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# What are the differences of linear and non-linear equation?

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# Was it difficult to shoot the part of the pilot in which Bruce's parents die?

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# What is the method of finite differences for linear equations?

Intuitive derivation Finite-difference methods approximate the solutions to differential equations by replacing derivative expressions with approximately equivalent differen…ce quotients. That is, because the first derivative of a function f is, by definition, then a reasonable approximation for that derivative would be to take for some small value of h. In fact, this is the forward difference equation for the first derivative. Using this and similar formulae to replace derivative expressions in differential equations, one can approximate their solutions without the need for calculus. [edit] Derivation from Taylor's polynomial Assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem, where n! denotes the factorial of n, and Rn(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. Again using the first derivative of the function f as an example, by Taylor's theorem, f(x0 + h) = f(x0) + f'(x0)h + R1(x), which, with some minor algebraic manipulation, is equivalent to so that for R1(x) sufficiently small, [edit] Accuracy and order The error in a method's solution is defined as the difference between its approximation and the exact analytical solution. The two sources of error in finite difference methods are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference between the exact solution of the finite difference equation and the exact quantity assuming perfect arithmetic (that is, assuming no round-off). The finite difference method relies on discretizing a function on a grid. To use a finite difference method to attempt to solve (or, more generally, approximate the solution to) a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image to the right). Note that this means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner. An expression of general interest is the local truncation error of a method. Typically expressed using Big-O notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity f'(xi) − f'i if f'(xi) refers to the exact value and f'i to the numerical approximation. The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. Using the Lagrange form of the remainder from the Taylor polynomial for f(x0 + h), which is , where x0 < ξ < x0 + h, the dominant term of the local truncation error can be discovered. For example, again using the forward-difference formula for the first derivative, knowing that f(xi) = f(x0 + ih), and with some algebraic manipulation, this leads to and further noting that the quantity on the left is the approximation from the finite difference method and that the quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation error. A final expression of this example and its order is: This means that, in this case, the local truncation error is proportional to the step size. [edit] Example: ordinary differential equation For example, consider the ordinary differential equation The Euler method for solving this equation uses the finite difference quotient to approximate the differential equation by first substituting in for u'(x) and applying a little algebra to get The last equation is a finite-difference equation, and solving this equation gives an approximate solution to the differential equation. [edit] Example: The heat equation Consider the normalized heat equation in one dimension, with homogeneous Dirichlet boundary conditions (boundary condition) (initial condition) One way to numerically solve this equation is to approximate all the derivatives by finite differences. We partition the domain in space using a mesh x0,...,xJ and in time using a mesh t0,....,tN. We assume a uniform partition both in space and in time, so the difference between two consecutive space points will be h and between two consecutive time points will be k. The points will represent the numerical approximation of u(xj,tn). [edit] Explicit method The stencil for the most common explicit method for the heat equation. Using a forward difference at time tn and a second-order central difference for the space derivative at position xj ("FTCS") we get the recurrence equation: This is an explicit method for solving the one-dimensional heat equation. We can obtain from the other values this way: where r = k / h2. So, knowing the values at time n you can obtain the corresponding ones at time n+1 using this recurrence relation. and must be replaced by the boundary conditions, in this example they are both 0. This explicit method is known to be numerically stable and convergent whenever . The numerical errors are proportional to the time step and the square of the space step: [edit] Implicit method The implicit method stencil. If we use the backward difference at time ti + 1 and a second-order central difference for the space derivative at position xj ("BTCS") we get the recurrence equation: This is an implicit method for solving the one-dimensional heat equation. We can obtain from solving a system of linear equations: The scheme is always numerically stable and convergent but usually more numerically intensive than the explicit method as it requires solving a system of numerical equations on each time step. The errors are linear over the time step and quadratic over the space step. [edit] Crank-Nicolson method Finally if we use the central difference at time tn + 1 / 2 and a second-order central difference for the space derivative at position xj ("CTCS") we get the recurrence equation: This formula is known as the Crank-Nicolson method. The Crank-Nicolson stencil. We can obtain from solving a system of linear equations: The scheme is always numerically stable and convergent but usually more numerically intensive as it requires solving a system of numerical equations on each time step. The errors are quadratic over the time step and formally are of the fourth degree regarding the space step: However, near the boundaries, the error is often O(h2) instead of O(h4). Usually the Crank-Nicolson scheme is the most accurate scheme for small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive. The implicit scheme works the best for large time steps. (MORE)

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# How does solving linear inequalities differ from solving linear equations?

