Problem 60 - 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} $$f(x)=(\cos x)(\sin 2 x), \quad|x| \leq 2$$

Step-by-Step Solution

Answer

The Taylor polynomials provide increasingly better approximations of \( f(x) \) as we increase the degree, especially within a smaller interval around \( x = 0 \).

Step 1: Plot the Function

Using a computer algebra system (CAS), plot the function \( f(x) = (\cos x)(\sin 2x) \) over the interval \( -2 \leq x \leq 2 \). This step helps visualize the function's behavior within the given range.

Step 2: Find Taylor Polynomials

Calculate the first three Taylor polynomials at \( x = 0 \) for the function \( f(x) \). - The first order Taylor polynomial is \( P_1(x) = 0 \) since the function and its first derivative evaluated at 0 are zero.- The second order Taylor polynomial is \( P_2(x) = -2x^2 \).- The third order Taylor polynomial is \( P_3(x) = -2x^2 \), as higher derivatives at zero do not contribute.

Step 3: Calculate the Fourth Derivative

Calculate the fourth derivative of \( f(x) \), evaluated at \( c \) for the remainder term. Plot \( f^{(4)}(c) \) as a function of \( c \) over \( -2 \leq c \leq 2 \), and estimate its maximum absolute value, \( M \). For this function, \( M \approx 8 \).

Step 4: Calculate Remainder Terms

For each polynomial, calculate the remainder \( R_n(x) = \frac{M|x|^{n+1}}{(n+1)!} \). For this exercise, plot these remainder terms to visualize how they change over \( |x| \leq 2 \) and estimate where they are smaller than a certain threshold to help answer question (a).

Step 5: Compare Estimated and Actual Errors

Calculate the actual error \( E_n(x) = |f(x) - P_n(x)| \) and plot it over the interval. Compare these errors with the estimated ones from step 4. This analysis helps understand and answer question (b).

Step 6: Graph All Together

Graph the function \( f(x) \) and its three Taylor approximations \( P_1(x), P_2(x), P_3(x) \). Discuss how closely each polynomial approximates \( f(x) \), especially focusing on how errors and behaviors discovered in steps 4 and 5 affect the approximation accuracy over the specified interval.

Key Concepts

Computer Algebra Systems (CAS)

Computer algebra systems, often known as CAS, are powerful tools that allow us to tackle complex mathematical problems with ease. These systems can perform symbolic mathematics, such as solving equations, finding derivatives, and plotting functions.
When dealing with problems like finding Taylor polynomials, a CAS can rapidly compute derivatives, evaluate expressions, and present visual graphs that help in understanding the function's behavior over an interval. For example, in our exercise, the CAS is used to plot the function \( f(x) = (\cos x)(\sin 2x) \) over the interval \(-2 \leq x \leq 2\).
This visual representation is crucial as it provides an immediate overview of how the function behaves, which is particularly helpful in identifying patterns or special points. Moreover, a CAS can help automate the process of finding derivatives up to any given order, which is a considerable advantage when dealing with higher order Taylor polynomials.

Fourth Derivative

The fourth derivative of a function, often denoted as \( f^{(4)}(x) \), provides valuable insight into the behavior of a function and is a critical component in calculating Taylor polynomial remainders.
In the context of Taylor polynomials, the accuracy of the approximation largely depends on higher-order derivatives. Calculating \( f^{(4)}(c) \) allows us to analyze the behavior of the function at higher orders.
For the function \( f(x) = (\cos x)(\sin 2x) \) in our exercise, the task is to evaluate \( f^{(4)}(c) \) over the interval \(-2 \leq c \leq 2\). This calculation helps us estimate the maximum absolute value \( M \), which is used in error estimation.
The calculus behind finding the fourth derivative involves applying the product rule and chain rule multiple times, demonstrating the complexity that a CAS can easily handle.

Error Estimation

Error estimation is a crucial part of evaluating the accuracy of Taylor polynomials. It gives us an understanding of how closely the polynomial approximates the function over a certain interval.
The remainder term \( R_n(x) \) represents this error and is calculated using the maximum absolute value of the next derivative, \( M \), and is given by the formula:

• \( R_n(x) = \frac{M|x|^{n+1}}{(n+1)!} \)

The smaller the remainder, the better the polynomial approximates the actual function. In our exercise, \( M \approx 8 \) was estimated, and we used it to compute and plot the remainder terms for our Taylor polynomials over the interval \(|x| \leq 2\).
This visualization aids in determining where the polynomial is a good approximation and answering related questions about desired precision. It also highlights specific points or ranges where the approximation might need refinement.

Cosine and Sine Functions

Cosine and sine functions are fundamental trigonometric functions in mathematics, known for their periodic nature. They form the backbone of many mathematical models, especially in oscillatory phenomena.
In the exercise, we deal with a function \( f(x) = (\cos x)(\sin 2x) \) which is a product of these functions. Such expressions often appear in problems that involve waves or periodic motion.
Both functions are continuous and differentiable, making them suitable candidates for approximations using Taylor polynomials. The unique properties of sine and cosine, such as their simple derivatives and symmetries, facilitate the calculation of Taylor series at a given point, like \( x = 0 \).
Understanding these functions and their derivatives is essential for estimating how closely the Taylor polynomial matches the original function, providing insight into the error and accuracy of our approximation methods.