Date: 12/13/95 at 11:10:44 From: Anonymous Subject: Optimization in Calculus Dear teachers, I am a AP Caluculus student at a military school in Augsburg, Germany. On Wednesday, December 13, 1995, I took a chapter test in my class. Having had a problem on 2 of the questions, I was wondering if you could explain how to complete the problems. 1. Find the equation for the graph that passes through the pt. (-1,-4) with slope 2 given d^2y/dx^2 = 3x/4 I am totally lost on this one. 2. You are designing a poster to contain 1200 sq. in of printing with margins of 4 in. each at the top and bottom and 3 in. at each side. What overall dimensions will minimize the amount of paper used. On this one, I've figured to start it out by having: (x-6)(x-8)=1200 Help me out from here. Your answer will be greatly appreciated. Ara Donabedian
Date: 12/15/95 at 11:24:12 From: Doctor Ken Subject: Re: Optimization in Calculus Hello! >1. Find the equation for the graph that passes through the pt. (-1,-4) >with slope 2 given > > d^2y/dx^2 = 3x/4 > >I am totally lost on this one. Okay, well let's think about what that statement means. Essentially, the d^2y/dx^2 = 3x/4 part means that the second derivative of our function is 3x/4. So what we're going to have to do to find the answer is take two antiderivatives of 3x/4. Well, if you take one antiderivative you get 3x^2/8 + c, and if you take another you get x^3/8 + cx + d. Now you'll have to figure out how to use the information about slope and point that it goes through (hint: slope = derivative) to find c and d. >2. You are designing a poster to contain 1200 sq. in of printing with >margins of 4 in. each at the top and bottom and 3 in. at each side. >What overall dimensions will minimize the amount of paper used. > >On this one, I've figured to start it out by having: > >(x-6)(x-8)=1200 Actually, what you're going to want is (x-6)(y-8) = 1200, letting x and y be the length and width of the paper (it's not necessarily a square). Then the function that you want to minimize is the area of the paper, Area = xy. The way the first equation is going to fit into the picture is that you're going to solve for x in terms of y (or y in terms of x, it doesn't really matter) and then plug that into the Area equation, then minimize. Good luck! -Doctor Ken, The Math Forum
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