3.4 Applications of Minima and Maxima 1 Example: For a short time interval, the current i (in amperes) in a circuit containing an inductor is given by Determine the maximum current. Solution: We are looking for a maximum of the given function i(t) Critical values: In various applications, the question to be solved is to find an optimal solution to a problem. The technique we have developed for finding extreme values are very useful in such problems.

2 Example (cntd): The point t=0 is not in the interval we are given (t>0). Use the Second-derivative test to determine if t=5 is minimum or maximum: At t=5, Since the second derivative is negative at this point, it is a ________. Thus, t=1/5 is a maximum value, and the maximum current is i(1/5)=2/5 A. Exercise: The drag on an aircraft traveling at veloicity v is Where a and b are positive constants. At what speed does the airplane experience the least drag?

3 If the expression to be minimized or maximized is not given, it has to be obtained from the given information first. In this case 1.Introduce notation: a symbol for the quantity to optimize, for other quantities mentioned in the problem. 2.Draw a diagram, if appropriate. 3.Express the quantity to optimize in terms of all others. 4.Identify constant parameters and variables in this expression. 5.If the expression contains more that one variable, use the information given in the problem to find relationships between the variables, and then eliminate all but one variable using those relationships. 6.Use the standard technique to find an appropriate extremum of the function obtained in the previous step.

4 Example: A cylindrical can is to be made to hold a certain volume V of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. Solution: To minimize the cost of the metal, we minimize the total surface area of the cylinder. 1. Introduce notation: the quantity to minimize is the surface of the cylinder S, which depends on its height h and radius r. 2. Draw a diagram… 3. Express S in terms of h and r: 4. Both h and r are variables (the rest are constants). 5. To eliminate one of them, we use the condition that the volume of the can must be equal V, where V is another constant: Thus,, and. 6. Apply the standard technique to minimize this quantity…

5 Exercises: 1. A rectangular field is to be enclosed by a fence and separated into two parts by a fence parallel to one of the sides. If 600m of fence is available, what should the dimensions be so that the area is a maximum? 2. Find the point in the first quadrant on the curve xy=2 nearest the origin.

6 Homework Section 3.4: 1,5,11,13,15,19,21,25,29.