Problem 58 - Thomas Calculus

Question

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \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 } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { 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 approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion 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

Taylor polynomials approximate \( f(x) \) well initially; error exceeds thresholds near endpoint.

Step 1: Plot the Function

Use a computer algebra system (CAS) to graph the function \( f(x) = (1+x)^{3/2} \) over the interval \( -\frac{1}{2} \leq x \leq 2 \). This provides a visual reference for subsequent steps and helps understand the behavior of the function over the interval.

Step 2: Find the Taylor Polynomials

Calculate the Taylor polynomials \( P_1(x) \), \( P_2(x) \), and \( P_3(x) \) centered at \( x = 0 \) for the function \( f(x) = (1+x)^{3/2} \). This involves computing the derivatives at \( x = 0 \) and constructing the polynomials:\[ P_1(x) = 1 + \frac{3}{2}x \]\[ P_2(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 \]\[ P_3(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 + \frac{1}{16}x^3 \]

Step 3: Calculate the Derivative for the Remainder Term

Find the \( (n+1) \)-st derivative of \( f(x) \) which is needed to calculate the remainder term of each Taylor polynomial. For \( n = 1, 2, 3 \), compute derivatives up to the fourth order. Use CAS to plot these derivatives as functions of \( c \) over the interval and estimate their maximum absolute value, \( M \).

Step 4: Calculate and Plot Remainders

Using the maximum \( M \) from Step 3, determine and plot the remainders \( R_n(x) \) of each Taylor polynomial over the specified interval. Estimate where \( |R_n(x)| \leq \text{desired accuracy} \). This helps answer question (a) by providing insight into the interval the remainders remain within acceptable error bounds.

Step 5: Compare Actual and Estimated Errors

Compute the actual error \( E_n(x) = |f(x) - P_n(x)| \) for each polynomial and plot them over the specified interval. Compare these to the remainder estimates from Step 4 to quantify the accuracy of the Taylor polynomials and thus answer question (b).

Step 6: Graph Function and Taylor Approximations

Plot \( f(x) \) alongside \( P_1(x) \), \( P_2(x) \), and \( P_3(x) \) to visualize their accuracy over the interval. Discuss graphically how well the Taylor polynomials approximate \( f(x) \), considering the insights from the remainders in Step 4 and the actual errors in Step 5.

Key Concepts

Taylor Series

A Taylor Series is a way to represent a function as an infinite sum of terms that are calculated from the function’s derivatives at a single point. This method is incredibly powerful for function approximation. You start with a real function and then construct a polynomial, called a Taylor polynomial, that closely approximates the function around a point.

For example, for function \( f(x) = (1+x)^{3/2} \), the Taylor polynomials up to third degree are given as:

• \( P_1(x) = 1 + \frac{3}{2}x \)
• \( P_2(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 \)
• \( P_3(x) = 1 + \frac{3}{2}x - \frac{3}{8}x^2 + \frac{1}{16}x^3 \)

These polynomials are constructed by calculating derivatives of \( f(x)\) and then evaluating each derivative at the expansion point, in this case, \( x=0 \).

The more terms included, the closer the approximation to the function around the point.

Function Approximation

Function approximation is the process of estimating a function's values without computing the exact form of the function. Taylor Polynomials serve as a tool for approximating functions because they can simplify complex functions into polynomials, which are easier to work with.

For example, to approximate \( f(x) = (1+x)^{3/2} \) over the interval \(-\frac{1}{2} \leq x \leq 2 \), we construct simpler polynomial expressions like \( P_1(x), P_2(x), \) and \( P_3(x) \).

Taylor Polynomials are particularly useful in calculus and numerical analysis because they can provide excellent approximations over small intervals and are easier to evaluate and differentiate compared to complex functions.

Error Analysis

Error analysis is crucial when using Taylor Polynomials because it helps understand the difference between the actual function and its polynomial approximation. The remainder term in a Taylor series provides an estimate of this error.

The formula for the remainder \( R_n(x) \) is given by:\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \]where \( c \) is some number between \( a \) and \( x \).Using the calculated \( M \), the maximum value of the \((n+1)\)-st derivative over the interval helps in bounding the error:\[ |R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1} \]Plotting \( R_n(x) \) alongside \( E_n(x) = |f(x) - P_n(x)| \) helps compare estimated vs actual error, providing insights into how good the approximation is.

Calculus

In calculus, Taylor Polynomials are applied to find approximations and gain insight into the behavior of functions. Calculus techniques involve computing derivatives which play a key role in forming Taylor Polynomials.

Calculus allows us to break down a complex function into a finite sum of easier computations, which are the derivatives evaluated at a point. This approach provides powerful tools in solving differential equations, evaluating limits, and even in computing integrals.

For example, by repeatedly differentiating \( f(x) = (1+x)^{3/2} \) and evaluating at \( x=0 \), we derive the coefficients of the Taylor polynomials, elucidating not just the value but the rate and manner of changes of \( f(x) \) around the specified point.

Derivative Calculations

Derivatives are fundamental in calculating Taylor Polynomials. They capture the rate of change and the nature of curvature of the function around a point.

For Taylor Polynomials, derivatives are calculated until the desired degree of the polynomial is reached. In our case, up to the third derivative for the third-degree polynomial for \( f(x) = (1+x)^{3/2} \).

• First derivative provides the linear component.
• Second derivative contributes to the quadratic component.
• Third derivative shapes the cubic term.

These calculations, taking place at \( x=0 \), lead to approximations that mimic the original function’s behavior accurately, at least within a small region around the center point. The better these derivatives capture small changes in the function, the more effective the Taylor Polynomial will be.