![]() Support us and buy the Calculus workbook with all the packets in one nice spiral bound book. (Link will open in a new tab.) Stage I: Develop the function. 5.10 Introduction to Optimization Problems. In the example problem, we need to optimize the area A of a rectangle, which is the product of its length L and width W. At the endpoints of the domain we have A(0) = 0 and A(200) = 0. Calculus Solution We’ll use our standard Optimization Problem Solving Strategy to develop our solution. Step 1: Determine the function that you need to optimize. If A has a maximum value, it happens at x such that dA/dx = 0. To find the value of x that gives an area A maximum, we need to find the first derivative dA/dx (A is a function of x). ![]() ![]() We might consider the domain of function A(x) as being all values of x in the closed interval since x >= 0 and y = 200 - x ≥ 0 (if you solve the second inequality, you obtain x <= 0). As you change the width x in the applet, the area A on the right panel change.Įxpand the expression for the area A and write it as a function of x. We now now substitute y = 200 - x into the area A = x*y to obtain. A quick guide for optimization, may not work for all problems but should get you through most: 1) Find the equation, say f (x), in terms of one variable, say x. Let x ( = distance DC) be the width of the rectangle and y ( = distance DA)its length, then the area A of the rectangle may written: We now look at a solution to this problem using derivatives and other calculus concepts. Find the length and the width of the rectangle. Problem You decide to construct a rectangle of perimeter 400 mm and maximum area. ![]() Then an analytical method, based on the derivatives of a function and some calculus theorems, is developed in order to find an analytical solution to the problem. An interactive applet (you need Java in your computer) is used to understand the problem. A problem to maximize (optimization) the area of a rectangle with a constant perimeter is presented. Optimization with Calculus 1 Fundraiser Khan Academy 7.76M subscribers 732K views Calculus Find two numbers whose products is -16 and the sum of whose squares is a minimum. ![]()
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