Linear inequalities are equations, but instead of an equal sign, it has either a greater than, greater than or equal to, less than, or a less than or equal to sign. Both can b…e graphed. Solving linear equations mainly differs from solving linear inequalities in the form of the solution. 1. Linear equation. For each linear equation in x, there is only one value of x (solution) that makes the equation true. Example 1. The equation: x - 3 = 7 has one solution, that is x = 10. Example 2. The equation: 3x + 4 = 13 has one solution that is x = 3. 2. Linear inequality. On the contrary, a linear inequality has an infinity of solutions, meaning there is an infinity of values of x that make the inequality true. All these x values constitute the "solution set" of the inequality. The answers of a linear inequality are expressed in the form of intervals. Example 3. The linear inequality x + 5 < 9 has as solution: x < 4. The solution set of this inequality is the interval (-infinity, 4) Example 4. The inequality 4x - 3 > 5 has as solution x > 2. The solution set is the interval (2, +infinity). The intervals can be open, closed, and half closed. Example: The open interval (1, 4) ; the 2 endpoints 1 and 4 are not included in the solution set. Example: The closed interval [-2, 5] ; the 2 end points -2 and 5 are included. Example : The half-closed interval [3, +infinity) ; the end point 3 is included. (MORE)

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# What makes linear equations different than linear inequalities?

An equation is a statement that two quantities are equal, or the same, identical, in value. It is expressed by putting an equal sign (=) between the two quantities. An inequal…ity is a statement that two quantities are not equal, or more specifically, that one is less than the other, or less than or equal to the other. It is expressed with the unequal sign (an equal sign with a slash through it), a less than sign (), or a less than or equal sign or greater than or equal sign. A less than or equal sign looks like a less than sign with an underscore; similarly for the greater than or equal sign. Answer 1 A linear equation may be represented by all the points on a straight line. A linear inequality would be represented by all points in the plane on one side or the other of the line which is determined by the corresponding equation. The line itself may or may not be part of the solution. (MORE)

# Linear and Atmospheric Perspective in Landscape Painting

Landscape painters have two powerful means of suggesting distance in painting: linear perspective and atmospheric (aerial) perspective.… (MORE)

# The Equation for Cellular Respiration

The summary equation for cellular respiration shows which molecules are used by cellular respiration and which are produced. By counting the number of type of atoms on each si…de of the reaction arrow, you can see that the reaction is balanced: every atom that enters the reaction also leaves it. (MORE)

# How Do Critical Thinking and Logic Differ?

Critical thinking and logic can seem very similar - both can lead to informed decisions about a wide variety of topics, and both can be used within various classes to enhance …the academic experience. Nevertheless, critical thinking and logic differ on multiple counts, and they are different for the following reasons:Even though critical thinking and logic are commonly thought to be very similar, they are by definition quite different. Critical thinking is defined as the careful analysis of a topic in order to arrive at an opinion. Critical thinking typically involves dialogue that serves to broaden the conversation, ultimately leading to the best perceived answer based on a variety of relevant factors. On the other hand, logic is defined as reasoning that occurs under specific guidelines. You can tell by their definitions that logic and critical reasoning use different capabilities to arrive at an answer. Moreover, when you analyze something according to firm and known principles, you likely are using logic. When you analyze something by discussing principles that could have an effect on your answer, you are likely using critical thinking.Both critical thinking and logic can be applied in a variety of different situations, and sometimes one method of thinking can be better suited to a specific topic than the other. Critical thinking is often applied to situations that have fluid and changing principles associated with them. Take choosing a major, for example. You should choose your major based on your organic interests, your college's curriculum, and your potential job prospects. Any or all of those elements might be changing at any given time, so thinking critically about your priorities and interests would serve you well in this situation. Logic is most often applied to situations that have static principles associated with them, like which physics equation to use to solve a problem in science class. Logically, it is likely that there is only one equation best suited for that specific physics problem.Logic assumes that there is always a good and a bad way to reason. On the other hand, critical thinking assumes that any position with the best argument at the time can be the best and is associated with how people go about solving problems in reality. Critical thinking thus is open to a vast array of possibilities at any given time that can be considered more or less important at any given moment. While logic assumes an absolute nature to solving problems, critical thinking assumes a relative nature to solving problems.Logical thinkers are extremely methodical in their approach to solving problems. They may create pro/con lists, use equations, variables, and systematically track any changes within the environment to use now or in the future to resolve an issue. Critical thinkers are attuned to the world around them. They also take note of any changes and use that knowledge to solve problems in the moment, rather than in the future. If any significant amount of time goes by during the decision-making process, critical thinkers re-evaluate all angles of the problem again, as most of the factors they work with are constantly changing. (MORE)

