2 Given f(x)nd1i/20, i 0,1,20. Note that o1 22 A. Write the formula of the piecewise polynomialS(...
Given f(x) = 1 and x¡ = 1 + i/20, i = 0,1, 20. Note that ao = 1, x20 = 2. A. Write the formula of the piecewise polynomial S(x) on the interval [ri,Ti+11 for i = 0,1, B. Use the error estimate theorem to show that , 19. ()S) 1600 C. Verify the error estimate with r = 1.42
Given f(x) = 1 and x¡ = 1 + i/20, i = 0,1, 20. Note that ao = 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)...
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Q1 (a) Given the function f(x)= x - 5x² - 2x +10. (1) Prove that there at least a root in the interval [1,3] by using Intermediate Value Theorem. (2 marks) (b) (i) Find the root of f(x) by using Bisection method. Iterate until i = 5. (8 marks) Prove the Lagrange interpolating polynomial of second degree for data of (0,1), (1,2) and (4,2) is P2(x) = -* x2 + x + 1. (5 marks)...
The second derivative of f(x) is given below: F"(x) = 22 – 2 – 30 Find the largest open interval where f(x) is concave down. (-6,5) (5,6) (-5,6) (1,00) (-6,-5) (-0,1) Submit Answer Tries 0/99 Communication Blocked Send Feedback Type here to search O e 9 A a 49 12:39 PM 1/9/2020 6
(3.2) Consider the data given in the following table 05 1 15 f(x) 0 2 0 6 1 2 20 (4) (a) Approximate f with a function of the form q (x) = kxm (4) (b) Approximate f with a function of the form g2(x) = be Which approximation between q and g2 1s more appropriate for the given data? Justify your (3) (c) answer < In, and a piecewise cubic polynomial Consider a set of points (I,) Such that...
2. Graph the functions f(x)x(x 1)(x-2) ..(x- k) for k- 1,2,..,10. (These are examples of the polynomials occurring in the error formula for polynomial interpolation.) We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [O,T/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h-/2n...
Please explain the solution and write clearly for nu, ber 25.
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25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
Verify Property 2 of the definition of a probability density function over the given interval. f(x)=3, [03] Next, determine F(x). First, find the antiderivative off. (3 dx = 3x 3x+C Let C = 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over the far right side of the formula for theprea. 0-0 [0,1] using area =
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...