Question

{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].

a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.

b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:
q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)

Show that q(x) = −(4/π²)x² + (4/π)x.

c. Graph f, p₂, and q on [0, π].

d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.

e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.

Accepted Answer

a. To write the quadratic Taylor polynomial $$p_2$$ for $$f(x) = \sin x$$ centered at $$x = \frac{\pi}{2}$$, start by computing the function value and the first two derivatives at $$x = \frac{\pi}{2}. $$Recall the Taylor polynomial formula up to degree 2:

$$p_2(x) = f\left(\frac{\pi}{2}ight) + f'\left(\frac{\pi}{2}ight)(x - \frac{\pi}{2}) + \frac{f''\left(\frac{\pi}{2}ight)}{2}(x - \frac{\pi}{2})^2$$

Calculate $$f\left(\frac{\pi}{2}ight)$$, $$f'\left(\frac{\pi}{2}ight)$$, and $$f''\left(\frac{\pi}{2}ight)$$ explicitly. b. For the quadratic interpolating polynomial $$q(x) = ax^2$ + bx + c$$, use the given conditions:

$$q(0) = f(0), \quad q\left(\frac{\pi}{2}ight) = f\left(\frac{\pi}{2}ight), \quad q(\pi) = f(\pi)$$

Substitute these points into $$q(x) to $$form a system of three equations:

$$c = f(0)$$

$$a\left(\frac{\pi}{2}ight)^2 + b\left(\frac{\pi}{2}ight) + c = f\left(\frac{\pi}{2}ight)$$

$$a\pi^2 + b\pi + c = f(\pi)$$

Solve this system for $$a$$, $$b$$, and $$c to $$find the explicit form of $$q(x). c. To $$graph $$f(x) = \sin x$$, $$p_2(x$$)$$, and q(x$$)$$ on the interval $$[0, \pi]$$, plot each function using the expressions found in parts (a) and (b). This visual comparison helps to see how well each polynomial approximates $$\sin x$$ over the interval. d. To find the error at points x$$ = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi$$, compute the absolute difference between f(x$$)$$ and each approximation:

$$	ext{Error for } p_2 = |f(x) - p_2(x)|$$

$$	ext{Error for } q = |f(x) - q(x)|$$

Calculate these values for each specified x$ to quantify the accuracy of each polynomial at those points. e. To determine which polynomial, p_2$ or q$, is a better approximation on $$[0, \pi]$$, compare the errors computed in part (d) across the points. Consider the nature of each approximation: p_2$ uses local derivative information at $$\frac{\pi}{2}$$, while q$ uses interpolation at three points. Discuss which method yields smaller errors overall and why, based on the error behavior and the construction of the polynomials.

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Key Concepts

Taylor Polynomial Approximation

Polynomial Interpolation

Error Analysis in Approximation