Consider the following and write a response that is no less than 10 sentences.
What is optimization? What rules of Calculus can we use to solve optimization problems? Why is this method of problem solving significant? Where is it found in todays world/society?
Here are some good practice problems:
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html
http://cda.morris.umn.edu/~mcquarrb/CI/HW/4.7.pdf
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Optimization problems are problems where you need to find a maximum or a minimum. One main calculus applications is finding the determination of minimum and maximum values. Some real world uses of this method include finding the greatest profit, least cost, least time, greatest voltage, optimum size, least size, greatest strength, greatest distance(text p218). These methods are very valuable when working for a manufacturing company and trying to get the best "bang for your buck." To solve optimization problems you must first make a primary equation. Then you have to reduce this equation to an equation with a single variable. This can involve making secondary equations. After that you must determine the feasible domain of the primary function. Finally you can find the desired maximum or minimum by using either the first or second derivative tests (text p219). Using optimization is important because you can't always use a calculator. How do you think the programmers of the "TI" calculators put the information in there? They had to have used calculus. This method is extremely valuable in solving problems that would otherwise seem extremely difficult.
Optimization can be used to solve problems where the minimum and maximum points have to be found. There are rules and guidelines that should be followed when solving the problems. First, you need to identify all which quantities are given and which quantities need to be found. It might be helpful to draw it out. Second, you need to write an equation, called the primary equation, for the quantity that is supposed to be minimized or maximized. Third, reduce the equation until it has a single independent variable. Next, determine which values make the problem make sense. And finally, determine the maximum or minimum value. You can find the minimum and maximum values by using the Mean Value Theorem, the First Derivative test, and the Second Derivative test. This method of problem solving is important because it is one of the main focuses of Calculus. It is also helpful to know as it can be found throughout the world. Businesses use it to figure out which monthes had their least or greatest profits. It can also be used to find out the maximum or minimum area/volume of shapes and figures. Distance can also be calculated using this method.
(source: Calculus of a Single Variable, pages 218-219)
Optimization, according to Answers.com, is often referred to 'mathematical programming.' It "refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. For example, in our text books (Calculus of a Single Variable, 8th Edition) pages 218-222, they use basic geometric formulas (such as volume, area, perimeter, distance, etc.) These specific formulas can help find the maximum or minimum equation that you are looking for. Also, differentiating functions can also be extremely helpful in finding these maximum and minimum answers. Drawing pictures and diagrams, though they seem trivial and childish, can actually help you out a great deal in solving the problem. Another example of helpful rules in Calculus is that of the first and second derivatives test. Our method will be taking the derivative of a function using only two variables. These methods of this kind of problem solving are significant to everyday life. Businesses may use this kind of calculus to figure out their maximum/minimum profits for the month. Also, landscapers may want to find the total area of a section of land so that they understand the potential, means, and resources of the land itself. Calculus is either indirectly or directly in the world as we know it. Optimization problems are even as simple as being given a certain amount of material, and you want to create the dimensions that will give it the biggest volume. Being informed and educated about these type of problems not only benefits out businesses and companies, but it benefits our knowledge of the world around us.
1.Optimization is determining the maximum or minimum values in a function. 2) You can use Calculus rules of identifiying all given quantities and quantities to be determined. 3)Then you need to write a primary equation for a quantity that needs to be maximized or minimized. 4)Next, you must reduce the primary equation to have a single independent variable, which may need a secondary equation. 5)Then you have to figure out the values that satisfy the problem. 6)So, to determine the maximum or minimum values, take the first derivitave to figure out the intervals and see which way the graph is going, whether you have found a maximum or minimum value. 7)If it is a maximum, the first test value will give a positive value when it is plugged into the function, and the second test value will give a negative result. 8)If it is a minimum, the first test value will be a negative result and the second will give a positive result, which means the graph was going down, reached a minimum at the point you found, and then started going up. 9)If it gives a closed interval, you also need to determine whether the end points of the interval are maximum or minimum values; this is called the first derivitave test(Calculus of a Single Variable 8th edition). 10)This problem solving method is significant because you can apply it to real life such as the maximum height a ball reached when it is thrown from the ground. 11) Or, a business can figure out how many units of something they should produce to maximize their profits and minimize their spendings.
