11/7/2023 0 Comments Common core sas geometry![]() The three students I had in the morning were confused and thought that today would begin with third period instead of fourth (since apparently, Tuesday began with second). This is likely why the teacher has first period conference - otherwise he'd be stuck with two homerooms. The film teacher's homeroom is actually the same as his "ninth" (zero) period, so that the class can give announcements during homeroom. Since homeroom was the class that was skipped, having first period be the class before the walkout means that it's the homeroom teacher who leads the classes out to the field. It's likely that the same is true at today's school as well - the homeroom kids are the first period kids. Recall that at the middle school I subbed at on Tuesday, homeroom is actually first period, in that the students in both classes are the same. But the gun walkout changed everything - not only was the schedule changed to allow for the 10:00 walkout, but the rotation was changed to start with sixth period instead of third. Yesterday the rotation started with sixth period, so today's rotation starts with - fourth period? Well, it turns out that yesterday, the rotation was supposed to start with third period. I won't do any more "Day in the Life" posts for this subbing assignment, but there are a few things I want to say. Meanwhile, today is the second of four days of subbing in the digital film class. It is the level set for 0, which passes through the point (2, 0).Lesson 12-9 of the U of Chicago text is called "The AA and SAS Similarity Theorems." In the modern Third Edition of the text, the AA and SAS Similarity Theorems appear in Lesson 12-7. The following call to the ODE subroutine computes a contour for the quadratic function. The trajectories are level sets of the "total energy function," which is the sum of the potential and kinetic energies for those mechanical systems. Although I didn't say it at the time, the phase portraits for the simple harmonic oscillator and the pendulum are plots that show contours. (For contours that are not closed curves, you also need to integrate "backwards" by using the vector field -G, which is also perpendicular to the gradient field.)īy a fortunate coincidence, I blogged last week about how to solve differential equations in SAS. Then the contour that passes through ( x 0, y 0) is exactly the same as the trajectory of G with initial condition ( x 0, y 0). Now the interesting fact about the predictor-corrector algorithm is this: the contour-tracing algorithm described above is the same as the predictor-corrector algorithm that is used to solve differential equations! Let G be a vector field that is perpendicular to the gradient field. * normalized vector field that is perpendicular to the gradientįield (-df/dy, df/dx) / norm(gradient) */ * gradient of function (df/dx, df/dy) */ This means that the perpendicular field is undefined at the critical points of f, where the gradient vanishes. The perpendicular field is normalized so thatĮach vector has length one. The following SAS/IML program defines the quadratic function, the gradient of the function, and the vector field that is perpendicular to the gradient field. For this function, the contour is an ellipse that passes through the point (2, 0). To give a simple example, suppose that you are interested in tracing the level set where f is the quadratic function, f(x,y) = x 2 + 4y 2 - 4. You now have a new point on the contour, so you can repeat the process.īecause the algorithm uses gradient information, it is often possible to form the tangent vector analytically. Usually, the step takes you off the contour, so you need to re-acquire the contour (the corrector step). ![]() ![]() You can then take a small step in the tangent direction (the predictor step). The gradient at ( x 0, y 0) is perpendicular to the contour at that point, so you can compute the tangent to the contour. Start with an initial point ( x 0, y 0) that is on the contour. The continuation method is illustrated by the graphic to the left. There are several algorithms for computing contours, but this article describes a technique known as a continuation method, or a predictor-corrector algorithm. However, sometimes you don't just want to see a picture of the contours, you actually want to compute a sequence of points along a specific contour. For example, I've previously written blogs that use contour plots to visualize the bivariate normal density function and to visualize the cumulative normal distribution function. Like many other computer packages, SAS can produce a contour plot that shows the level sets of a function of two variables. ![]()
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