{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.
a. Find p₅ and q₅
b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?
c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

Verified step by step guidance

1. Step 1: Find the fifth-order Taylor polynomial \(p_5(x)\) for \(f(x) = \sin x\) centered at 0. Recall that the Taylor series for \(\sin x\) at 0 is \(\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}\). The fifth-order polynomial includes terms up to \(x^5\), so write out \(p_5(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}\).
2. Step 2: Find the fifth-order Taylor polynomial \(q_5(x)\) for \(f(x) = \sin x\) centered at \(\pi\). To do this, compute the derivatives of \(\sin x\) at \(x=\pi\) up to the fifth derivative, then use the Taylor polynomial formula centered at \(\pi\): \(q_5(x) = \sum_{n=0}^5 \frac{f^{(n)}(\pi)}{n!} (x - \pi)^n\). Write out the polynomial explicitly by substituting the values of the derivatives.
3. Step 3: To compare the approximations, graph \(f(x) = \sin x\), \(p_5(x)\), and \(q_5(x)\) on the interval \([-\pi, 2\pi]\). Observe where each polynomial closely follows the sine curve. The polynomial centered at 0 (\(p_5\)) will approximate better near 0, and the polynomial centered at \(\pi\) (\(q_5\)) will approximate better near \(\pi\). Identify the intervals where each polynomial is a better approximation by visually comparing the graphs.
4. Step 4: Complete the error table by calculating the absolute errors \(|\sin x - p_5(x)|\) and \(|\sin x - q_5(x)|\) at the given points \(x = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\). For each \(x\), evaluate \(p_5(x)\) and \(q_5(x)\) using the polynomials found in steps 1 and 2, then subtract from \(\sin x\) and take the absolute value.
5. Step 5: Analyze the completed table to determine at which points \(p_5\) is a better approximation (smaller error) than \(q_5\), and vice versa. Also, identify any points where the errors are equal. Explain these observations in terms of the distance of each \(x\) value from the centers 0 and \(\pi\), since Taylor polynomials approximate best near their center points.

Key Concepts

Taylor Polynomials

Taylor polynomials approximate functions near a specific point using derivatives at that point. The nth-order Taylor polynomial of a function f centered at a point a uses derivatives up to order n to create a polynomial that closely matches f near a. For example, p₅ is the 5th-order Taylor polynomial of sin x centered at 0.

Error in Taylor Polynomial Approximation

The error between a function and its Taylor polynomial approximation measures how closely the polynomial matches the function. It is given by the absolute difference |f(x) - pₙ(x)|. Understanding this error helps determine the interval where the approximation is accurate and which polynomial (p₅ or q₅) better approximates the function at different points.

Interval of Convergence and Approximation Accuracy

Taylor polynomials approximate functions best near their center of expansion. The interval of convergence is where the polynomial closely matches the function. For p₅ centered at 0 and q₅ centered at π, each polynomial will better approximate sin x near its center, and the graph and error table help identify these intervals.