Question

{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.

Accepted Answer

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)!}$$. The fifth-order polynomial includes terms up to x^5$, so write out p_5$(x) = x - \frac{x^3$}{3!} + \frac{x^5$}{5!}. $$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. 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. 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. 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.

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Key Concepts

Taylor Polynomials

Error in Taylor Polynomial Approximation

Interval of Convergence and Approximation Accuracy