Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline(Optimization II)Friday, February 29Announcements1. Grades for midterm 2 will be posted on Coursework by Sunday evening.2. Quiz 7 on Monday.Optimization Problems - On Open IntervalsIt often happens that you are asked to optimize a function on an open interval (or the wholereal line) instead of on a closed interval. How does one go about solving such a problem?Happily, much of the technique for optimizing functions on closed intervals will carry overto optimizing functions on open intervals.The main difference in technique is that in finding absolute maxima or minima we willneed to use a variant of the first derivative test instead of the techniques we developed onMonday to find the absolute maxima and minima for continuous functions on closed inter-vals. This is our variation of the first derivative test:1st Derivative Test for Absolute Minima and Maxima Suppose f (x) is a contin-uous function defined on (a, b) (where a = −∞ and b = ∞ are possible). Let x = c be acritical point of f (x). Then:1. If f0(x) < 0 for all x < c and f0(x) > 0 for all x > c, then f has an absolute minimum(in its domain) at c.2. If f0(x) > 0 for all x < c and f0(x) < 0 for all x > c, then f has an absolute maximum(in its domain) at c.Notice that this is very similiar to the 1st derivative test for local min/max. The differ-ence is that in this test we require information about the sign of the derivative for all x < cand all x > c; in the local test, we needed to only know this for values of x near c.Note:1. To use the 1st derivative test for absolute min/max directly, you should be in thesitutation where you have exactly one critical point in your domain. If you have morethan one critical point in your domain, you probably have to do more work.12. If you’re in the situtation where you have exactly one critical point (say, x = c) in yourdomain, then to test the sign of the derivative for values x < c it is enough to computethe sign of the derivative for one value of x < c. Similarly for checking the sign whenx > c.The technique then for solving optimization problems for continuous functions on anopen interval is:1. Read and re-read the problem until you understand it. In particular, make sure youknow the quantity you’re being asked to maximize or minimize.2. Draw and label a picture which gives the relevant information.3. Write equations that describe(a) the quantity that you’re attempting to maximize/minimize in terms of the othervariables which appear in your drawing, and(b) the constraints that your variables must ob ey.4. Solve Equation (b) for one of the variables, and plug this result back into Equation(a).5. Determine the domain of your function. This is very important!6. Use the first derivative test for absolute minima and maxima to finish the problem.Examples1. Find two numbers whose sum is 23 and whose product is maximum.2. A cylindrical can is to be made to hold 1000cm3of oil. Find the dimensions that willminimize the cost of the metal to manufacture the
View Full Document
Unlocking...