# Headache: Battle of the Sexes

Men and women are biologically different. We all know hormones like estrogen and testosterone differ between the sexes, but there are many other important differences between …men and women. Interestingly, men and women experience pain differently. Women are more sensitive to pain and are more likely to develop chronically painful conditions, including headaches.The hormone estrogen has important influences on the brain and nervous system. As estrogen levels cycle, nerves change. Amazingly, nerves grow more receptive areas to receive signals from other nerves when estrogen levels increase. Estrogen levels are linked to the release of enkephalins, the body's natural pain reliever. They also affect levels of a wide range of brain chemicals important for signaling headache, including serotonin, dopamine, and norepinephrine.When pain sensitivity is tested in animals, female animals detect pain earlier and react more strongly to pain than males. Females without working ovaries respond like males and males injected with estrogen become more sensitive to pain.Research consistently shows that women are more sensitive to pain than men. In research experiments, healthy men and women will be tested by stimulating them with a temperature device or a small electric shock. As the temperature changes and the shock gets stronger, people will eventually find that the sensation they feel changes from just a touch to something painful. Women consistently feel less extreme changes as painful compared with men.The [American Migraine Prevalence and Prevention]() study was a large study surveying over 160,000 people in the United States about headaches. Women were twice as likely to have severe headaches as men (24 percent of women vs. 11 percent of men). Migraine occurred in 17 percent of women and 6 percent of men. Both men and women were most likely to have one migraine per week or fewer. Compared with men, women were 34 percent more likely to have more disabling headaches. Women were also more likely to experience a variety of other symptoms during a migraine attack, including an aura, blurred vision, sensitivity to noises and lights, and nausea or vomiting.Experimental pain experiments show that women have different pain relief after taking an analgesic medication compared with men. In an interesting experiment in mice, treatment with an anti-inflammatory drug decreased inflammation similarly in both genders, but the [analgesic effect was different](). In a study in humans, treatment with ibuprofen was more effective in reducing experimental pain in men compared with women]( http://www.ncbi.nlm.nih.gov/pubmed/9620515). Pain medicines or narcotics also work differently in men and women. For example, narcotics like morphine give better pain relief in men than women, while women respond better than men to opioids like butorphanol. One study testing relief of pain after surgery found that morphine doses needed to be [30 percent higher in women than men to get the same degree of pain relief]().Women are more likely to experience pain relief from [exercise and massage]() compared with men.Men and women experience pain and headaches differently. Don't be surprised if your headaches have different symptoms or respond differently to treatment compared with a spouse or sibling of the opposite sex. Women are more sensitive to pain than men and tend to experience more severe and disabling headaches. Medications that may work well in men, such as ibuprofen, may not be as effective when used to treat women. (MORE)

# Differences Between Computer Science and Computer Engineering

As the world continues to become more digital, it is reasonable to assume that the number of employment opportunities for those in the technology and computer industries will …continue to grow. Indeed, computer science and engineering are among the top-paying fields with room for growth. But with so many different technology and computer-related fields in existence, it can be hard to figure out what's what and to determine which one best suits your interests or skills.In general, these positions and fields of study tend to fall into one of two areas: computer science and computer engineering. Although these two fields tend to overlap each other in a number of different ways, they are in fact separate areas of study with separate applications.In the simplest terms, computer science is the study of how computers operate and can be used. Computer science covers areas like programming, coding, or other practical processes performed by computers as they send and receive data.For example, if you wanted to develop a program to increase the speed and efficiency of downloading large files from the internet, you would do so using a number of the skills that you've learned through computer science courses. Similarly, if you had an interest in video game design and production, you would need to have a base of knowledge that included several technical aspects of computer science, like programming, various coding languages, networking, and so on.Considering that computer science is largely focused on the functioning of software, the field requires individuals to have a strong background in mathematics. This is because the processing and transmission of data works through a series of equations and algorithms that are fundamentally rooted in math.Unlike computer science, which focuses on what the computer can do, computer engineering is much more focused on what the computer can be. Computer engineering often has similar goals as computer science, but it is pursued through the lens of electrical engineering. In the most basic terms, computer engineering is about the actual design and building of computers and technological devices. Say, for example, that you decide that someday you want to build the world's smallest cellphone or a piece of wearable technology. In order to do this, you would probably need to pursue a degree in computer engineering, which can teach you the skills you'll need to know in order to construct the individual pieces of the device.Like computer science, computer engineering involves a large amount of math in order to form equations and algorithms that will power the devices you are building. Because computers involve a number of small parts working together, you would need to develop a firm understanding of both math and science in order to succeed in the field.The most significant difference between these two is that one is for developing uses for the computer, while the other is for developing the hardware itself. Despite these separate objectives, computer scientists and computer engineers often work collaboratively in the design and building of new digital technology. This is because one can rarely work without the other.Ultimately, the decision to pursue one field over the other will depend on your areas of interest and goals, but it is probably worth keeping in mind that having a degree in one or the other will not preclude you from working on projects in both fields of study. (MORE)

