![]() ![]() Suppose we are told that the product of two positive numbers is 192 and the sum is a minimum. Let’s look at a few problems to see how our optimization problem-solving strategies in work. While this may seem difficult at first, it’s really quite straightforward as we are simply finding two equations, plugging one equation into the other, and then taking the derivative. Step 4: Verify our critical numbers yield the desired optimized result (i.e., maximum or minimum value). Step 3: Take the first derivative of this simplified equation and set it equal to zero to find critical numbers. Step 2: Substitute our secondary equation into our primary equation and simplify. Step 1: Translate the problem using assign symbols, variables, and sketches, when applicable, by finding two equations: one is the primary equation that contains the variable we wish to optimize, and the other is called the secondary equation, which holds the constraints. Solving Optimization Problems (Step-by-Step) ![]() ![]() It is our job to translate the problem or picture into usable functions to find the extreme values. Optimization is the process of finding maximum and minimum values given constraints using calculus.įor example, you’ll be given a situation where you’re asked to find: Or, on the flip side, have you ever felt like the day couldn’t end fast enough?īoth are trying to optimize the situation! ![]() Most students who take calculus at a university are planning to go into one of these fields, so calculus will be relevant in their lives-specifically in their future studies and in their professions.Jenn, Founder Calcworkshop ®, 15+ Years Experience (Licensed & Certified Teacher) That is, it's useful for all the things that make our society run. What calculus is useful for is science, economics, engineering, industrial operations, finance, and so forth. Even in a class full of future farmers, the fence problem would still be bad, because farmers don't use calculus to plan their fences. But it's not because the students aren't farmers, or wire-cutters, or architects. I agree-none of these problems are relevant. Of course, it's neat that you can use calculus to solve this problem precisely, but this is more of a curiosity than a legitimate application.Ĭhris specifically mentions the farmer fence problem, the wire-cutting problem, and the Norman window problem as not relevant to the students' lives. you are buying a ladder), the thing to do would be to draw the situation on paper and then use a ruler to estimate the minimum length. If you don't have a specific ladder in mind (e.g. The proper response to this question is: who cares? Is there any reason to calculate this length precisely? Why would anyone ever use calculus to compute this? If you have an actual building and an actual ladder, you could just try it and see if the ladder fits. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Consider the following problem from Stewart's Calculus: Concepts and Contexts.Ī fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. Students know this, and you'll have trouble convincing them otherwise.īecause of this, I've always found "everyday"-style calculus problems a little artificial. With few exceptions, mathematics beyond basic arithmetic is simply not useful in everyday life. To some extent, I agree with this comment. But I'm not sure if that's exactly what you mean. Mathematics beyond basic arithmetic is simply not useful in ordinary life. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I thought that Jack M made an interesting comment about this question: ![]()
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