Step 5: To determine the domain of consideration, let’s examine Figure 4.64. Now let’s apply this strategy to maximize the volume of an open-top box given a constraint on the amount of material to be used. This step typically involves looking for critical points and evaluating a function at endpoints. Locate the maximum or minimum value of the function from step 4.Identify the domain of consideration for the function in step 4 4 based on the physical problem to be solved.Use these equations to write the quantity to be maximized or minimized as a function of one variable. Write any equations relating the independent variables in the formula from step 3.This formula may involve more than one variable. Write a formula for the quantity to be maximized or minimized in terms of the variables.Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time).If applicable, draw a figure and label all variables. Problem-Solving Strategy: Solving Optimization Problems Differentiating the function A ( x ), A ( x ), we obtain
Since the area is positive for all x x in the open interval ( 0, 50 ), ( 0, 50 ), the maximum must occur at a critical point. These extreme values occur either at endpoints or critical points. Īs mentioned earlier, since A A is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. Maximize A ( x ) = 100 x − 2 x 2 A ( x ) = 100 x − 2 x 2 over the interval.
Therefore, we consider the following problem: If the maximum value occurs at an interior point, then we have found the value x x in the open interval ( 0, 50 ) ( 0, 50 ) that maximizes the area of the garden. Therefore, let’s consider the function A ( x ) = 100 x − 2 x 2 A ( x ) = 100 x − 2 x 2 over the closed interval. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. We do not know that a function necessarily has a maximum value over an open interval. Therefore, we are trying to determine the maximum value of A ( x ) A ( x ) for x x over the open interval ( 0, 50 ). Therefore, we need x > 0 x > 0 and y > 0. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. A ( x ) = x īefore trying to maximize the area function A ( x ) = 100 x − 2 x 2, A ( x ) = 100 x − 2 x 2, we need to determine the domain under consideration. Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. For example, in Example 4.32, we are interested in maximizing the area of a rectangular garden. However, we also have some auxiliary condition that needs to be satisfied. We have a particular quantity that we are interested in maximizing or minimizing. The basic idea of the optimization problems that follow is the same. Solving Optimization Problems over a Closed, Bounded Interval
#OPTIMIZATION REAL LIFE EXAMPLES CALCULUS HOW TO#
In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. For example, companies often want to minimize production costs or maximize revenue. One common application of calculus is calculating the minimum or maximum value of a function. 4.7.1 Set up and solve optimization problems in several applied fields.
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