748: ME Optimum Design of Mechanical Elements and Systems Fall 2008; HW-1 Due: 17th Sept 2008 (Wed), 5 pm. Problem 1: Find the largest rectangle that can fit inside the ellipse:
x y + =1 a b by posing a one variable optimization problem in a standard form, and solving for the minimum analytically.
2 2
Problem 2: Find and classify the stationary points for: (a) f (x ) = x 3 + 6x 2 15x + 2 , (b) f (x ) = x 2e x , (c) f (x ) = x + x 1 Problem 3: Consider a beam of sectional modulus EI and length L that is pinned at both ends. A load P is applied at a distance L/3 from the left end and a load P/2 is applied at the center. Find the location and value of maximum deflection. You may use the following equation for the deflection.
Problem 4: (Refraction Law of Optics). Let p and q be two points on the plane that lie on opposite sides of a horizontal axis. Assume that the speed of light from p to the horizontal axis is v, and from the horizontal axis to q is w. Find the fastest path from p to q (the path that a light ray will take). Pose as a one variable optimization problem and solve analytically. Problem 5: Prove the following: Suppose a function f (x ) is infinitely differentiable and d n f (x * ) = 0 for n = 1, 2, 3,..., 2m 1 dx n d f 2m (x * ) >0 dx 2m then x * is a local minimum.
Problem 6: Use the above Lemma to decide whether 0 is a local minima for f (x ) = x n for n = 1,2, 3,... Problem 7: Suppose f (x 1 ) = f1; f (x 2 ) = f2 ; f (x 3 ) = f3 , where x 1 < x 2 < x 3 , and f1 > f2 < f3 . Find an estimate for the local minimum x * via parabolic interpolation. Your expression should involve the three points x 1, x 2 , x 3 and the three function values f1, f2 , f3 . Problem 8: Write a Matlab function to find the minima of a 1-D function via the Golden Section method (you may use a 'while loop' instead of recursive calls). Your code should be of the form:
function [xmin, fmin,nIterations] = fminViaGoldenSection_LastName(f,x0,x1,alpha,xtol,fTol,maxIteration...