Problem 58 - 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} $$f(x)=e^{x / 3} \sin 2 x, \quad|x| \leq 2$$

Step-by-Step Solution

Answer

Calculate Taylor polynomials, estimate the remainder, compare actual errors, and graph them together to determine accuracy within \(|x| \leq 2\).

Step 1: Plot the function

We need to plot the function \(f(x) = e^{x/3} \sin(2x)\) over the interval \(|x| \leq 2\). This will visually represent how the function behaves over this range. Use the computer algebra system (CAS) to accomplish this.

Step 2: Find the Taylor Polynomials

Calculate the Taylor polynomials \(P_1(x)\), \(P_2(x)\), and \(P_3(x)\) at \(x = 0\). These are the linear, quadratic, and cubic approximations of the function respectively. Use the formulas for Taylor series:\[P_1(x) = f(0) + f'(0)x\]\[P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2\]\[P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3\].Compute these derivatives using the CAS.

Step 3: Calculate the (n+1)st Derivative

For each Taylor polynomial, compute the \((n+1)\)-st derivative at point \(c\), denoted as \(f^{(n+1)}(c)\). Use the remainder term as part of the Taylor series expansion: \ \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1} \]. \ Plot the derivative versus \(c\) over the interval and estimate its maximum absolute value, \(M\).

Step 4: Calculate the remainder

Utilize the estimated \(M\) from Step 3 to calculate the remainder \(R_n(x)\) for each polynomial. Plot \(R_n(x)\) over the specified interval \(|x| \leq 2\). This visualization helps determine for what \(x\)-values the function's error in approximation stays below \(10^{-2}\) to answer question (a).

Step 5: Compare with actual error

Compute the actual error \(E_n(x) = |f(x) - P_n(x)|\) by plotting \(E_n(x)\) over the interval. Comparing this with the estimated error from Step 4 will help determine the maximum error, aiding in answering question (b).

Step 6: Graph together and discuss

Create a graph that includes the original function \(f(x)\) and its three Taylor approximations \(P_1(x)\), \(P_2(x)\), and \(P_3(x)\). Analyze the behavior of these graphs, considering how closely each approximation matches \(f(x)\). Your analysis from Steps 4 and 5 informs this discussion, highlighting effectiveness and error ranges at different \(x\)-values.

Key Concepts

Taylor polynomial

A Taylor polynomial is an approximation of a function around a specific point. This approach simplifies complex functions by using polynomials, which are easier to work with.
The Taylor polynomial of order \( n \), denoted as \( P_n(x) \), is derived by taking the first \( n \) derivatives of the function at a given point, typically referred to as \( a \). For our exercise, this point is \( a = 0 \). You can remember the expansion with the following general form:
\[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n \]
By substituting \( a = 0 \), we simplify to:
\[ P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n \]

• \( P_1(x) \) is a linear approximation using just the slope.
• \( P_2(x) \) adds curvature with a quadratic term.
• \( P_3(x) \) extends further with a cubic term for more precision.

By using Taylor polynomials, we simplify calculations for functions that are otherwise too difficult to handle directly.

error approximation

Error approximation assesses how close a Taylor polynomial \( P_n(x) \) is to the actual function \( f(x) \). This concept helps us understand and control the precision by which we approximate a function using polynomials.
When we approximate a function, there's always a degree of inaccuracy, referred to as error. For a Taylor polynomial, the error is calculated by comparing the function against its polynomial, which is expressed as:
\[ E_n(x) = |f(x) - P_n(x)| \]
In our exercises, we're particularly interested in keeping this error below a certain threshold, like \( 10^{-2} \), to ensure the approximation is sufficiently accurate. When plotting these errors, students should observe where the approximation often deviates most from the actual curve, typically at the boundaries of the interval.
Understanding error approximation enables us to choose the correct order of the Taylor polynomial for our desired precision.

remainder term

The remainder term, also known as the error term, is crucial for understanding how well a Taylor polynomial approximates a function. It provides a mathematical form of the error.
Given a polynomial \( P_n(x) \), the remainder term \( R_n(x) \) tells us how much of \( f(x) \) isn't captured by the polynomial \( P_n(x) \). For the Taylor series, it's expressed as:
\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1} \]
where \( c \) is some value between the starting point \( a \) and \( x \). This term is critical because it dictates the approximation's accuracy.
Using the maximum absolute value of \( f^{(n+1)}(c) \), denoted as \( M \), allows students to estimate the error bounds over the given interval, ensuring an error below the specified threshold, like \( 10^{-2} \). This real-world application shows how the remainder influences the effectiveness of approximations.

derivative plotting

Plotting the derivatives of a function and its approximations aids in visualizing how closely a Taylor polynomial matches the function. This step involves computing and graphing the derivative necessary for the remainder term calculations.
The plot of the (n+1)st derivative \( f^{(n+1)}(x) \) helps determine the maximum value \( M \) used in estimating the remainder term. By observing this plot, students can pinpoint at which intervals the approximation error may peak.
Moreover, derivative plotting isn't just for error estimation; it also provides insights into the behavior and characteristics of the function. For instance,

• First derivatives show slope (or rate of change).
• Second derivatives convey curvature (or concavity).
• Higher-order derivatives further detail complex behaviors.

These graphs make understanding the dynamics of function approximation more intuitive, reinforcing the practical application of Taylor series in mathematical analysis.