Optimization means to optimize something, or use something at its best. In calculus optimization refers to the maximum and minum value of something. In order to find the answer to a problem u must first write out an equation and get it into 2 variable form. After u have that then u take the derivavtive of the equation. The set the derivative to 0. Then u plug in that answer into the original equation and u can find the maximum or the minimun. This method is significant because it will help you find the minimum or maximum of something. Calculus is significant to solveing optimization problems because optimization problems have to do with maximum and minimum values. In order to find these values we use the derivative rule from calculus. Optimization is found everywhere in todays world and society. For example, if a pizza parlor wants to maximize profit and will need to make a box with the greatest volume he would use calculus to find the answer.
Optimization is a mathematical technique for finding a maximum or minimum value of a function of several variables subject to a set of constraints, as linear programming or or systems analysis. (Dictionary.com) There are five specific guidelines that can be used when solving these problems: 1) Identify all given quantities and quantities to be determined. 2) Write a primary equation for the quantity that is to be maximized or minimized. 3)Reduce the primary equation to one having a single independent variable. 4)Determine the feasible domain of the primary equation. 5)Determine the desired maximum or minimum value by using the Calculus rules and techniques. (Textbook) The rules of Calculus that can be used to solve optimization problems include Rolle's Theorem, the Mean Value Theorem, and the rules for finding the derivative of a function, such as the product, quotient, and power rules. This method of problem solving is significant because we see many examples of this in the real world every day when we hear or read about greatest profit, least cost, least time, greatest voltage, optimum size, least size, greatest strength and greatest distance. (Textbook) Optimization can be found in many aspects of todays world, such as area, volume, strength, and distance.
Optimization is a mathematical technique for finding a maximum or minimum value of a function of several variables subject to a set of constraints, as linear programming or systems analysis. We can use the deriviation, first deriviative test, and the rolle's theorem to solve the optimization problems. To solve the optimization Problems: First, we need to find a extrema of a function. Since the rolle's theorem, the x-value of a differentiated function is the point where the relative maximum or minimum of a original function in an interval. second, figure out whether the extrema is point of a maximum or a minimum by using first derivative test
Third, substitute the critical value in the original function. Fourth, calculate the y-value, then the y-value is the maximum or the minimum. Today, the optimization is being used in the companies and the factories. They use the optimization to find out the minimum cost, or the maximum profit.
(Sources: Dictionary.com; en.wikipedia.com)
Optimization problems call for you to find the greatest or least value that can be used in a function. Although these problems seem similar to the absolute extrema problems we worked on earlier this year, one major difference exists between the two. While absolute extrema problems ask you to find the greatest or least values of a function on a certain interval, optimization problems usually ask you to find the greatest or least value given some sort of limitation (which is usually given in the problem itself).
Of course, all aspects of calculus are connected; rules and principles that were studied weeks ago have a painful tendency of reappearing in later sections. However, I think Rolle’s Theorem and the First Derivative Test will be especially useful in this section. Rolle’s Theorem requires us to find the x-values for which f’(x) = 0. These x-values can then be used to determine the relative extrema of the function. The First Derivative Test will prove helpful because it determines whether any of the critical numbers of a function are relative extrema. Since you need to find the derivative of the given equation in optimization problems, finding the critical numbers and using the First Derivative Test may prove to be a simple way of finding the greatest or least values of a function.
Optimization problems are significant because they are an easy way of cracking difficult mysteries. Optimization is also necessary. In the world we live in today, challenges arise in which we are constantly required to find the greatest value for something or the least value for something. For example, in mainstream economics businesspeople constantly need to assess ways in which they can make the greatest profit. Conversely, they would want to designate scenarios that would yield the least profit and make sure they avoid them.
