Question 2. Find all second-order three-step methods in the form y+=ay+ay+a₂yn-2" +h[b_f (t+1Y+) +bf (ty)+bf (t-19yn-1)+b₂f (1-2yn-2)], n ≥2 Explain completely how we can obtain the coefficients of this relation.

a.) Calculate L6 for f(x)= (squareroot (9x+1)) over [1,3] Give answer to three decimal places L6= b.) Calculate R5 for f(x)=3x^2+5x over [-1,1] (Give an exact answer. Use symbolic notation and fractions where needed.) R5= c.) Calculate M6 for f(x)=(squareroot(2x)) over [2,5] (Use decimal notation. Give your answer to two decimal places.) M6=

(5b) Prove that if x5 – 3x* + 2x³ – x² + 4x – 1 > 0, then x > 0.

Let P0" (x) denote the quadratic polynomial that interpolates the data {(xo, yo), (x1, yı), (x2, y2)}; let P"."(x) denote the quadratic polynomial that interpolates the data {(x1, yı), (x2, y2), (x3, Y3)}. Finally, let P3(x) de- note the cubic polynomial interpolating the data{(xo, yo), (x1, yı), (x2, y2), (x3, y3)}. Show that 14. (a) (x3 – x)P0.2) (x) + (x – xo) P!.3) (x) P3 (x) = (1,3) X3 - Xo How might this be generalized to constructing P(x), interpolating {(xo, yo), (Xn, Yn)}, from interpolation polynomials of degreen – 1? (b) ....

= mx + b. 4. Let m and b be real numbers, and consider the following three functions: f(x) = 2x + 1, g(x) x + 2, and h(x) A. If function f has a codomain of (5, 7) U (7, 9), the largest its domain can be chosen is (2, 3) U (3, 4). Explain. B. If function g has a codomain of (−1, 1) U (1, 3), the largest its domain can be chosen is (-3, 3) U (3, 9). Explain. C. Within the context of the e- definition of a limit, your result from part A suggests that if € is equal to 2, the largest that can be chosen for function f(x) is 1. Explain. =

Q.1.Find the domain of f+g, f-g, fg and f/g where f(x)=- Vx +4x-60 and x-2 8(x)=2/x-1

13