AP Calculus BC Review Taylor Polynomials

Taylor polynomials are certain polynomials that can approximate functions. The more terms in the Taylor polynomial, the greater the accuracy of the approximation. Brook Taylor, one of the originators of Taylor polynomials In this article we will define Taylor polynomials and work out a number of examples similar to those you might see on the AP Calculus BC exam. Taylor Polynomials Before we define the Taylor polynomial, lets remember what a normal, everyday kind of polynomial is. Polynomials A polynomial is any expression of the following form. Here, n is a whole number. Each coefficient a0, a1, a2, , an, must be constant. Furthermore, we always insist that an is nonzero. (Otherwise, we really wouldnt have an nth term at all, right?) Then we say that the degree of the polynomial is n. Often, youll see the terms written in the opposite order, from high degree down to low, which is called standard form. But a Taylor polynomial is usually written starting from low degree and then progressing to higher degrees with each term. However you slice it, a polynomial is simply the sum of a terms, each of which involving a whole number power of the variable. Here are a few example polynomials: Here, f has degree 6, g has degree 8, and h has degree 11. Polynomials and Derivatives The nice thing about a polynomial is that its very easy to take its derivative. All you need are three simple rules. The Sum/Difference Rule the Constant Multiple Rule and the Power Rule Now, even if you replace x by (x c) in a polynomial, you still get a polynomial. Lets explore the derivatives of such a function. As you keep taking higher-order derivatives, notice how the degree of the polynomial steps down. Another curiousity is how the constant terms behave. By plugging x = c into the function and each successive derivative, you can recover each constant term. Take a look at the pattern that emerges. Original function: f(c) = a0 First derivative: f(c) = a1 Second derivative: f(c) = 2a2 Third derivative: f(c) = 6a3 By the time you get all the way to the nth derivative, theres nothing left but the last term. Each derivative step tacks on another factor, until you end up with all of the counting numbers starting from n down to 1 multiplied to the final coefficient: You might recognize that excited notation, n!, as the factorial of n. Taylor Coefficients Now I bet youre wondering what all of this have to do with Taylor polynomials. Its simple actually! We get to treat any given function f(x) as if it were a polynomial, and use the formula above to find out what each coefficient should be if it really were a polynomial. Well, first we have to solve for the Taylor coefficients an: (Remember that 0! = 1.) Now lets see a formal definition. Definition of the Taylor Polynomial Let n 0 be a whole number. Suppose that f is a function with derivatives of order k for each k = 1, 2, , n. If c is a real number, then the Taylor polynomial of order n centered at c is: Examples The easiest (and most common) kinds of Taylor polynomials are those centered at c = 0. In fact, these ones have a special name: Maclaurin polynomials. Example: Maclaurin Polynomials for ex Find a 3rd degree Maclaurin polynomial for f(x) = ex, and use it to approximate the value of e. Solution I like to organize my work in a table. Dont forget to include the 0th derivative, which is the original function. The Taylor coefficients are as follows. kf(k)(x)f(k)(0)/k! 0f(x) = exe0/0! = 1 1f(x) = exe0/1! = 1 2f(x) = exe0/2! = 1/2 3f(x) = exe0/3! = 1/6 Therefore, the 3rd-degree (or 3rd-order) Taylor polynomial for ex is: Finally, to approximate e = e1, just plug x = 1 into the Taylor polynomial and evaluate. (The actual value of e is 2.71828, so our estimate is not too far off!) Example: Taylor Polynomial for ln(x) Use the 5th degree Taylor polynomial for f(x) = ln(x), centered at 1, to approximate the value of ln(2). Solution Again, Ill use a table to record the Taylor coefficients. kf(k)(x)f(k)(1)/k! 0f(x) = ln xln(1) = 0 1f(x) = 1/x1/1 = 1 2f(x) = -1/x2(-1)/2! = -1/2 3f(x) = 2/x32/3! = 2/6 = 1/3 4f(4)(x) = -6/x4(-6)/4! = -6/24 = -1/4 5f(5)(x) = 24/x524/5! = 24/120 = 1/5 So our 5th degree polynomial is: Now, to approximate ln(2), we plug x = 2 into this expression. Notice that each time you will get a factor of (2 1) to some power. Of course, 2 1 = 1, so those factors wont matter. It all boils down to an alternating sum of fractions in the end. ln 2 1 1/2 + 1/3 1/4 + 1/5 = 47/60 0.78333 Example: Maclaurin Polynomials for sin x Find the Maclaurin polynomials for sin x up to degree 9, and plot them on the same set of axes. Solution Fortunately, there is a repeating pattern in the derivatives of sin x. The derivatives repeat in blocks of four, sin x cos x -sin x -cos x Furthermore, when you plug in x = 0, half of the Taylor coefficients will become zero because youre plugging into sin x. The remaining terms end up alternating in sign. Keep in mind, cos 0 = 1. kf(k)(x)f(k)(1)/k! 0f(x) = sin xsin 0 = 0 1f(x) = cos xcos 0 = 1 2f(x) = -sin x(-sin 0)/2! = 0 3f(x) = -cos x(-cos 0)/3! = -1/6 4f(4)(x) = sin x(sin 0)/4! = 0 5f(5)(x) = cos x(cos 0)/5! = 1/120 6f(6)(x) = -sin x(-sin 0)/6! = 0 7f(7)(x) = -cos x(-cos 0)/7! = -1/5040 8f(8)(x) = sin x(sin 0)/8! = 0 9f(9)(x) = cos x(cos 0)/9! = 1/362880 Lets label the Taylor polynomial of degree n by Tn. According to the table, each T2 will be the same as T1 because the degree two coefficient is zero. This is true for all of the even cases. So Im just going to list the odd degree cases: Below, you can see how they progressively get closer to the graph of y = sin x as the degree goes up. The graphs of the Taylor polynomials of a function tend to match the original functions graph closer and closer as the degree increases. Whats Next? Taylor polynomials are just that: polynomials. And no matter how high the degree, the polynomial will not match the original function f exactly (unless f is itself a polynomial). Check back on this blog to see what happens when you allow the number of terms to go to infinity. Then we can get an exact match for f in a lot of cases. These ideas are incredibly important in fields like engineering and physics. But thats a discussion for another day!

