6. A CLAMPED cubic spline s for a function f is defined by \begin{equation*} s(x) = \begin{cases} s_0(x) = 1 + Bx + 2x^2 - 2x^3, & \text{if } 0 \le x \le 1, \\ s_1(x) = 1 + b(x - 1) - 4(x - 1)^2 + 7(x - 1)^3, & \text{if } 1 \le x \le 2. \end{cases} \end{equation*} Find $f'(0)$ and $f'(2)$.
Added by Diamond B.
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For 0 ≤ x ≤ 1: -5x = 1 + Bx + 2x^2 - 2x^3 Rearranging the equation: 2x^3 - 2x^2 - Bx - 5x + 1 = 0 For 1 ≤ x ≤ 2: 5x = 1 + bx - 1 - 4(x - 1) + 7x - 13 Simplifying the equation: 5x = bx - 4x + 4 + 7x - 13 5x = bx + 3x - 9 Rearranging the equation: (b + 3)x - 5x Show more…
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A natural cubic spline S on [0, 2] is defined by S(x) = { S0(x) = 2 - x + 3x^3, if 0 <= x < 1 S1(x) = a + b(x - 1) + c(x - 1)^2 + d(x - 1)^3, if 1 <= x < 2 } Find a,b,c,d.
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7. (5 points) Determine the coefficients a, b, and c such that S is a cubic spline function S(x) = { x^3, 0 <= x <= 1 1/2(x - 1)^3 + a(x - 1)^2 + b(x - 1) + c, 1 <= x <= 3 } A. a = 1, b = 2, c = 1. B. a = 3, b = 3, c = 1. C. a = 3, b = 1, c = 1. D. a = 1, b = 3, c = 3.
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