SUNY Geneseo Department of Mathematics
Monday, March 26
Math 223 04
Spring 2018
Prof. Doug Baldwin
There’s now a set of sample exam questions with solutions available on Canvas.
There’s also a table showing how I converted numeric grades to letter grades last time I taught this course; the conversion won’t be exactly the same this semester, but historical information might give you some sense of how you’re doing.
On the chain rule for partial derivatives, directional derivatives, gradients.
See the handout for details.
For z = x3/3 - x2 + y2 + 2y on -3 ≤ x ≤ 3, -2 ≤ y ≤ 2
Critical points:
Candidate extreme values on boundary:
What more do we need? Values of z at the corners!
Now look for the minimum and maximum z values in all these points, and get...
The process for finding absolute extremes; in particular remember to check boundaries and corners.
Section 4.8.
A method for solving optimization problems with a constraint, i.e., problems in which you want to minimize or maximize something, but have to do it while preserving some relationship between the variables. In calculus 1 you may have seen some such problems, in particular ones where the constraint lets you express the problem as a single-variable one. Lagrange multipliers allow you to solve some optimization problems that can’t be turned into single-variable problems in this way.
Find the points on the hyperbola x2 - (y-1)2 = 1 that are closest to the origin.
While not described as such in the problem, there is a function here to minimize (i.e., an “objective function”), namely the distance from the origin to point (x,y), and a constraint on x and y.
So we can use the method of Lagrange multipliers, as described in the reading, to solve this. (We could also use the constraint to deduce that x2 = 1 + (y-1)2 and solve it as a single-variable problem, but we’ll use it as an opportunity to look at Lagrange multipliers instead.) Work it out in class or at least before Wednesday’s class, and come with a solution or questions we need to discuss.
Lagrange multipliers as a technique for solving optimization problems.
Finish Lagrange multipliers.
Review questions / requests for the exam.