Optimization is applied to minimum and maximum problems such as ones dealing with the greatest profit and the least amount of money spent. (Calculus of a Single Variable: Eighth Edition) Rules of calculus that we can use to solve optimization problems are the following:
1. Read each problem slowly and carefully.
2. Sketch a drawing of the problem to be solved.
3. Define variables to be used and label.
4. Write down all equations which are related to the problem.
5. Before differentiating, make sure that the optimization equation is a function of only one variable.
6. Make sure the result is a maximum or minimum value using the first or second derivative test for extrema.
(http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html) This method of problem solving is significant because optimization problems are in the form of word problems. So it is more difficult to decipher what the question is compared to regular numerical problems. By using these steps as a guide it makes it much easier to follow somewhat of a pattern while solving these minimum and maximum problems. These kinds of problems are found in our daily lives. For example we can use these methods in order to find maximum volume of a room, least amount of time of travel, and even greatest voltage of a light bulb.
Optimization is the process of trying to find an assignment of values to the other variables (called parameters) that produces a maximum or minimum value of the target variable, called the optimum or optimal value.
In optimization problems we are looking for the maximum value or the minimum value that a function can take.
There are three methods in the calculus. First is finding aboulute extrema.We can solve some opotimizing problems but sometimes we can't when it doesn't have finite endpoints at both ends or it is not continuous.
Then we use second method which is First Derivative Test and in this case, the endpoints do not need to finite. The third method is Second Drivative Test.
These methods of this kind of problem solving are significant to everyday life. Businesses may use this kind of calculus to figure out their maximum/minimum profits for the month.Optimization is found everywhere in todays world and society. For example,in the businese world many people try to find the greatest profit points.
Optimization is used to solve applied minimum and maximum problems. It refers to finding the minimum or maximum values of a function within a specified part of its domain (Sparknotes). To solve optimization problems, first write a primary equation for the quantity that is to be minimized or maximized. Next, the primary equation must be reduced so that it has only one independent variable. A domain needs to be determined so you can decide what values for the problem make sense. To fin the max or min value, methods such as Rolle’s Theorem and the Mean Value Theorem can be applied. Finding critical numbers and using the First Derivative Test can also be simple ways of finding the greatest or least values of a function. Optimization is significant because it is useful for many things in the real world. Optimization can be used to solve problems such as greatest profit, least cost, least time, greatest voltage, optimum size, least size, greatest strength, and greatest distance (Pg 218). Optimization can also be applied to find things such as area, volume, and distance.
Optimization is the process in which one doing the math problem chooses from a set of intervals to determine the extrema (maximum or minimum) of a real function. This means finding the best answer to complete and satisfy the question asked or problem presented. Many Calculus topics can be applied to
these mathematical programming problems, such as the first and second derivative tests, Rolle's Theorum, The Mean Value Theorum, The Extreme Value Theorum, and
Concavity. Other areas of math are also necessary to calculate optimization problems, such as ordinary algebra, domain, primary and secondary equations, and the very first part of math learned
by everyone: Operations of
addition, subtraction, multiplication and division. The method of solving
mathematical equations is significant because it combines and applies all math techniques learned by the student (or
person) to real everyday issues that could not be solved otherwise. Calculus is needed for the largest step of whichever problem is being solved: the ending. Precalculus and algebra can be used up until the last few steps, where Calculus is needed to intervene and finish correctly. In life, this utilization comes in handy very often, especially in professions such as architecture, or any job requiring building (finding the maximum area and
minimum cost and supplies), manufacturing (to assess the minimum cost and maximum profit and volume of product), farming (maximum yeild, area, and profit, and minimum effort of the farmer, and cost of growing), and being a Calculus teacher (maximum learning, minimum sleeping). To further prove why this method is significant enough to warrant the label 'an essentiality to everyday life', imagine a world without optimization. Houses, roads, buildings, food and otehr products would virtually be destroyed.
