Question

Graph \(y=\cos x\) and the Taylor polynomial $$ T_{2 M}(x)=\sum_{n=0}^{M} \frac{(-1)^{n} x^{2 n}}{(2 n) !} $$ on the interval \((-2 \pi, 2 \pi)\) for \(M=1,2,3, \ldots,\) until you find a value of \(M\) for which there's no perceptible difference between the two graphs.

Short answer

Answer: The value of M for which the Taylor polynomial approximation provides a close approximation to the original cosine function on the interval (-2π, 2π) is approximately M ≈ 4 to 6. This may vary depending on the level of precision or the graphical representation of the functions.

Step-by-step solution

Recalling the Taylor Polynomial formula

The Taylor polynomial \(T_{2M}(x)\) is given by $$ T_{2 M}(x)=\sum_{n=0}^{M} \frac{(-1)^{n} x^{2 n}}{(2 n) !}. $$ This is an approximation of \(\cos x\) using even powers of \(x\) and alternating positive and negative terms.

Graphing y = cos x

First, we need to graph the function \(y=\cos x\) on the interval \((-2\pi, 2\pi)\). This will serve as a reference for comparing the Taylor polynomial approximations. The graph of \(y = \cos x\) will be a wave with period \(2\pi\), starting at a maximum value of 1 at \(x = 0\), and oscillating between 1 and -1.

Graphing the Taylor Polynomial approximation

Next, we will graph the Taylor polynomial approximation \(T_{2M}(x)\) on the same interval, for different values of \(M\). Start with \(M=1\) and increment \(M\) until there is no noticeable difference between the graphs of \(y = \cos x\) and \(T_{2M}(x)\). - For \(M=1\): $$ T_{2}(x)=1-\frac{x^2}{2!}. $$ This graph will be a parabola with a vertex at \((0,1)\) opening downward. - For \(M=2\): $$ T_{4}(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}. $$ This graph will resemble the cosine function more closely, with additional turning points. - Continue incrementing \(M\) and graphing the Taylor polynomial approximation: - For \(M=3\): $$ T_{6}(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}. $$ - For \(M=4\): $$ T_{8}(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}. $$ - Keep incrementing \(M\) until no perceptible difference between graphs.

Determine the value of M

By visually comparing the graphs of \(y = \cos x\) and the Taylor polynomial approximation, we can estimate the value of \(M\) for which there is no noticeable difference between the two graphs. This value of \(M\) may vary depending on the level of precision required or the graphical representation of the functions. Nonetheless, in general, a relatively small value of \(M\) (such as \(M \approx 4\) to \(6\)) should provide a close approximation to the original cosine function on the interval \((-2\pi, 2\pi)\).

Key concepts

Cosine Function

The cosine function, denoted by \(y = \cos x\), is a fundamental trigonometric function that represents the x-coordinate of a point on the unit circle as the angle x varies. The graph of \(\cos x\) appears as a continuous wave that oscillates between the values of 1 and -1.
The wave completes one full cycle over an interval of \(2\pi\), which is called its period. At the very start of the graph, \(\cos x\) reaches its maximum value of 1 at \(x = 0\), and then goes through zero, reaching its minimum value at \(\pi\) before returning back to the maximum value at \(2\pi\).
This regular oscillation makes the cosine function very useful in various applications, especially those dealing with periodic phenomena like sound and light waves.
Understanding the behavior and properties of the cosine function is crucial when analyzing how well approximations capture its shape over specific intervals.

Approximation

Approximations are crucial in mathematics when an exact function may be too complex to handle or compute. In this context, we aim to approximate the cosine function using Taylor polynomials. Taylor polynomials are valuable mathematical tools that approximate functions as sums of polynomial terms based on derivatives at a specific point, usually around zero (Maclaurin Series).

• The polynomial we use is written in the form \(T_{2M}(x) = \sum_{n=0}^{M} \frac{(-1)^{n} x^{2n}}{(2n)!}\).
• This particular Taylor series involves only even powers of \(x\), since the cosine function is even.
• The terms alternate in sign, reflecting the role derivatives of the cosine function play at each step, which are also alternating.

Each added polynomial term improves the approximation's accuracy, particularly closer to the center of the interval. Understanding and calculating these approximations helps illustrate how complex functions can be represented robustly in simpler forms, which is especially important in computing and engineering contexts.

Graphing Techniques

Graphing plays a critical role in understanding how well one function approximates another. In this exercise, the goal is to visually assess the accuracy of Taylor polynomial approximations against the original \(\cos x\) over the interval \((-2\pi, 2\pi)\).
To do this:

• Begin by graphing the cosine function itself, which provides the reference wave pattern.
• Successively overlay the graphs of the Taylor polynomial approximations \(T_{2M}(x)\) starting from low degrees \(M=1\) upward, noting how each successive overlay better mimics the cosine wave.
• With \(M=1\), the Taylor polynomial yields a simple parabolic curve: \(T_2(x) = 1 - \frac{x^2}{2!}\).
• As \(M\) increases, the graph starts resembling the cosine wave more closely, capturing more of its cycles and amplitude.

Through this visual method, one can see not only the convergence of the polynomial to \(\cos x\) but also the respective roles each polynomial term plays in shaping the approximation.

Polynomial Degrees

The degree of a Taylor polynomial significantly affects the quality of approximation it provides. In this exercise, each increase in degree offers a new level of precision in approximating \(\cos x\) within the interval \((-2\pi, 2\pi)\).

• For \(M=1\), a low-degree polynomial captures only the basic symmetrical drop around zero.
• When \(M=2\), the polynomial \(T_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}\) starts resembling the cosine curve more closely, adding subtle undulations.
• With \(M=3\), adding one more term \(-\frac{x^6}{6!}\) refines the approximation further, smoothing discrepancies observed with previous approximations.
• Beyond \(M=4\), the added terms provide diminishing but still valuable returns on approximation quality.

The decision on which degree polynomial to use often depends on the balance between desired precision and computational effort. In practice, polynomial degree is chosen based on the application's precision needs while maintaining efficiency.