For the given functions fx let x0 1 x1 125 and x2 16

Question

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.

x	f(x)
8.3	17.56492
8.6	18.50515

9. What is the Taylor series of the function:

f(x) = x⁵ - 2x⁴ + 3x³ - 4x² - 10x - 5 , at the point c =17.

Answer 1

1. Utilize the Bisection technique to discover p3 for f (x) = vx - cos x on [0, 1]. 2. Discover an approximation to v3 correct to inside 10-4 using the Bisection Algorithm. [Hint: Consider f(x) = x2 - 3.] 3. Let f(x) = x2 - 6 and p0 = 1. Use Newton's process to find p2. 4. Let f(x) = x2 - 6 and p0 = 3 and p1 = 2, find p3 a) Utilize the Secant technique. b) Employ the process of False Position c) Which of a or b is closer to v6 5. Use Newton's process to discover solutions accurate to within 10-5 to the following problem. a) x2 - 2xe-x + e-2x = 0, for 0 = x = 1. 6. For the specified functions f(x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most 2 to estimated f(1.4) and find the absolute error. a) f(x) = sinpx