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Article / Updated 09-22-2022

Each quantum state of the hydrogen atom is specified with three quantum numbers: n (the principal quantum number), l (the angular momentum quantum number of the electron), and m (the z component of the electron’s angular momentum, How many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers n, l, and m? Well, the actual energy is just dependent on n, as you see in the following equation: That means the E is independent of l and m. So how many states, |n, l, m>, have the same energy for a particular value of n? Well, for a particular value of n, l can range from zero to n – 1. And each l can have different values of m, so the total degeneracy is The degeneracy in m is the number of states with different values of m that have the same value of l. For any particular value of l, you can have m values of –l, –l + 1, ..., 0, ..., l – 1, l. And that’s (2l + 1) possible m states for a particular value of l. So you can plug in (2l + 1) for the degeneracy in m: And this series works out to be just n2. So the degeneracy of the energy levels of the hydrogen atom is n2. For example, the ground state, n = 1, has degeneracy = n2 = 1 (which makes sense because l, and therefore m, can only equal zero for this state). For n = 2, you have a degeneracy of 4: Cool.

View ArticleArticle / Updated 09-22-2022

When g'(x) = f(x), you can use the substitution u = g(x) to integrate expressions of the form f(x) multiplied by h(g(x)), provided that h is a function that you already know how to integrate. Variable substitution helps to fill the gaps left by the absence of a Product Rule and a Chain Rule for integration. Here’s a hairy-looking integral that actually responds well to substitution: The key insight here is that the numerator of this fraction is the derivative of the inner function in the denominator. Watch how this plays out in this substitution: Declare u equal to the inner function in the denominator and make the substitution: Here’s the substitution: Differential du = (2x + 1) dx: The second part of the substitution now becomes clear: Notice how this substitution hinges on the fact that the numerator is the derivative of the inner function in the denominator. (You may think that this is quite a coincidence, but coincidences like these happen all the time on exams!) Integration is now quite straightforward: You take an extra step to remove the fraction before you integrate: Substitute back x2 + x – 5 for u: Checking the answer by differentiating with the Chain Rule reveals how this problem was set up in the first place: Here’s another example where you make a variable substitution: Notice that the derivative of x4 – 1 is x3, off by a constant factor. So here’s the declaration, followed by the differentiation: Now you can just do both substitutions at once: At this point, you can solve the integral simply. Similarly, here’s another example: At first glance, this integral looks just plain horrible. But on further inspection, notice that the derivative of cot x is –csc2 x, so this looks like another good candidate: This results in the following substitution: Again, this is another integral that you can solve.

View ArticleArticle / Updated 09-22-2022

The PMP Certification Exam will expect you to be able to manage large projects. The larger your project, the more time you will spend managing stakeholders and their engagement level, and the more critical this process is to the success of your project. Imagine a project with city and county government, the public, the department of transportation, police, and various and sundry other stakeholders. Do you think that you’re going to spend your time looking at a schedule? No. You’re going to spend your time juggling the various needs and interests of all these very influential stakeholders! The cross-cutting skill of relationship management is particularly relevant to the Manage Stakeholder Expectations process. To maintain satisfied stakeholders, you want to do three things: Engage stakeholders. Talk to your stakeholders throughout the project to ensure a common understanding of the project scope, benefits, and time and cost estimates. You will want to ensure their continued support of the project. Actively manage stakeholder expectations. You can do a lot of this by seeking stakeholder input while you’re planning the project. The project management plan and the project documents are written records that communicate all aspects of the project. These are an effective method of managing expectations. Address stakeholder concerns before they escalate. You might need to have individual conversations with some stakeholders to ensure that they understand situations, such as why they can’t have what they want in the time frame they want it, or why you require their staff for specific periods of time. By addressing concerns before they become issues, you will save a lot of time and hassle. Clarify and resolve issues in a timely manner. If a stakeholder has a concern that rises to the level of an issue that you need to document and address, do so as soon as is reasonable. Sometimes, this results in a change request to alter the schedule, the scope, or the resources. Manage Stakeholder Engagement. Communicating and working with stakeholders to meet their needs/expectations, addressing issues as they occur, and fostering appropriate engagement in project activities throughout the project lifecycle. Managing stakeholders can get very political. Make sure you know whom you are talking to and what their agenda is! Manage Stakeholder Engagement: Inputs During project planning, you create a stakeholder management plan that you will use and update throughout the project. Because your main method of managing expectations is communication, you will use your communication management plan as well. It would be nice to assume that everything will go smoothly and as planned, but because this never happens on projects, you also need a change log. Many change logs are based on templates or information from past projects — in other words, organizational process assets (OPAs). Manage Stakeholder Engagement: Tools and Techniques Communication skills are applied when managing your stakeholders. As you manage stakeholders, pay particular attention to their needs, promoting open communication and building trust. You can build trust by employing active listening. The nature of projects is that they have changes along the way. Some stakeholders will be resistant to the changes, but you have to help them overcome their resistance. You will also use conflict resolution skills. Some interactions with stakeholders will be in one-on-one meetings. Other times, you will be speaking to a large group and making presentations. Still others will require you to communicate via report or other written methods. All these situations are considered management skills.

