91Ó°ÊÓ

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} $$f(x)=\frac{1}{\sqrt{1+x}}, \quad|x| \leq \frac{3}{4}$$

Step by step solution

01

Plot the Function

First, plot the function \( f(x) = \frac{1}{\sqrt{1+x}} \) over the interval \(|x| \leq \frac{3}{4}\). This gives us a visual understanding of the function within the specified range.

02

Find Taylor Polynomials

Find the Taylor polynomials \( P_1(x), P_2(x), \) and \( P_3(x) \) for the function \( f(x) \) at \( x=0 \). These are derived by considering terms up to \( n=1, 2, \) and \( 3 \) respectively:- \( P_1(x) = 1 - \frac{1}{2}x \)- \( P_2(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 \)- \( P_3(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 \).

03

Compute the Derivative and its Maximum Value

Calculate the relevant derivative for the remainder term of each polynomial. Specifically, determine the \((n+1)\)st derivative, which are:- For \( P_1(x) \), use \( f''(c) \)- For \( P_2(x) \), use \( f'''(c) \)- For \( P_3(x) \), use \( f^{(4)}(c) \).Plot these derivatives over the specified interval to find their maximum absolute values, \( M \).

04

Calculate Remainders and Estimate Errors

Calculate the remainder \( R_n(x) \) for each polynomial using the maximum values found:- \( R_1(x) = \frac{M}{2}x^2 \)- \( R_2(x) = \frac{M}{6}x^3 \)- \( R_3(x) = \frac{M}{24}x^4 \).Plot these remainders over the interval and estimate where \( |R_n(x)| < 10^{-2} \).

05

Compare Estimated Error with Actual Error

Compare \( R_n(x) \) with the actual error \( E_n(x) = |f(x) - P_n(x)| \) by plotting \( E_n(x) \) over the specified interval. This helps verify the adequate range for each approximation with maximum error less than \( 10^{-2} \).

06

Graph the Function and Approximations

Graph the original function \( f(x) \) along with its Taylor approximations \( P_1(x), P_2(x), \) and \( P_3(x) \). Discuss how well each approximation fits the function, especially noting where the error falls below the tolerance level based on insights from Steps 4 and 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linearization

Linearization is the simplest form of approximation used in calculus, derived from Taylor's formula when we take the first-degree polynomial. It provides a straight-line approximation to the function at a particular point, typically close to where we perform the calculation. In this exercise, the linearization of the function \( f(x) = \frac{1}{\sqrt{1+x}} \) at \( x=0 \) is given by the polynomial \( P_1(x) = 1 - \frac{1}{2}x \). This linear approximation is often useful because:

• It simplifies analysis since it's easier to work with straight lines than with curves.
• It provides a good approximation when \( x \) is small.

However, the linearization is only valid near the point of expansion, and its accuracy decreases as we move further from this point. Plotting the function alongside its linear approximation, we can see how well it captures the behavior of the function close to \( x=0 \). For any x-value where the error in approximation is acceptable (less than \(10^{-2}\)), this method is deemed useful.

Error Estimation

Error estimation in the context of Taylor series approximations involves determining how closely a Taylor polynomial approximates a function. In particular, this exercise focuses on quantifying the error using remainder terms for each level of approximation: linear, quadratic, and cubic. As we calculate these remainders:

• \( R_1(x) = \frac{M}{2}x^2 \) for linear approximation \( P_1(x) \)
• \( R_2(x) = \frac{M}{6}x^3 \) for quadratic approximation \( P_2(x) \)
• \( R_3(x) = \frac{M}{24}x^4 \) for cubic approximation \( P_3(x) \)

The value \( M \), determined from the maximum of the derivatives \( f''(c), f'''(c), \) or \( f^{(4)}(c) \) within the interval, helps in evaluating the size of these remainders. Error estimation is crucial because:

• It allows us to set bounds on the interval where the approximation remains valid within an acceptable error margin.
• It provides insights into which higher-degree approximation should be used, based on the precision needed.

In practical applications, such errors tell us whether a linear or more complex approximation would be most beneficial within specified constraints.

Cubic and Quadratic Approximations

Cubic and quadratic approximations extend the idea of linearization by including higher-order terms in the polynomial expressions. These approximations are generated by taking the Taylor series expansion to the second or third degree:

• Quadratic approximation: \( P_2(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 \)
• Cubic approximation: \( P_3(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 \)

By including these extra terms:

• The polynomial becomes a better fit near the point of expansion, providing greater accuracy over a broader interval.
• The error in approximation decreases significantly, especially for values of \( x \) further from zero.

These approximations become particularly useful when the linear model isn't sufficient, and when balanced between computational simplicity and the need for precision. By plotting both quadratic and cubic approximations together with the original function, students can visually comprehend how each added term in the Taylor series affects the approximation's fit and accuracy across the specified interval.