Method 1 : Use the method used in Finding Absolute Extrema. This is the method used in the first example above. If these conditions are met then we know that the optimal value, either the maximum or minimum depending on the problem, will occur at either the endpoints of the range or at a critical point that is inside the range of possible solutions. There are two main issues that will often prevent this method from being used however. First, not every problem will actually have a range of possible solutions that have finite endpoints at both ends.
Method 2 : Use a variant of the First Derivative Test. However, in this case, unlike the previous method the endpoints do not need to be finite. This will not prevent this method from being used. However, suppose that we knew a little bit more information.
Nowhere in the above discussion did the continuity requirement apparently come into play. Also, the function is always decreasing to the right and is always increasing to the left. There are actually two ways to use the second derivative to help us identify the optimal value of a function and both use the Second Derivative Test to one extent or another. What it does do is allow us to potentially exclude values and knowing this can simplify our work somewhat and so is not a bad thing to do. Suppose that we are looking for the absolute maximum of a function and after finding the critical points we find that we have multiple critical points.
We could do a similar check if we were looking for the absolute minimum. Doing this may not seem like all that great of a thing to do, but it can, on occasion, lead to a nice reduction in the amount of work that we need to do in later steps. The second way of using the second derivative to identify the optimal value of a function is in fact very similar to the second method above.
In fact, we will have the same requirements for this method as we did in that method. As we work examples over the next two sections we will use each of these methods as needed in the examples. In some cases, the method we use will be the only method we could use, in others it will be the easiest method to use and in others it will simply be the method we chose to use for that example. With some examples one method will be easiest to use or may be the only method that can be used, however, each of the methods described above will be used at least a couple of times through out all of the examples.
We may need to modify one of them or use a combination of them to fully work the problem. There is an example in the next section where none of the methods above work easily, although we do also present an alternative solution method in which we can use at least one of the methods discussed above. Next, the vast majority of the examples worked over the course of the next section will only have a single critical point.
Problems with more than one critical point are often difficult to know which critical point s give the optimal value. There are a couple of examples in the next two sections with more than one critical point including one in the next section mentioned above in which none of the methods discussed above easily work. In that example you can see some of the ideas you might need to do in order to find the optimal value. This was done to make the discussion a little easier.
We want to minimize the cost of the materials subject to the constraint that the volume must be 50ft 3. Note as well that the cost for each side is just the area of that side times the appropriate cost. As with the first example, we will solve the constraint for one of the variables and plug this into the cost. Now we need the critical point s for the cost function. We are constructing a box and it would make no sense to have a zero width of the box.
Secondly, there is no theoretical upper limit to the width that will give a box with volume of 50 ft 3. The third method however, will work quickly and simply here. This example is in many ways the exact opposite of the previous example. In this case we want to optimize the volume and the constraint this time is the amount of material used. If you can do one you can do the other as well. Note as well that the amount of material used is really just the surface area of the box. In this case we can exclude the negative critical point since we are dealing with a length of a box and we know that these must be positive.
Do not however get into the habit of just excluding any negative critical point. There are problems where negative critical points are perfectly valid possible solutions. Now, as noted above we got a single critical point, 1. In both examples we have essentially the same two equations: volume and surface area.
However, in Example 2 the volume was the constraint and the cost which is directly related to the surface area was the function we were trying to optimize. In Example 3, on the other hand, we were trying to optimize the volume and the surface area was the constraint. This is one of the more common mistakes that students make with these kinds of problems. They see one problem and then try to make every other problem that seems to be the same conform to that one solution even if the problem needs to be worked differently.
Keep an open mind with these problems and make sure that you understand what is being optimized and what the constraint is before you jump into the solution. Also, as seen in the last example we used two different methods of verifying that we did get the optimal value. Do not get too locked into one method of doing this verification that you forget about the other methods.
This will in turn give a radius and height in terms of centimeters. In this problem the constraint is the volume and we want to minimize the amount of material used. Here is a quick sketch to get us started off. The volume is just the area of each of the disks times the height. Similarly, the surface area of the walls of the cylinder is just the circumference of each circle times the height. By identifying positive exceptions, you gather information you can use to solve your problem. Creativity is combining existing elements to create something new.
You bring two or more things together which have not yet been brought together in this context and, as a result, a new situation arises. You can combine anything from techniques and materials to processes and ideas. Using random words or images is a simple and effective technique that enables you to quickly generate delightfully original approaches.
In other words: the further away a problem seems, the easier it is to solve. This, however, does not mean that you are helpless in handling your own challenges. You simply need to trick your brain. There are ways to make your own problem seem more distant.
Imagine you are looking back on the problem you had and how you solved it. What actions did you take? Asking your older self for help is one way to create a little distance, but there are more. How would you solve the problem if it was bugging a villager in a country on the other side of the world? Even after generating a brilliant idea for tackling your problem, things could go wrong.
Implementation is tough and executing an idea is often harder than anticipated. To prevent inconvenient surprises, prepare for failure. Imagine you are looking back at the project, realising it completely failed. Being prepared for the worst will make a successful implementation much more likely.
Of course, there is much more to tell about the road from problem to solution. Are you curious, or do you have any questions? We respect your privacy. His background in both law and photography enables him to shift smoothly between very different worlds and might explain his fascination with unusual combinations. Recommended Posts Why you should gather problems The Transformers technique — how to use a verb to solve your problem The 5 Whys — identifying the root cause of your problem Leave a Comment Cancel reply Save my name, email, and website in this browser for the next time I comment.
Buy Problem Solving Tips: Read Kindle Store Reviews - lirodisa.tk This book list problems of our home and health that happen in our life very often, solving tips in every list, just click the problem to find the solving tips.
Or is it merely a symptom of a deeper lying cause… One simple technique to quickly list possible causes is the 5 Whys technique. Ask the right question Every problem can be solved. Change your perspective Generating ideas to solve a problem is not very hard. Embrace the bizarre Not only can unusual ideas be very effective, they can also spark brilliant insights. Find the positive exception Sometimes part of the solution already exists. Make strange combinations Creativity is combining existing elements to create something new. Use a pre-mortem Even after generating a brilliant idea for tackling your problem, things could go wrong.
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