AP Calculus Review Differential Equations

A differential equation is a relation that involves an unknown function and its derivative. There are many kinds of differential equations and tons of specialized techniques we may use to solve them. Fortunately, on the AP Calculus exams you will only encounter a handful of the most basic kinds. Basic Differential Equations: Integration You have probably worked out hundreds of differential equations without even realizing it! Its true! Let me give you an example. What makes this a differential equation? Well think about what the notation means. You know that integration is the opposite of differentiation. So what were looking for here is a function f(x) whose derivative is equal to 2x. In other words, we have to solve: f  (x) = 2x Thats a relation involving an unknown function f and its derivative. But this differential equation is trivial to solve! Just use the power rule for integrals (or guess-and-check). f(x) = x2 + C (For a full review of integration, check out: AP Calculus Exam Review: Antidifferentiation or AP Calculus Exam Review: Integrals.) Initial Value Problems Now suppose you have more information about an unknown function, such as its value at a certain point. Then you may be able to solve for the function explicitly, rather than getting stuck with an unknown constant of integration at the end. An initial value problem typically gives a derivative expression along with a function value. The goal is to produce the original function. Example 1: Initial Value Problem If f  (x) = 1/x2 for x 0, and f(1) = 5, find the expression for f(x) for x 0. Solution Because the derivative expression is given, we integrate to find the original function. Dont forget about your constant of integration! Next, plug in the known info and solve for the unknown constant of integration. This implies that the original function must be:   Ã‚   f(x) = -1/x + 6. Separation of Variables Of course not every differential equation (or DE) problem can be handled by simple integration. Often more advanced techniques must be used, especially if both x and y show up in the equation. For example, the following DE cannot be cracked by integration (not in its current form, that is). The key is to somehow break up the dy and dx. The derivative notation dy/dx looks like a fraction. However, its not really a fraction, because the two quantities dy and dx (called differentials) are supposed to represent the idea of taking Δx to zero in the limit. In other words, if you had to identify the numerical value of a differential, it would make sense to say dx = dy = 0. And we all know that 0/0 is undefined. But those differential gadgets are more subtle than that. They really dont carry a numerical value. Instead, they stand for a limiting process. And if were very careful, we can work with dx and dy individually. The method of Separation of Variables works by manipulating dy/dx as if it were a fraction, and then using integration to get rid of the differentials. Now lets take another look at our example above. Example 2: Separation of Variables If   Ã‚  Ã‚  , and y(0) = -3, find an equation for y in terms of x. Solution First multiply dx to both sides. Then multiply or divide as necessary so that only expressions of y are on the left, and only those with x are on the right. Next, we use the magic of calculus! All you have to do is to apply the integral symbol to each side of the equation. Now you have two separate integrals to work out. Now even though both sides generated a constant of integration, those can be combined into a single constant on the right hand side. The next few steps are just for isolating y. Notice that the quantity eC is replaced by another constant k. This is because C is still arbitrary (and unknown). So just think of k as a related, but still arbitrary constant. Finally we solve for k using the known information, y(0) = -3. (By the way, we will drop the  ± notation at this point because k itself can take care of that choice.) Therefore, replacing k by its computed value -3, we obtain the final form of the function. Setting Up Differential Equations Fortunately you wont encounter any DE problems on the AP Calculus exam that cant be handled by either integration or separation of variables. However, you may be asked to set up and/or analyze a DE (without solving it). Example 3: Setting up a DE A certain bacteria culture has P cells at time t and grows in proportion to the square root of the amount present. Set up a differential equation that models A(t). Solution Because the culture grows at a certain rate with respect to time, we know well be working with the derivative dA/dt. Let k be the constant of proportionality. Then translate the given information into mathematics: dA/dt = k Ãâ€" √(A) Remember, we dont have to solve for A(t); just set up the DE that could be used to solve for it. So were done at this step! Example 4: Analyzing a DE The number y of people infected with the flu in a certain town at time t can be modeled by the differential equation, dy/dt = 0.00002y(21500-y), where y(0) = 1. Determine the limit of y as t → ∞. Solution It would be difficult to solve this DE for y explicitly. Instead, look closely at the equation. It has the form of a logistics equation. That means we can answer this question with no work at all! You just have to remember that logistics equations always have solutions that tend toward the carrying capacity M as t → ∞. Here, M = 21500. Thats all you need to answer the question! The limit will be 21500. (For more details about the logistics equation, I recommend: AP Calculus BC Review: Logistics Growth Model.) Conclusion Differential equations show up only sparingly on an AP Calculus exam. But its important to be aware of the techniques for solving them, setting them up, and analyzing them. Remember the basic methods of integration and separation of variables. Know how to set up a differential equation. Look for situations in which you may avoid solving the DE.

