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# 524 x 2 2 00000 x 4 00000 x 5 2 5 43656 x 54 x 2

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5)(2)]4 x 2 + 2 . 00000 x * 4 . 00000( x - 0 . 5) 2 + 5 . 43656( x - 0 . 5)4 x 2 Since calculating (as opposed to simplifying) is the point, this final answer is fine. 4. The data in Exercise 2 were generated using the following functions. Use the polynomials constructed in Exercise 2 for the given value of x to approximate f ( x ), and calculate the absolute error.

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4 BENJAMIN JOHNSON a. f ( x ) = e 2 x ; approximate f (0 . 43). solution: Plugging 0 . 43 into the polynomial we got for Exercise 2 part a, we obtain H 3 (0 . 43) = 2 . 36207. Since f (0 . 43) = e 0 . 43 * 2 = 2 . 36316, the absolute error is 0.00109. section 3.4 4. Construct the free cubic spline for the following data. b. x f ( x ) - 0 . 25 1.33203 0 . 25 0.800781 solution: Since there are only two points, there will only be one cubic function S 0 in the cubic spline. This function must have the form S 0 ( x ) = ax 3 + bx 2 + cx + d . Our constraints are that S 0 ( - 0 . 25) = 1 . 33203, that S 0 (0 . 25) = 0 . 800781, and that S 00 0 ( - 0 . 25) = S 00 0 (0 . 25) = 0. From these constraints we need to determine a, b, c and d . We begin by considering the constraints on the second derivative. Starting with our assumed form for S 0 , we have S 0 0 ( x ) = 3 ax 2 + 2 bx + c . Then S 00 0 ( x ) = 6 ax + 2 b . Since this is a linear degree one equation, the only way it can be zero at two distinct points x = - 0 . 25 and x = 0 . 25 is if it is the zero equation, which gives a = b = 0. We thus have that S 0 is a linear function. The slope of this function is 0 . 800781 - 1 . 33203 0 . 25 - ( - 0 . 25) = - 1 . 0625, and so c = - 1 . 0625. Now we can solve for d using (for example) c · 0 . 25 + d = 0 . 800781, obtaining d = 1 . 06641. S 0 ( x ) = - 1 . 0625 x + 1 . 06641 6. The data in Exercise 4 were generated using the following functions. Use the cubic splines constructed in Exercise 4 for the given value of x to approximate f ( x ), and f 0 ( x ) and calculate the actual error.
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• Fall '10
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• Polynomials, Polynomial interpolation, Spline interpolation, benjamin johnson, cubic spline

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