View ArticleArticle / Updated 09-22-2022

You can use chord patterns to track chord progressions in the open position on the guitar, although doing so takes some extra work and requires that you identify the actual note name of each chord. To play in the key of G using common open chords, visualize the 6th string chord pattern starting on G at the 3rd fret and replace each barre chord with an open chord. Here’s how: Visualize the 1st barre chord, the I chord (G), but play an open G chord instead. Visualize the 2nd chord, the ii chord (Am), but play an open Am instead. Because there’s no open chord iii (Bm), play Bm at the 2nd fret of the 5th string close to the open position instead. Use common open chords to play chords IV, V, and vi (C, D, and Em). Play through all the chords forward and backward, calling out the numbers as you go. Follow along with your eye using the 6th string chord pattern even though you’re not using its barre chords. After you get the hang of playing like this in G, you can move the chord pattern and use open chords in other keys like F and A. When you do this, play open chords when you can and use the barre chords to fill in the rest, staying as close to the open position as possible. For example, in the key of F, you can play the Am, C, and Dm as open chords, but you have to use the barre chords to play the rest. In the key of A, you can play A, D, and E as open chords, Bm and C♯m at the 2nd and 4th frets of the 5th string, and F♯m at the 2nd fret of the 6th string. Do the same thing with the 5th string chord pattern starting in the key of C. In C, you can play all the chords as open chords except for F, which guitarists usually play as a partial barre chord when it’s paired with open chords. Move the chord pattern and use open chords in other keys; just remember to stay as close to the open position as possible when you need to fill in with barre chords.

View ArticleArticle / Updated 09-22-2022

You can calculate a confidence interval (CI) for the mean, or average, of a population even if the standard deviation is unknown or the sample size is small. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is numerical, most people want to estimate the mean (average) value for the population. You estimate the population mean, by using a sample mean, plus or minus a margin of error. The result is called a confidence interval for the population mean, In many situations, you don’t know so you estimate it with the sample standard deviation, s. But if the sample size is small (less than 30), and you can’t be sure your data came from a normal distribution. (In the latter case, the Central Limit Theorem can’t be used.) In either situation, you can’t use a z*-value from the standard normal (Z-) distribution as your critical value anymore; you have to use a larger critical value than that, because of not knowing what is and/or having less data. The formula for a confidence interval for one population mean in this case is is the critical t*-value from the t-distribution with n – 1 degrees of freedom (where n is the sample size). The t-table The t*-values for common confidence levels are found using the last row of the t-table above. The t-distribution has a shape similar to the Z-distribution except it’s flatter and more spread out. For small values of n and a specific confidence level, the critical values on the t-distribution are larger than on the Z-distribution, so when you use the critical values from the t-distribution, the margin of error for your confidence interval will be wider. As the values of n get larger, the t*-values are closer to z*-values. To calculate a CI for the population mean (average), under these conditions, do the following: Determine the confidence level and degrees of freedom and then find the appropriate t*-value. Refer to the preceding t-table. Find the sample mean and the sample standard deviation (s) for the sample. Multiply t* times s and divide that by the square root of n. This calculation gives you the margin of error. Take plus or minus the margin of error to obtain the CI. The lower end of the CI is minus the margin of error, whereas the upper end of the CI is plus the margin of error. Here's an example of how this works For example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond. You take a random sample of 10 fingerlings and determine that the average length is 7.5 inches and the sample standard deviation is 2.3 inches. Because you want a 95 percent confidence interval, you determine your t*-value as follows: The t*-value comes from a t-distribution with 10 – 1 = 9 degrees of freedom. This t*-value is found by looking at the t-table. Look in the last row where the confidence levels are located, and find the confidence level of 95 percent; this marks the column you need. Then find the row corresponding to df = 9. Intersect the row and column, and you find t* = 2.262. This is the t*-value for a 95 percent confidence interval for the mean with a sample size of 10. (Notice this is larger than the z*-value, which would be 1.96 for the same confidence interval.) You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. This means Multiply 2.262 times 2.3 divided by the square root of 10. The margin of error is, therefore, Your 95 percent confidence interval for the mean length of all walleye fingerlings in this fish hatchery pond is (The lower end of the interval is 7.5 – 1.645 = 5.86 inches; the upper end is 7.5 + 1.645 = 9.15 inches.) Notice this confidence interval is wider than it would be for a large sample size. In addition to having a larger critical value (t* versus z*), the smaller sample size increases the margin of error, because n is in its denominator. With a smaller sample size, you don’t have as much information to “guess” at the population mean. Hence keeping with 95 percent confidence, you need a wider interval than you would have needed with a larger sample size in order to be 95 percent confident that the population mean falls in your interval. Now, say it in a way others can understand After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.” In this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 5.86 and 9.15 inches, based on my sample data.” (Always be sure to include appropriate units.)