Allotrope Definition and Examples Chemistry Glossary

The term allotrope refers to one or more forms of a chemical element that occur in the same physical state. The different forms arise from the different ways atoms may be bonded together. The concept of allotropes was proposed by Swedish scientist Jons Jakob Berzelius in 1841. The ability for elements to exist in this way is called allotropism. Allotropes may display very different chemical and physical properties. For example, graphite and diamond are both allotropes of carbon  that occur in the solid state. Graphite is soft, while diamond is extremely hard. Allotropes of phosphorus display different colors, such as red, yellow, and white. Elements may change allotropes in response to changes in pressure, temperature, and exposure to light. Examples of Allotropes To continue the carbon example, in diamond, the carbon atoms are bonded to form a tetrahedral lattice. In graphite, the atoms bond to form sheets of a hexagonal lattice. Other allotropes of carbon include graphene and fullerenes. O2 and ozone, O3, are allotropes of oxygen. These allotropes persist in different phases, including the gas, liquid, and solid states. Phosphorus has several solid allotropes. Unlike the oxygen allotropes, all phosphorus allotropes form the same liquid state. Allotropism Versus Polymorphism Allotropism refers only to the different forms of pure chemical elements. The phenomenon in which compounds display different crystalline forms is called polymorphism.

Compare and Contrast Autocratic and Democratic Leadership...

Autocratic leadership is a classical leadership style with the following characteristics: manager seeks to make as many decisions as possible, manager seeks to have the most authority and control in decision making, manager seeks to retain responsibility rather than utilise complete delegation , consultation with other colleagues in minimal and decision making becomes a solitary process managers are less concerned with investing their own leadership development, and prefer to simply work on the task at hand. Whereas Democratic Leadership is the leadership style that promotes the sharing of responsibility, the exercise of delegation and continual consultation. The style has the following characteristics: manager seeks consultation on all†¦show more content†¦Many leaders who start pursuing leadership development are often trying to improve upon their organisations autocratic leadership style. Each of these leadership styles has its own benefits (advantages) disadvantages and they are as follows: Advantages of autocratic leadership Style Advantages democratic leadership style Reduced stress due to increased control: where the manager ultimately has significant legal and personal responsibility for a project, it will comfort them and reduce their stress levels to know that they have control over their fate. Positive work environment: a culture where junior employees are given fair amount of responsibility and are allowed to challenge themselves is one where employees are more enthused to work and enjoy what they do. A more productive group ‘while the leader is watching’: the oversight that an autocratic manager exerts over a team improves their working speed and makes them less likely to slack. Successful initiatives: the process of consultation and feedback naturally results in better decision making and more effective operations. Improved logistics of operations: having one leader with heavy involvement in many areas makes it more likely that problems are spotted in advance and deadlines met. Creative thinking The free flow of ideas and positive work environment is the perfect catalyst for creative thinking. Faster decision making: when only one person makesShow MoreRelatedLeadership Styles Of Captain William Bligh1076 Words   |  5 PagesCompare and Contrast the Leadership Styles of Captain William Bligh verses Lieutenant Fletcher Christian. The topic of leadership evokes curiosity about our leaders and their approaches in decision making, leadership styles and the effectiveness of their leadership. At time leaders are critiqued for their actions or views on different business affairs. In today’s working environment leaders set the tone, vision, and goals of any organization. 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Object and Data Modeling Business Development and Business Strategies

