Question 2 (20 Points) (1) Use the Bisection method to find solutions accurate to within 10-2...
(1) Use the Bisection method to find solutions accurate to within 10-2 for x3 – 7x2 + 14x – 6 = 0 on the interval [3.2, 4]. Using 4-digit rounding arithmatic. (2) Consider the function f(x) = cos X – X. (a). Approximate a root of f(x) using Fixed- point method accurate to within 10-2 . (b). Approximate a root of f(x) using Newton's method accurate to within 10-2. Find the second Taylor polynomial P2(x) for the function f(x) =...
Question 4 (16 Points) Use Neville's method to approximate V11 with the following function and values. Use this polynomial to approximate f(11). Using 4-digit rounding arithmatic. (a). f(x) = (x and the values Xo = 6, xı = 8, x2 = 10, x3 = 12 and X4 = 13. (b). Compare the accuracy of the approximation in parts (a).
Question 1 (20 Points) Find the second Taylor polynomial P2(x) for the function f(x) = ex cos x about Xo = 0. Using 4-digit rounding arithmatic. (a). Use P2(0.7) to approximate f(0.7). (b). Find the actual error. (c). Find a bound for the error f(x) - P2(x) in using P2(x) to approximate f(x) on the interval [0, 1].
[20 Marks] Question 2 a) Given f(x)= x - 7x2 +14x-6 i) Show that there is a root a in interval [0,1] (1 mark) ii) Find the minimum number of iterations needed by the bisection method to approximate the root, a of f(x) = 0 on [0,1] with accuracy of 2 decimal points. (3 marks) iii) Find the root (a) of f(x)= x - 7x² +14x6 on [0,1] using the bisection method with accuracy of 2 decimal points. (6 marks)...
1) Use the bisection method to find the third approximation of 2 starting with the initial interval [1,2], and find the corresponding absolute error. Also, compute the number of iterations needed to achieve an approximation accurate to within 10 Then, use the suitable one to compute the second approximation of the root using xo,and find an upper bound for the corresponding error. 1) Use the bisection method to find the third approximation of 2 starting with the initial interval [1,2],...
3. Use Newton's method to find solution accurate to within 10-3 for x3 + 3x2 – 1 = 0 on (-3,-2]. Use po -2,5. 4. Use Secant method to find the solution P4 for In(x - 1) + cos(x - 1) = 0 on [1.3,2]. Use po 1.3 and p1 = 1.5. 5. Use False position method to find the solution P4 for 3x – e* = 0 on [1,2]. Use - Ро 1 and P1 2.
5. For each of the following functions, and the corresponding initial interval, tell whether Bisection method can be applied to find a root in the interval, and if so, how many iterations are required to achieve the associated accuracy. Recall 10-G1 (b-a). (a) f(x) = sin(x), [-1, 1], E = 2-16 (b) f(x) = sn'(x), [-1, 1], € = 2-16 (c) f(z) = cos(x), [-1, 1], ε = 2-16 7. Show that Newton's method for finding the root of a...
Find the smallest positive root for the given function by using the bisection method with accuracy 10^-3 f(x) = 2x5 – x3
Find the second Taylor polynomial P2(x) for the function f(x) = ex cos x about xo = 0. Using 4-digit rounding arithmatic. (a). Use P2(0.7) to approximate f(0.7). (b). Find the actual error. (c). Find a bound for the error |f(x) – P2(x) in using P2(x) to approximate f(x) on the interval [0, 1].
Question 3 ( 14 Points) (a). Use the numbers (called nodes) Xo = 2.0, x1 = 2.4, and x2 = 2.6 to find the second Lagrange interpolating polynomial for f(x) = sin(In x). Using 4-digit rounding arithmatic. (b). Use this polynomial to approximate f(1). Using 4-digit rounding arithmatic.