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.

Risposta video verificata per un problema simile:

Concetti chiave

Taylor Polynomial Approximation

Polynomial Interpolation

Error Analysis in Approximation