Question: Discuss about theObject and Data Modelingfor Business Development and Business Strategies. Answer: Introduction The report depicts the importance of implementing cloud computing technology solution for the development of business and its overall strategies. The taken case study to prepare this report is Evaluating Cloud Technology Solutions for Business Development and Business Strategies. The advanced business functionalities will help different business organizations to gain competitive advances from the competitive marketplace. Cloud computing acts as a tool and it provide mobility to the businesses. In addition to this, the report will illustrate the resources that are required to collect data for the business development. Though, technology gives extreme commercial benefit to organizations but at the same time it put certain negative impact to the businesses also (Julfathna, Babu Jayakameswaraiah 2014). After complementary study of the business it will analyze different business risks that might hamper the system. The outcome of the business processes will be elaborated at the end of the report. Business Case The cloud computing technology, gives many possible advantages to the organization. It helps an organization to build a flexible virtual office that can connect the business to other technical advances. The technology gives effective business environment in terms of digital devices and due to this adaptation the users can access data from wherever and whenever they want (Paroutis, Bennett Heracleous, 2014). However, all time it will not help to scale up the business rater sometimes, lack of management system it might scale down the business growth. Cloud computing technology provides positive ambiance in terms of backup resilience, disaster recovery. It means once the data get stored in the database of an organization it cannot be deleted automatic, because cloud possesses a vast storage. If any information regarding previous applications is required, then due to the backup resilience one can fetch information from the storage easily (Ward Peppard, 2016). Moreover, it can be said t hat, cloud computing gives wide range of economical profit as it is capable to acquire different IT based capacities. Cloud computing can be acquired as a service model, that provides shared and configurable resources. Analysis of the Business Functionalities The business case has much functionality that is associated to cloud computing technologies. In order to instantiate advanced, flexible and scalable business model cloud computing technology is required to be adopted (Oliveira, Thomas Espadanal, 2014). Cloud computing serves various functions that are beneficial from the business perspectives. Some features of cloud computing, help to enhance the business ability of an organization. The functions are as follows: Infrastructure as a service (IaaS): In order to increase the business abilities most of the businesses from medium to large are eager to adopt IaaS platform (Paroutis, Bennett Heracleous, 2014). This is referred to an obvious choice as it is cot saving and it helps to develop and deploy the platform application. Private and hybrid cloud: The business organizations those want to host large IT infrastructures, can adopt hybrids cloud where they will be getting large storage capacity. Again the public cloud will help to protect information from different external attacks (Julfathna, Babu Jayakameswaraiah 2014). This particular case study implies that, more than 763 Billion cumulative finance supports can be achieved between the time periods of 2010 to 2012by adopting the hybrid, public and private cloud technology. Testing and development: This is one of the best possible scenarios provided by cloud computing technology needed to develop the business models (Gupta, Seetharaman Raj, 2013). With the help of secured budget estimation, manpower support and physical assets a specific environment can be created for the business. A business model supported by cloud service does not have any particular boundary rather; it often helps to combine the real and virtual world together. Big data analytics: Big data acts as a vital part of computing technology. It is referred to as an analytical tool, and helps to take important decision for business organizations. In order to develop the quantity and quality of the business infrastructure big data analytics is found to be very helpful (Ward Peppard, 2016). Information System is a combination of different business functionalities, and act in an integrated manner that includes customer relationship management and supply chain management also. On the other hand, backup and disaster recovery are the other functionalities associated to cloud computing. Required Resources In order to create a supportive set up for cloud computing in business model development the main required resources are computing resources that include- network, servers, storage, application and different services (Khanagha, Volberda Oshri, 2014). Most of the computing resources based on the service level agreement require datacenter resources with different cloud components. The resources are selected and monitored by the organizational business managers. The IT resources are also needed to develop the business model and business strategy at the same time. Broad Feasibility of the Project Cloud computing solution helps to create innovative business models. The development team of the business focuses on the infrastructure management system which is based on innovation (Oliveira, Thomas Espadanal, 2014). The main businesses professionals are focused on the cost benefit analysis program. The main aim is to evaluate a successful and cloud based collaborative platform that might reduce the pre-estimated cost. As cloud does not need to depend on any particular software and hardware even, no regular update policy is associated to it (DaSilva et al., 2013). Thus, they need to balance the production and delivery system. In online processing, the production is customer focused, and the regulatory management is focused on the prevention and promotion of the products. The project is enough feasible. Risk Analysis In this case study the most critical and highlighted issue is referred to as the data security. Most of the consumers were avoiding net banking as they felt that the system is not at all safe and secured. The basic key components oriented to risk analysis include availability, confidentiality, traceability and integrity (Repschlaeger et al., 2012). Other kinds of risks are associated to authentication. Unauthorized users might hack confidential data from the storage and misuse those is another issue. Outcome of the Business Project After analysis of the complete business model it has been found that, after adaptation of cloud computing solution, one traditional model can become more innovative and advantageous in nature. The business model will give competitive advantages to the consumers (Paroutis, Bennett Heracleous, 2014). The business project gave innovative business strategies in terms of application and actions. Conclusion From the overall discussion it can be concluded that the Information System gives effective profit to the consumers and to the organization as well. The report has mentioned the business background including the business functionalities. The cloud computing solution gave effective outcome. Currently, most of the organizations are providing strict attention to new technological advances such as cloud computing, digital platform and big data. The report analyzed the business case including its functionalities. The required resources are also discussed in this report. At the same time, many business risks are also associated to it those are also analyzed. References DaSilva, C. M., Trkman, P., Desouza, K., Lindi?, J. (2013). Disruptive technologies: a business model perspective on cloud computing.Technology Analysis Strategic Management,25(10), 1161-1173. Gupta, P., Seetharaman, A., Raj, J. R. (2013). The usage and adoption of cloud computing by small and medium businesses.International Journal of Information Management,33(5), 861-874. Julfathna, N., Babu, G. A., Jayakameswaraiah, M. (2014). Evaluating Cloud Technology Solutions for Business Development and Business Strategies.International Journal of Scientific and Research Publications, 49. Khanagha, S., Volberda, H., Oshri, I. (2014). Business model renewal and ambidexterity: structural alteration and strategy formation process during transition to a Cloud business model.RD Management,44(3), 322-340. Oliveira, T., Thomas, M., Espadanal, M. (2014). Assessing the determinants of cloud computing adoption: An analysis of the manufacturing and services sectors.Information Management,51(5), 497-510. Paroutis, S., Bennett, M., Heracleous, L. (2014). A strategic view on smart city technology: The case of IBM Smarter Cities during a recession.Technological Forecasting and Social Change,89, 262-272. Repschlaeger, J., Zarnekow, R., Wind, S., Turowski, K. (2012). Cloud Requirement Framework: Requirements and Evaluation Criteria to Adopt Cloud solutions. InECIS(p. 42). Ward, J., Peppard, J. (2016).The Strategic Management of Information Systems: Building a Digital Strategy. John Wiley Sons.