# Optimum Design of Mechanical Elements and Systems

**Problem 1:** For the following functions:

find the Taylor series about (1,1) up to quadratic term.
Then, using the Taylor series,

predict the function value at (1.1, 0.8). Compare against exact value.

**Problem 2**: For the generic quadratic function:

classify the stationary point (minimum, maximum, saddle,
'other') based on the

coefficients. Using this general expression, classify the stationary points of

**Problem 3:** For the following quadratic functions:

find the stationary point(s) (using MATLAB's ezmesh,
ezcontour, fminunc, …), and

classify.

**Problem 4:** Consider the function:

Find the minimum using fminunc. Now find the minimum
within the region

0 < u < 0.5 and 0 < v < 0.1 using a naïve random search using K = 1,00,0000

(maximum) random searches. Plot the error (distance from exact to result from

random search) for K =100, K =1000, K = 10000, K = 100,000 and K =

1,00,0000. There is no need to submit your Matlab code; just submit the plot.

**Problem 5:** Consider the function f (x,y) = 3x^2−xy
+ 4y^2−x −y . Find the

minimum by performing line-search in two directions, first in x then in y,
starting at

(0,0). (You need to do this problem by hand, but you are welcome to confirm your

answers with the line-search code.)

**Problem 6:** Consider the function f (x,y) = x^2 −xy
+ 4y^2 + x −3 . Suppose a

direction d = (1,1) is given. Find the corresponding conjugate direction. Using
this

pair show that you can find the minimum from the origin in two steps.

**Problem 7:** Write a Matlab function to find the
minima of an arbitrary N-D function

via the Powell's method (you may use the line-search method provided, but at
your

own risk!). Your code should be of the form:

function [xMin,fMin,iter] = ...

Powell_LastName(f,xBar0,xTol,fTol,maxIterations)

where the function parameters are as before. Test your
code against a variety of

quadratic and non-quadratic functions. Email only the Powell code (and not the
test scripts).