Problem 62 - 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} $$f(x)=e^{x / 3} \sin 2 x, \quad|x| \leq 2$$

Step-by-Step Solution

Answer

Use a CAS to plot and analyze the function and its Taylor polynomials, comparing approximation errors over \([-2, 2]\). Graph different approximations to visualize accuracy.

Step 1: Plot the Function

Use the computer algebra system (CAS) to plot the function \( f(x) = e^{x/3} \sin(2x) \) over the interval \( -2 \leq x \leq 2 \). Observe the behavior of the function in this interval and note any critical points or patterns.

Step 2: Find Taylor Polynomials

Calculate the Taylor polynomials of degree 1, 2, and 3 for the function \( f(x) = e^{x/3} \sin(2x) \) at \( x = 0 \). Use the formula for Taylor series expansions: \[ P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k \] Find the derivatives \( f'(0) \), \( f''(0) \), and \( f'''(0) \) to calculate \( P_1(x) \), \( P_2(x) \), and \( P_3(x) \).

Step 3: Calculate (n+1)th Derivative

Find the 4th derivative of the function \( f(x) \) and plot it over the interval \( -2 \leq x \leq 2 \). Estimate the maximum absolute value \( M \) of this derivative over the interval. This \( M \) will assist in calculating the remainder in the next step.

Step 4: Calculate and Plot the Remainder

Compute the remainder \( R_n(x) \) using \[ R_n(x) = \frac{M}{(n+1)!} |x|^{n+1} \]for each Taylor polynomial. Use the value \( M \) estimated from Step 3. Plot \( R_n(x) \) over the interval and estimate values of \( x \) where the remainder is significant.

Step 5: Compare Estimated and Actual Error

Plot the actual error \( E_n(x) = |f(x) - P_n(x)| \) over the interval \( -2 \leq x \leq 2 \). Compare this with the remainder estimates from Step 4 to see how accurate your estimates were in predicting the behavior of the Taylor approximations.

Step 6: Graph Function and Taylor Approximations

Draw graphs of the original function \( f(x) \) and its Taylor polynomial approximations \( P_1(x) \), \( P_2(x) \), and \( P_3(x) \) over the interval. Analyze how closely the Taylor polynomials approximate the function within the interval and at which points the approximation diverges from the actual function.

Key Concepts

Taylor Polynomials

Taylor Polynomials are polynomial approximations of functions that are especially useful for functions that are otherwise difficult to compute. In essence, these polynomials represent the function as a sum of its derivatives at a specific point, typically around zero, known as the center of the expansion. The general formula for a Taylor polynomial of degree \( n \) is: \[ P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k \] This formula means you take successive derivatives of the function, evaluate them at the point of interest (commonly zero), and use these values to create terms of increasing power in \( x \).

• \( f^{(0)}(0) \) is just the function value \( f(0) \).
• \( f^{(1)}(0) \) is the first derivative evaluated at zero.
• And so on, for as many terms as needed.

Taylor polynomials provide increasingly accurate approximations of smooth functions as more terms are added.

Error Estimation

Error Estimation is crucial in understanding how closely a Taylor polynomial approximates a function. When using Taylor polynomials, it’s important to estimate how much the polynomial deviates from the actual function. The error can be estimated using the remainder term \( R_n(x) \): \[ R_n(x) = \frac{M}{(n+1)!} |x|^{n+1} \] Here, \( M \) is the maximum absolute value of the \((n+1)\)-th derivative over the interval of interest. Error estimation helps check the reliability of a Taylor series approximation by examining how closely the calculated polynomial matches the original function.

• The smaller the remainder \( R_n(x) \), the better the approximation.
• Helps identify the interval where the approximation is most effective.

The estimation requires understanding the derivative's behavior over the given interval and using graphical or analytical methods to ensure precision.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is an invaluable tool for handling complex mathematical calculations like plotting functions, finding derivatives, and performing symbolic mathematics.

Using a CAS helps in multiple ways, especially when dealing with Taylor polynomials:

• Automatically calculates derivatives of any order.
• Plots functions and their approximations for visual comparison.
• Simplifies the manipulation of symbolic math expressions.

For instance, in the given problem, a CAS would aid in plotting the function \( f(x) = e^{x/3} \sin(2x) \) and its Taylor polynomials over the interval \(-2 \leq x \leq 2\). This automation reduces potential human errors and enhances understanding by providing precise graphical and numerical outputs.

Derivative Calculation

Derivative Calculation is fundamental to constructing Taylor polynomials and estimating errors. In the context of Taylor series, you'll need to compute derivatives up to the \((n+1)\)-th order:

• The first derivative \( f'(x) \) provides the initial slope.


• Higher-order derivatives \( f''(x), f'''(x), \text{etc.} \) reveal curvature and finer changes in the function around the center point.

For the function \( f(x) = e^{x/3} \sin(2x) \), it's essential to calculate derivatives like \( f'(0) \), \( f''(0) \), \( f'''(0) \), etc., to accurately form the corresponding Taylor polynomials. Using the results, you can construct polynomials like \( P_1(x) \), \( P_2(x) \), and so forth, each adding more terms for better approximation. Derivatives also play a pivotal role in error estimation by determining the maximum \((n+1)\)-th derivative value, \( M \), used in calculating the remainder term.