If a farmer thinks he can only produce x number of crops, or he cannot figure out the greatest amount able to be produced with the lowest cost, he will only
grow x number of crops. This situation, if affecting all other farmers and food producers, would
catastrophically devastate the food industry and leave many people to hunger or starve. Now imagine if this confusion were applied to all other companies and social orders, and think of the chaos the world would be in without optimization. Thanks to you, optimization, we are at least out of that horrific scenario.
Optimization problems involve solving for the maximum and minimum values of a function. To optimize an equation, you must first make it so all the variables are the same. Next, decide on intervals that would suit the equation. Then, you take the derivative of the equation. Next, you set the derivative equal to zero and solve. This gives you the critical numbers of the equation. The critical numbers must fit within the intervals that you found earlier. You can use either the Mean Value Theorem, Rolle's Theorem, or the first or second derivative test to solve for the max. and min. Optimization is used in landscaping. It is also very important in aerospace engineering problems.
Optimization is the procedure or procedures used to make a system or design as effective or functional as possible, especially the mathematical techniques involved (Ask Jeeves). It is a mathematical discipline that concerns the finding of minima and maxima of functions that originated in the 1940s when it was used to generate programs, such as training timetables and schedules, for military application(FriarTuck). The Mean Value Theorem and Its Applications Derivatives are two rules of calculus that can be used to solve optimization problems (Differential and Integral Calculus). In todays world/society, the "linear programming" techniques and their descendents were applied to a wide variety of problems, from the scheduling of production facilities, to yield management in airlines (Friar Tuck). Optimization comprises a wide variety of techniques from Operations Research, artificial intelligence and computer science, and is used to improve business processes in practically every business (Friar Tuck). These methods are very valuable when working for a manufacturing company. To solve optimization problems you must first make a primary equation. Then you have to reduce this equation to an equation with a single variable. This can involve making secondary equations. After that you must determine the feasible domain of the primary function. Finally you can find the desired maximum or minimum by using either the first or second derivative tests (text p219).
Optimization is hard to put into words. It uses algebra and calculus to find the maximum or minimum of something, such as the area of a yard. For optimization problems, there are a couple of rules; one big group is the differentiation rules. You must be able to differentiate different kinds of problems to be able to find the min or max. You must also know Rolle’s Theorem. After differentiating the equation, you must use Rolle’s Theorem to be able to find the min or max. This method of problem soliving is significant because most people use it in everyday life. It can be used for anything from fencing in a garden to folding a cardboard box to where to sit at the movies. This makes optimization problems very significant for everyone. In today’s society it can be found at home and at work, even in leisurely activities. At home if you are fencing in a garden, you use optimization to figure out the dimensions of the garden. At work if your putting in a window, and you want the largest area with the preset perimeter. You can even use optimization problems when you go to the movies. Choosing the best place to sit so that you don't get a cramp in the neck at the end of the movie is a great optimization problem.
All the examples i used came from practice problems from the following website:
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html
Optimization deals with finding the maximum and minimum values in functions. It was originated in the1940s by George Danzig for military application (friartruck.net). In order to find the max and min, a primary equation has to be formed by the given quantities or the to be determined quantities. You can use other calculus rules to solve optimization problems. Once a one variable equation is formed, the derivative can be taken to find the critical numbers of the function. Also, such methods as extrema on an interval, increasing/decreasing functions, first and second derivative tests, and concavity can all be used to help solve optimization problems.
This method of problem solving is important because it helps in linear programming. The variables occurring in the optimization function can take only a finite number of discrete values (friartruck.net). It is also useful when a calculator is not available. Optimization can be used in businesses to help find out the maximum number of patients that can be seen in a day if x number of nurses are available, or to find out how to maximize monthly profits. Optimization can also be used to find the minimums, such as the minimium or least time for traveing.
Optimization is the study of problems in which you minimize or maximize a real function. By doing this you systematically choose variables, which can be real or integers, from within an allowed set(en.wikipedia.org). Almost all optimization problems have a single objective function. If there are no variables, you cannot define the obective function and the problem constraints(new.mcs.anl.gov). To solve these problems you can start out with a primary equation, than to get a secondary equation you need to reduce the primary equation. Next, you should determine the feasible domain. Last, you find the critical number or numbers of the equation by setting the derivative equal to zero. You can also use max and min problems to figure it out. It is found in many things in today's society. Many are the profit, cost, time, distance, strength, or size of an object or what ever is being used(text pg. 218).
