Elementary Numerical Analysis : Gaussian Numerical Integration

1. Show that if an integration formula of the form In ( f ) = ∑ wjf(xj) is exact when integrating 1, x, x2, …, xm, then it is exact for all polynomials of degree ≤ m. Please see attached for proper format of question.

Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Consider approximating integrals of the form... in which f(x) has several continuous derivatives on [0, 1] a. Find a formula... which is exact if f(x) is any linear polynomial. b. To find a formula... which is exact for all polynomial of degree ≤ 3, set up a system of four e ...continues

Gaussian Quadrature

1. Consider approximating integrals of the form I ( f ) = ∫ √x f(x)dx in which f(x) has several continuous derivatives on [0, 1] a. Find a formula ∫ √x f(x)dx ≈ w1 f(x1) ≡ I1( f ) which is exact if f(x) is any linear polynomial. b. To find a formula ...continues

Newton-Cotes formula

Derive all the weights for closed Newton-Cotes formula. Please see attached for full question.

Newton-Cotes formula for n=4

Derive all of the weights for the closed Newton-Cotes formula for n=4(the so-called Milne rule).

Cubic polynomial interpolation

1. a) Consider the problem of cubic polynomial interpolation p(xi) = yi, I = 0,1,23 with deg(p) ≤ 3 and x0, x1, x2, x3 distinct. Convert the problem of finding p(x) to another problem involving the solution of a system of linear equations. b) Express the system from (a) in the form Ax = b, i ...continues

Key Metrics and Ratios for Abercrombie and Fitch Company

Using Abercrombie and Fitch Company (data attached): You are interested in learning moe about from an investment standpoint. Identify KEY METRICS and RATIOS of the company that will give a good indication of how "investment worthy" it is. 1) List the key financial metrics and ratios for Abercrombie and Fitch for that comp ...continues

Histograms, Frequency Polygons, Mean, Median, Mode, Class Mid-Point, Pie and Bar Charts

Q1. The masses of 50 similar castings are measured correct to the nearest 0.1 kg, and the results are shown below. 8.1 7.7 7.4 8.6 7.8 8.1 8.6 8.0 7.3 9.0 9.0 7.3 8.5 7.7 8.3 7.9 7.5 8.1 7.1 7.8 9.1 7.6 8.2 8.4 8.5 9.0 7.8 7.6 8.4 7.7 8.2 9.0 7.2 8.3 7.4 8.1 8.3 8,5 8.7 7.9 7.5 8.9 7.7 7.1 8.2 9.1 7.1 8.8 8.0 8.8 (a) From ...continues

Maximum-Minimum Theorem, Limits, Continuity and Function Composition

1) Let f, g be defined on R and let c in R. Suppose that lim f = b and that g is continuous at b. Show that lim g 0 f = g(b) Note: R: real numbers g 0 f means composition of f and g 2) Let A = [0, 1) U (1,2]. Let B = [0, 1] U [2, 3]. Does the conclusion of the maximum-minimum theorem always hold for a function f: A ...continues

Maximum-Minimum Theorem : Continuity and Bounded Functions

(d) Does the conclusion of the Maximum-Minimum Theorem always hold for a bounded function f : R --> R that is continuous on R? Prove or give a counterexample. (a) Fix a, b E R, a < b. Prove that if f [a, b] -->R is continuous on [a, b] and f(x)≠0 for all x E [a, b], then 1/f(x) is bounded on [a, b]. (b) Find a, b E R, a ...continues