24-311 NUMERICAL METHODS Fall 03 Carnegie Mellon University PROBLEM SET 7 Issued: 10/10/2003 Due: 10/17/2003 4:00PM @ HH B127 Weight: 3% of total grade PS7-1 Directional Derivative Find the directional derivative of 22(, ) 2fxy x y=+ at 2x= and 2y= in the direction of 32=+hij. (Read the textbook pp. 360-363 for the definition of the directional derivative.) PS7-2 Gradient Vector and Hessian Matrix Find the gradient vector and Hessian matrix for 2(, ) 2 3xyfxy xy e=+ PS7-3 Necessary Conditions for Min/Max Given 22(, ) 2 1.5 1.25 2fxy xy y x y=+ − −, construct and solve a system of linear algebraic equations, or a matrix equation, that maximizes (, )fxy. Note that this is done by setting the partial derivatives of the function with respect to x and y to zero.PS7-4 2D Unconstrained Optimization--Equilibrium Position of a Spring System Consider the simple two-spring system shown in the following figure. The springs are assumed to be linearly elastic and the loads N51=p and N52=p are constant. This is a geometrically nonlinear problem because the resistance to the load is a function of the deformed position. The original length of the two springs are cm101=l and cm102=l , and two spring constants are N/cm81=k and N/cm12=k . (1) Find the total energy of the system ),( yxE as a function of x and y . (2) Use Mathcad to draw the surface plot of the function, ),( yxE . Use the “color” shading option. Your plot should look similar to Figure 2. (3) Use Mathcad to draw the contour plot of the function, ),( yxE . Draw at least 40 contours. Your plot should look similar to Figure 3. Figure 2. Surface plot Figure 3. Contour plot xyPS7 The first letter of __________________________ ____________________________ your LAST name First Name Last Name PS7-1 (20 pts) PS7-2 (20 pts) PS7-3 (20 pts) PS7-4 (40 pts) Total (100 pts) 24-311 NUMERICAL METHODS Fall 03 Carnegie Mellon University PROBLEM SET 7 Issued: 10/10/2003 Due: 10/17/2003 4:00PM @ HH B127 Weight: 3% of total