# How Do ACT and SAT Math Differ?

When it comes to deciding between which test to take - the American College Testing assessment (ACT) or the Scholastic Aptitude Test (SAT) - the decision can be a tricky one. …For many years the majority of colleges in the United States only accepted the SAT, so that was the test most college-bound students took without question. However, recently many universities began accepting either the ACT or SAT, giving students the ability to decide which test works best for them; additionally, some states require its students to take one of the tests. Since the math section of these tests is generally what makes students the most nervous, here are some of the key differences between the ACT and SAT math sections to help in the decision-making process.Both the ACT and SAT have problems that cover geometry, pre-algebra, and algebra I and II, but the ACT includes some trigonometry questions at the end. While this is not a huge portion of the section, students who struggle with trigonometry may wish to steer clear of the ACT if they believe this section will bring down their grade.The ACT contains more straightforward math questions that test whether students can remember important equations and theorems and apply them correctly to a variety of problems. The SAT contains mostly word and story problems, wanting students to take the time to comprehend what the question is asking, draw a diagram if necessary, and then solve the problem. For some people, the straightforward nature of the ACT is preferred, while many students are confident when solving story problems and feel the SAT will better demonstrate their math knowledge. Understanding where you fall on this spectrum will greatly enable you to make a decision toward which test is best for you.The SAT contains three math sections that have a combined total of 54 questions that students are given 70 minutes to complete. The ACT has one math section with 60 questions and 60 minutes to solve them. While it may seem strange that the SAT gives more time with each question, while the ACT only gives students about one minute per problem, remember that the questions are structured differently. Since the SAT focuses on critical thinking, students will need more time to work through each problem, while the ACT questions may be more straightforward and require less time to solve. The SAT is graded with each section being worth 200-600 points, and the points earned are combined into the final score. For the ACT each section is worth 1-36 points, and the composite score is reached by averaging the points earned in each section. So with the ACT, even if you don't do well on one section you may still earn a good overall score, while the SAT reflects how you actually performed in each section.These are the key differences between the ACT and SAT math portions. When debating which test to take, students are encouraged to consider their preferred learning and testing styles, the requirements of the university they are interested in attending, and the types of math they feel most confident in. Deciding between these two tests can be a confusing process, but these points may add some clarity. (MORE)

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# Whats Difference between linear equation to linear function?

A linear equation contains only the first power of the unknown quantity. Thus, 5x - 3 = 7 and x/6 = 4 are both linear equations. Linear equations have only one solution which …is the value of the unknown that when substituted in the equation , makes the left hand side equal to the right hand side. Linear functions have the same limitation in terms of only containing the first power of the unknown quantity. They yield graphs that are straight lines and thus the name 'linear' is used. A simple linear function is f:x →2x + 1. This can also be written as f(x) = 2x + 1 or another identifying letter used such as y = 2x + 1. Consequently, for different values of the unknown quantity (in this case 'x') then the function also yields a different value. (MORE)

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# How does solving a literal equation differ from solving a linear equation?

Because linear equations are based on algebra equal to each other whereas literal equations are based on solving for one variable.

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# What are the difference between linear equation and quadratic equation?

A quadratic equation must be able to be written in the form: y = ax2 + bx + c where a is not equal to zero. The graph will be a parabola. There must be a "squared" …term and no larger exponent than "2". A linear equation will consist of variables only to the first power and the graph will be a straight line. Y = mx + b is an example of a liear equation where m will represent the slope and b will represent the y-intercept. (MORE)