f(0.5) ≈ 0.375(0) - 0.25(0.8414709848079) + 0.0625(0.9092974268257) ≈ 0.479425538.
f(x) ≈ L0(x) f(x0) + L1(x) f(x1) + L2(x) f(x2)
Use the bisection method to find a root of the equation x^3 - 2x - 5 = 0. First Course In Numerical Methods Solution Manual
Evaluating these expressions at x = 0.5, we get:
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L0(0.5) = 0.375, L1(0.5) = -0.25, L2(0.5) = 0.0625.
The bisection method involves finding an interval [a, b] such that f(a) and f(b) have opposite signs. In this case, we can choose a = 2 and b = 3, since f(2) = -1 and f(3) = 16. The midpoint of the interval is c = (2 + 3)/2 = 2.5. Evaluating f(c) = f(2.5) = 3.375, we see that f(2) < 0 and f(2.5) > 0, so the root lies in the interval [2, 2.5]. Repeating the process, we find that the root is approximately 2.094568121971209. Using Lagrange interpolation
Using Lagrange interpolation, we can write the approximate value of f(x) as:
Use Lagrange interpolation to find an approximate value of the function f(x) = sin(x) at x = 0.5, given the data points (0, 0), (1, sin(1)), and (2, sin(2)).
f(0) = 0, f(1) = sin(1) ≈ 0.8414709848079, f(2) = sin(2) ≈ 0.9092974268257.