MTH 2001: Project 2 Instructions Each group must choose one problem to do, using material from chapter 14 in the textbook. Write up a solution including explanations in complete sentences of each step and drawings or computer graphics if helpful. Cite any sources you use and mention how you made any diagrams. Write at a level that will be comprehensible to someone who is mathematically competent, but may not have taken Calculus 3. Use calculus, but explain your method in simple terms. Your report should consist of 8090% explanation and 1020% equations. If you find yourself with more equations than words, then you do not have nearly enough explanation. See the checklist at the end of this document. One person from each group must present the work orally to Naveed or Ali. Presenters must make an appointment. Visit the Calc 3 tab: http://www.fit.edu/mac/group_projects_presentations.php Submit written work to the Canvas dropbox for Project 2 by October 7 at 9:55PM. The deadline for the oral presentation is October 7 at 2PM. Problems . One method for finding extreme values of a function f of two (or more) variables is gradient descent (or ascent). The idea is to guess a point in the domain that could be a minimum, and follow a certain path in the domain to reach a much more precise approximation of the location of a minimum. In order to do this, we find the direction of steepest decline, take a small step in that direction, and repeat over and over until we reach what seems to be a minimum. The method would look like the following on a contour map (i.e. taking the most direct path between level curves). Gradient ascent uses the same idea to find maxima, but follows the steepest incline rather than decline. (a) Assuming we are able to find partial derivatives of f, use ideas from 14.6 to write down the steps for carrying out a gradient descent strategy. How can we know when to stop (i.e. how do we know if we have reached a minimum)? What is different for the method of gradient ascent? [Hint. The inputs to our method will be f, its partial derivatives, and a point in the domain we guess is a minimum.] (b) Suppose we have function f with continuous partial derivatives, but we are not able to calculate them. How can we modify the process in part (a) so that it can still work? (How could we determine in which direction f has a steepest decline without the partial derivatives?) (c) Suppose our method converges to a relative minimum that is not an absolute minimum. How canwe check to see if there are other minima? (d) Use a gradient ascent method to find the 3 maxima for part (d) of problem 2 above. (e) Use a gradient descent and ascent method to approximate the absolute minimum and maximumof where 2 x 2, 2 y 2. Below is f on the domain, with red colors corresponding to high points and blue colors low points