1. Use the Bisection method to find p3 for f (x) = √x - cos x on [0, 1].
2. Find an approximation to √3 correct to within 10-4 using the Bisection Algorithm. [Hint: Consider f(x) = x2 - 3.]
3. Let f(x) = x2 - 6 and p0 = 1. Use Newton's Method to find p2.
4. Let f(x) = x2 - 6 and p0 = 3 and p1 = 2, find p3
a) Use the Secant method.
b) Use the method of False Position
c) Which of a or b is closer to √6
5. Use Newton's method to find solutions accurate to within 10-5 to the following problem.
a) x2 - 2xe-x + e-2x = 0, for 0 ≤ x ≤ 1.
6. For the given functions f(x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f(1.4) and find the absolute error.
a) f(x) = sinπx
7. Determine the natural cubic spline S that interpolates the data f (0) = 0, f (1) = 1, and f (2) = 2.
8. Construct the natural cubic spline for the following data.
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x
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f(x)
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8.3
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17.56492
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8.6
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18.50515
|
9. What is the Taylor series of the function:
f(x) = x5 - 2x4 + 3x3 - 4x2 - 10x - 5 , at the point c =17.