View ArticleArticle / Updated 09-22-2022

You can use the sum and difference formulas for cosine to calculate the cosine of the sums and differences of angles similarly to the way you can use the sum and difference formulas for sine, because the formulas look very similar to each other. When working with sines and cosines of sums and differences of angles, you're simply plugging in given values for the variables (angles). Just make sure you use the correct formula based on the information you're given in the question. Here are the sum and difference formulas for cosines: The sum and difference formulas for cosine (and sine) can do more than calculate a trig value for an angle not marked on the unit circle (at least for angles that are multiples of 15 degrees). They can also be used to find the cosine (and sine) of the sum or difference of two angles based on information given about the two angles. For such problems, you'll be given two angles (call them A and B), the sine or cosine of A and B, and the quadrant(s) in which the two angles are located. Use the following steps to find the exact value of cos(A + B), given that cos A = –3/5, with A in quadrant II of the coordinate plane, and sin B = –7/25, with B in quadrant III: Choose the appropriate formula and substitute the information you know to determine the missing information. then substitutions result in this equation: To proceed any further, you need to find cos B and sin A. Draw pictures representing right triangles in the quadrant(s). Drawing pictures helps you visualize the missing pieces of info. You need to draw one triangle for angle A in quadrant II and one for angle B in quadrant III. Using the definition of sine as opp/hyp and cosine as adj/hyp, this figure shows these triangles. Notice that the value of a leg is missing in each triangle. To find the missing values, use the Pythagorean theorem. The length of the missing leg in Figure a is 4, and the length of the missing leg in Figure b is –24. Determine the missing trig ratios to use in the sum or difference formula. You use the definition of cosine to find that cos B = –24/25 and the definition of sine to find that sin A = 4/5. Substitute the missing trig ratios into the sum or difference formula and simplify. You now have this equation: Follow the order of operations to get this answer: This equation simplifies to cos(A + B) = 4/5.

View ArticleArticle / Updated 09-22-2022

You can find a confidence interval (CI) for the difference between the means, or averages, of two population samples, even if the population standard deviations are unknown and/or the sample sizes are small. The goal of many statistical surveys and studies is to compare two populations, such as men versus women, low versus high income families, and Republicans versus Democrats. When the characteristic being compared is numerical (for example, height, weight, or income), the object of interest is the amount of difference in the means (averages) for the two populations. For example, you may want to compare the difference in average age of Republicans versus Democrats, or the difference in average incomes of men versus women. You estimate the difference between two population means, by taking a sample from each population (say, sample 1 and sample 2) and using the difference of the two sample means plus or minus a margin of error. The result is a confidence interval for the difference of two population means, There are two situations where you cannot use z* when computing the confidence interval. The first of which is if you not know In this case you need to estimate them with the sample standard deviations, s1 and s2. The second situation is when the sample sizes are small (less than 30). In this case you can’t be sure whether your data came from a normal distribution. In either of these situations, a confidence interval for the difference in the two population means is where t* is the critical value from the t-distribution with n1 + n2 – 2 degrees of freedom; n1 and n2 are the two sample sizes, respectively; and s1 and s2 are the two sample standard deviations. This t*-value is found on the following t-table by intersecting the row for df = n1 + n2 – 2 with the column for the confidence level you need, as indicated by looking at the last row of the table. To calculate a CI for the difference between two population means, do the following: Determine the confidence level and degrees of freedom (n1 + n2 – 2) and find the appropriate t*-value. Refer to the above table. Identify Identify Find the difference, between the sample means. Calculate the confidence interval using the equation, Suppose you want to estimate with 95% confidence the difference between the mean (average) lengths of the cobs of two varieties of sweet corn (allowing them to grow the same number of days under the same conditions). Call the two varieties Corn-e-stats (group 1) and Stats-o-sweet (group 2). Assume that you don’t know the population standard deviations, so you use the sample standard deviations instead — suppose they turn out to be s1 = 0.40 and s2 = 0.50 inches, respectively. Suppose the sample sizes, n1 and n2, are each only 15. To calculate the CI, you first need to find the t*-value on the t-distribution with (15 + 15 – 2) = 28 degrees of freedom. Using the above t-table, you look at the row for 28 degrees of freedom and the column representing a confidence level of 95% (see the labels on the last row of the table); intersect them and you see t*28 = 2.048. For both groups, you took random sample of 15 cobs, with the Corn-e-stats variety averaging 8.5 inches, and Stats-o-sweet 7.5 inches. So the information you have is: The difference between the sample means is 8.5 – 7.5 = +1 inch. This means the average for Corn-e-stats minus the average for Stats-o-sweet is positive, making Corn-e-stats the larger of the two varieties, in terms of this sample. Is that difference enough to generalize to the entire population, though? That’s what this confidence interval is going to help you decide. Using the rest of the information you are given, find the confidence interval for the difference in mean cob length for the two brands: Your 95% confidence interval for the difference between the average lengths for these two varieties of sweet corn is 1 inch, plus or minus 0.9273 inches. (The lower end of the interval is 1 – 0.9273 = 0. 0727 inches; the upper end is 1 + 0. 9273 = 1. 9273 inches.) Notice all the values in this interval are positive. That means Corn-e-stats is estimated to be longer than Stats-o-sweet, based on your data. The temptation is to say, “Well, I knew Corn-e-stats corn was longer because its sample mean was 8.5 inches and Stat-o-sweet was only 7.5 inches on average. Why do I even need a confidence interval?” All those two numbers tell you is something about those 30 ears of corn sampled. You also need to factor in variation using the margin of error to be able to say something about the entire populations of corn.

