2. Determine cı and c2, 1, and 2 such that the integration formula gives the exact...
J0 3. Determine cı and cz in f(x) dac1f(0) + cf(1) so that it is exact for all polynomials of as large a degree as possible. Find the degree of precision, on Pharnt.
Numerical Analysis: a) The basic form of the Gaussian quadrature formula is The integration formula using two points can be made exact when a polynomial of order 3 is integrated. i) Determine the weights c, c2 and the points , 2 e-radz.16] (ii) By find using a change of variable use Gaussian Quadrature to 0 a) The basic form of the Gaussian quadrature formula is The integration formula using two points can be made exact when a polynomial of order...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b) T3 finite element is defined over...
Exercise 4 Leta(c)-c1/2 and let c2 > cı > 0 be given. Let: π1c1+12c2. where π2 = 1-T1. (i) Sketch the function u and indicate in your sketch the points (C1, u(a), (c, u(c)), and (c2,u(c2)). (ii) Draw the line that connects the two points (ci, u(cı)) and (c2, u(c2)) and represent that line algebraically. Hint: Find the slope and intercept in terms of the two points, (c1, u(c) and (c,,u (сг)).] (iii) Use that algebraic result to show that...
dont ans this question Euler's method is based on the fact that the tangent line gives a good local approximation for the function. But why restrict ourselves to linear approximants when higher degree polynomial approximants are available? For example, we can use the Taylor polynomial of degree about = No, which is defined by P.(x) = y(x) + y (xo)(x – Xa) + 21 (x- This polynomial is the nth partial sum of the Taylor series representation (te) (x –...
1. Simpson's rule. Simpson's rule is a different formula for numerical integration of lºf (d.x which is based on approximating f(2) with a piecewise quadratic function. We will now derive Simpson's rule and relate it to Romberg integration: a. Suppose that (2) is a quadratic polynomial so that q(-h) = f(-h), q0) = f(0) and q(h) = f(h). Prove that 92 f(-h) + 4f(0) + f(h)). -h b. Suppose that the interval [a, b] is divided by a = 20,...
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC 4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC
Problem 4 Let f(x) be a cubic polynomial defined on interval [−1, 1]. Determine a Gaussian intergration formula with minimal number of nodes such that the integral formula Xn i=0 f(xi)wi is exact for cubic polynomials. Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,. Determine a Gaussian intergration formula with minimal number of nodes such that the integral formula is exact for cubic polynomials. Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,....
1. Harvey Habit's utility function is U (C1, C2) = min {c1, c2}, where cı is his consumption of bread in period 1 and c2 is his consumption of bread in period 2. The price of bread is $1 per loaf in period 1. The interest rate is 21%. Harvey earns $2,000 in period 1 and he will earn $1,100 in period 2. (a) Write Harvey's budget constraint in terms of future value. (b) How much bread does Harvey consume...
1 1. Find the exact sum of the following infinite series as indicated below: -1 1 1 1 1 + n(-4) 2(16) 3(64) 4(256) a. Let f(x) = 2n=1 (-1) x". I n a b. Find the power series for the derivative f'(x), and observe that it is a geometric series. Find its first term and common ratio. c. Use the formula 1-r to find an algebraic expression for f'(x). d. Integrate to find an algebraic expression for f(x). Make...