Optimization refers to finding the maximum or minimum values of a real function. This is done by systematically choosing the values of real or integer variables from within an allowed set (Answers.com). The rules of calculus which can be used to solve
optimization problems consist of the different ways of taking a derivative including the product rule, quotient rule, and power rule. I also feel that the First Derivative test and Rolle's Theorem will be very important in using optimization. In order to solve an optimization problem, first we make a primary equation for the quantity and reduce it to have a single independent variable. After this will determine the maximum or minimum by using one of the calculus techniques we have learned previously in this chapter (textbook). Optimization is significant because it allows one to find values such as greatest profit, least cost, least time, greatest voltage, optimum size, least size, greatest strength, and greatest distance (textbook). There are many situations in today's world which optimization can be applied. For example, to find the maximum flow rate of traffic on a highway on a given day, optimization can be used. Another way we can apply it is when finding out the minimum amount of fencing needed to put around your house.
Optimization is to find the minimum and maximum of the function. For example, by using optimization, one can find the biggest volume from the given information. To solve the optimization problem, find out the given quality and the function, primary function, which will apply for seeking maximum or minimum. Then, set another equation, secondary function, from the given information. With two equations, reduce the function into single variable equation. Find the feasible domain and the critical numbers of the function. Lastly, substitute the critical numbers into two functions to find the answer. Optimization is important because it provides a theory and an application that use in science, engineering, economics, and industry. It is necessary for students who study science, technology, and business. One can use the optimization to find the most profit of the company in order to minimize the risks. Mainstream economics also depend on the optimization.
Optimization is the process in which you determine the maximum or minimum values in a function. In the given function there will be two variables, although you only need to simplify the function down into a single variable function. If there is a secondary function in which you can plug into the equation that would be a means of simplifying. In optimization, there is a feasible domain in which what range of values will work in the particular problem. Take the single variable function, find the critical numbers using the first derivative test and then set up to find extremes. This in turn will help you conclude the maximum or minimum values of the function using optimization. Basically you need to use the first and second derivative tests, Rolle’s Theorum, and the extreme value theorem to solve optimization problems. This way of problem solving is significant because you can use it to solve two-variable functions and in daily life for work. In today's world, optimization can be used to find the distance between two different points or more. Optimizations helps businesses and calculus teachers alike, whether from max profits/costs to the dimensions of a cube or another shape.
The Optimization refers to the study of problem in which one finds the maximum or minimum values of a function. First, you need to find all equations that lead to to get the values; for example, to find out the maximum volume of a box, the eqution will be f(x)=wlh, which means width times leghth times height.The equation should have only two variables in order to take derivation of the equation. Then, you can derivate the equation and set it equals to zero because in graphs of maximums or minimums, the slope is zero. In that way, you can fine the critical numbers of the prime equation, which make the slope equals to 0. The derivation method solving problem is significant and useful because it generalize many possible values to obtain the best available values of the function. Optimization problem is very useful in real world; it can be used to find out the greaest profit, least profit, maximum volume of product or least volume of product, and ect.
Optimization refers to the finding of maximum or minimum values. We use optimization problems to find the best value in a given domain. To solve optimization problems, we use the rules of derivatives and other theorems. Rules of derivatives involve the Product and Quotient Rules and such theorems as Rolle's Theorem and the Mean Value Theorem. Solving optimization problems invloves writing a primary and second equation. Then, determine the domain and use the rules of Calculus already mentioned to find the maximum or minimum value. This method of problem solving is significant because it involves steps that people use in real world situations. Optimization problems are found in the real world from determining profits in businesses to explaining trade patterns between nations (Answers.com). Optimization problems are also used in the field of physics to find energy minimization(Answers.com). They are also useful in finding solutions to problems that seem difficult at first glance.
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