The Individuals With Disabilities Education Act Orders

The Individuals With Disabilities Education Act orders all states to give disabled students a free and appropriate education. Many schools in the United States mainstream disabled students and have succeeded in properly educating them, along with creating a safe environment where they are socially comfortable and accepted. However, there are too many incidents where teachers improperly handle these students and administrators do not give enough money to fund adequate supplies for these students. Although disabled students gain social benefits when mainstreamed, they ultimately do not thrive in such settings due to immoral methods and insufficient budgets. In academic settings, disabled students have access to more opportunities and many†¦show more content†¦Most importantly, â€Å"children with disabilities have the opportunities to observe, imitate and model the language use of their peers who do not have disabilities† (Grabmeier). By following their nondisabled fri ends, disabled students learn how to better communicate their thoughts, feelings and ideas which makes it easier for teachers to help them and understand them. Another beneficial skill disabled students receive through mainstreaming is positive thinking. According to Kathy Foust, the mainstreaming environment allows disabled students to learn what type of behavior is appropriate and what is not. They learn that listening to a teacher leads to little or no punishments while disobeying a teacher leads to more punishments. Foust also includes that mainstreaming stimulates self-esteem because the disabled know they are in regular classes, which makes them proud of themselves. These students believe that since they are in classrooms with nondisabled students, they are in higher and more advance classes. In other words, since students around the disabled are capable of extraordinary things, the disabled feel like they are extraordinary as well, thus having an improved self-esteem. â€Å"The Choice of Settings: The Pros and Cons of Mainstreaming Children With IntellectualShow MoreRelatedIndividuals with Disabilities Education Act1188 Words   |à ‚  5 PagesIndividuals with Disabilities Education Act The Individuals with Disabilities Education Act (IDEA) was initially passed in 1975. Prior to the passage of the IDEA, many people with disabilities were denied access to public education and many were relegated to institutions. The IDEA guarantees individuals with disabilities a free and appropriate education (FAPE) in a least restrictive environment (LRE). 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Alcohol as a Gateway Essays - 779 Words

Alcohol and heavy drinking throughout Canada plays a distinctive role in instigating other key addictions. Drinking and consumption abuse can be linked strongly to the abuse of illicit drugs. Binge drinking should be seen as a gateway or portal to the development of poly-drug users. As the gateway drug theory suggests, routine use of less harmful drugs, in this case alcohol, will lead to risk of abusing more serious drugs. Alcohol is so readily available and like any other psychoactive drug it can be very addictive. More than 600,000 Canadians are dependent on alcohol, and nearly 200,000, on illicit drugs. Alcohol consumption needs to be portrayed as a more dangerous activity. The possible risk factors in alcohol consumption can be†¦show more content†¦By establishing the patterns of heavy drinking we can then begin to see correlations with other sorts of rebellious behavior and risky activity. Binge drinking has been identified as a gateway to other illicit drugs. There is a strong link between alcohol consumption and the gateway drug theory. The gateway drug theory suggests that the routine use of less harmful drugs or substances may lead to a future risk of abusing more serious hard drugs. It has been found that alcohol precedes the use of cannabis as well as other drugs. Alcohol is the starting point in a vicious cycle. Cigarette and alcohol use is associated with the likelihood of marijuana use; marijuana use is in turn associated with the likelihood of other drug use. A study of a British Columbian sample group showed â€Å"43% of the sample reporting simultaneous use of both alcohol and marijuana on the previous weekend and 14% reporting simultaneous use of alcohol and cocaine† (Stockwell et al, 2007). Alcohol and drinking is observed as a fairly low risk activity among students; however, with the possible relationship to hard drugs, alcohol should be viewed m ore cautiously. 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Parts of Adobe Photoshop Essay Example For Students

Parts of Adobe Photoshop Essay It is where the tools you will apply directly to the canvas are found. Toolbox or the tools panel is likely the part tooth Photos workspace you will find yourself working the most, Palettes or panes ;The palettes or panes in ; Photos helps you monitor and modify image. All sorts Of information are displayed in panes. ;quickly change the view of ; Navigator Palette your artwork using a thumbnail display. The colored box in the Navigator (called the proxy view area) corresponds to the currently viewable area in the window. Provides a color ramp along the bottom ; Color Palette of the palette for quickly picking an approximate color. Displays the color values for the current foreground and background colors, using the sliders in the Color palette, you can edit the foreground and background colors using different color models, ;Undo or redo multiple steps. ;Temporary copy of the image, and use for experimenting. History Palette Layers Palette Can work on one element Without disturbing Others. Status Bar.. .. Displays the status of your created document or file. It also displays the settings you have done to your work as well as file properties tools used. ;Largest element of the ; Working Area/ Canvas workspace and is located in the middle of the application Space within the program window where a new or existing document appears and edited. The real power of Photos is in working with existing images. Typical task include treating and manipulation, composting, converting to different formats, printing, etc.

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The Characteristics of Essay Writing Topics for Grade 5

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