the function. Overall, calculus provides the essential toolset for rigorously understanding and implementing polynomial approximations.