Chapter 10.7, Problem 11P

2. Show that 8 xa S -dx (b² + 12) dr = 2 cos(π2) пра-1 a, b real and -1 0 Your solution should clearly explain the closed contour you are using, and state clearly any vanishing properties of integrals over contours that are being used. You are free to quote from the lectures, the appropriate results on such vanishing properties, without deriving these properties. Any residue calculations involved should be explained clearly.

1. Calculate the integral 500 x sin x (a² +x2)20 dx by using the residue theorem. You may assume that a is real and a > 0. Your solution should clearly explain the closed contour you are using, and state clearly any vanishing properties of integrals over contours that are being used.

Steel Production Planning: Let S  represent the amount of steel produced (in tons). Steel Production is related to the amount of labor used(L) and the amount of capital used ( C ) by the following function:S = 20L⁰˙³⁰C⁰˙⁷⁰In this formula L represents the units of Labor input and C the units of Capital input. Each unit of Labor costs $50, and each unit of Capital costs $100. a:Formulate an optimization problem that will determine how much labor and capital are needed to produce 50,000 tons of steel  at minimum cost.                                                                                                                                                                                             Q#3B: Solve the optimization problem you formulated in part (a). (Hint: When using Excel Solver, start with an initial L>0 and C>0.)…

Q/Draw the graph k 3,4, and extract perfect ma ching from it

π sin2x sin3x sin5x sin30x sinx - sin6x - sin10x - sin15x) 2 dx

f(x+T) = f(x) f(x) = sin² (x) + cos² (x) T=?

Q/ Give an example of a simple graph G Such that h (G) is Euler but G is not.

9 / prove that: Let G be agraph with n vertices, ny 2 و the h G has at Least two vertices which are not cut vertices.

/prove that:- Let G be agraph. Then X(G) 3 iff G has an odd cycle.