View ArticleArticle / Updated 09-22-2022

A matrix is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classified as: square, identity, zero, column, and so on. Where did matrices come from? For most of their history, they were called arrays. There are references to arrays in Chinese, French, Italian, and many other mathematical works going back many hundreds of years. American mathematician George Dantzig's work with matrices during World War II allowed for the coordination of shipments of supplies and troops to various locations. Matrices are here to stay. You may be familiar with a method used to solve systems of linear equations using matrices, but this application just scratches the surface of what matrices can do. First, just in case you're not familiar with solving equations using matrices, let me give just a quick description. If you want to solve the following system of equations: You write the matrix: And then you perform row operations until you get the matrix: From that matrix, you know that the solution of the system of equations is x = 1, y = -3, and z = -5. Pretty slick, don't you think? But uses for matrices don't stop there. You can solve traffic control problems, transportation logistics problems (how much of each item to send to various distribution centers), dietary problems (how much of each food product is needed to meet several different dietary requirements), and so on. Matrices work well in graphing calculators and computer spreadsheets — just set up the problem and let the technology do all the work.

View ArticleArticle / Updated 09-21-2022

The mapping of skin receptors to a specific area of neocortex illustrates one of the most fundamental principles of brain organization, cortical maps. The projection from the thalamus is orderly in the sense that receptors on nearby parts of the skin project to nearby cortical neurons. The figure shows a representation of the skin map on the somatosensory cortex. The fundamental idea about a given area of cortex being devoted to receptors in a given skin area is that activity in this area of cortex is necessary for the perception of the skin sense (along with other parts of the brain). You perceive activity in that bit of cortex area as a skin sensation not because that area has some special skin perceiving neurons, but because it receives inputs from the skin and has outputs that connect to memories of previous skin sensation and other associated sensations. In other words, the perception produced by activity in this and other areas of cortex is a function of what neural input goes to it and where the outputs of that area of cortex go. This skin map on the cortex is called a homunculus, which means "little man." However, the surface area of the somatosensory cortex onto which the skin receptors project is not really a miniature picture of the body; it is more like a band or strip, as established in the studies by Canadian neurosurgeon Wilder Penfield. Because of the difficulty in mapping a three-dimensional surface onto a two-dimensional sheet (think of how two-dimensional maps of the earth compare to the three-dimensional globes), the image is distorted, depending on choices that the "map-maker" makes about what is relatively more important to represent accurately versus what is less. Also note that some areas of the body, such as the hands and fingers, are located in the cortex map close to areas such as the face that are actually quite distant, body-wise. Some researchers suggest that the phantom limb feelings (including pain) that sometime occur after an amputation may occur because some neural projections from the face invade the part of the cortex that was being stimulated by the limb and cause sensations to be perceived as being located there, even when the limb is gone.

View ArticleCheat Sheet / Updated 09-21-2022

Whether you're talking about evolution — or any other element of science — you should understand the process of scientific investigation, which proves or disproves a scientific theory. Take a look at a chart of our hominid ancestors as discovered through fossil records, and learn some key terms to grasp the course of evolution.

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