# Rational approximation of vertical segments The root test Some infinite series can be compared to geometric series. The comparison test We compare infinite series to each other using inequalities.

The limit comparison test We compare infinite series to each other using limits. Absolute and Conditional Convergence. Absolute and Conditional Convergence The basic question we wish to answer about a series is whether or not the series converges. If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values.

This is the distinction between absolute and conditional convergence, which we explore in this section. Approximating functions with polynomials. Approximating functions with polynomials We can approximate smooth functions with polynomials. Power series Infinite series can represent functions. Introduction to Taylor series. Introduction to Taylor series We study Taylor and Maclaurin series.

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## Grade 8 Common Core Standards

Numbers and Taylor series. Numbers and Taylor series Taylor series are a computational tool. Calculus and Taylor series. Calculus and Taylor series Power series interact nicely with other calculus concepts. Differential equations Differential equations show you relationships between rates of functions. Separable differential equations. Separable differential equations Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation. Parametric equations We discuss the basics of parametric curves. Calculus and parametric curves We discuss derivatives of parametrically defined curves.

Introduction to polar coordinates. Introduction to polar coordinates Polar coordinates are coordinates based on an angle and a radius. Gallery of polar curves We see a collection of polar curves. Derivatives of polar functions. Derivatives of polar functions We differentiate polar functions. Integrals of polar functions. Integrals of polar functions We integrate polar functions.

## Mathematical Content Standards

Working in two and three dimensions. Working in two and three dimensions We talk about basic geometry in higher dimensions. Vectors Vectors are lists of numbers that denote direction and magnitude. The Dot Product The dot product is an important operation between vectors that captures geometric information. Projections and orthogonal decomposition Projections tell us how much of one vector lies in the direction of another and are important in physical applications.

The cross product The cross product is a special way to multiply two vectors in three-dimensional space. Jim Talamo and Bart Snapp.

Consider the segment of the curve from to :. The quantity is the slope of the secant line. The quantity is the slope of the tangent line.

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A little more formally: Assuming that the curve is piecewise differentiable, we can construct the slices so that the function is differentiable on each slice. For an arbitrary slice on , We can then invoke the Mean Value Theorem to guarantee that the slope of the secant line, , along the slice is equal to the slope of the tangent line at some point for. As the slice shrinks, , and hence we have that. We saw that the length of the curve on the interval is given by. Suppose the segment of a curve between the points on and in the -plane is defined by a sufficiently differentiable function.

Then, the length of this curve segment is.

Another important point that arises here is that we have complete freedom to express the infinitesimal arclength element in terms of or. Find the length of from to. Write with me Use the substitution to compute this integral. Trying to write down an integral with respect to here would be quite annoying. This problem will be much easier if we work with respect to. The limits of integration are We can now set up the integral. To proceed, we use the Pythagorean Trigonometric identity. Find the length of the segment of the curve from to. We have to differentiate carefully. The expression in the arclength formula simplifies nicely. Thus, the length of the curve segment can now be found.

Estimate the length of the curve from to. Truncate your answer at the hundredths place. Write with me. We have complete freedom to express the infinitesimal arclength element in terms of or. Thus, when choosing a variable of integration, consider only whether it would be easier to work with a description of the curve in terms of or. Most integrals involving radicals are difficult to evaluate and many of the integrands do not have elementary antiderivatives! Before using numerical software to approximate the integral check whether: The integrand is a square root of a linear function of.

If so, evaluate it by inspection or by using a substitution if necessary. The expression under the square root is actually a perfect square in disguise.

## The distance and midpoint formulas

This often requires careful algebra and differentiation. For a vertex p of P, dir p is a continuous range of directions. Convolution of Polygons I p dir p P Q. The convolution segments form closed cycles. Convolution of Polygons II 0 1 2. If P or Q are convex, we are done. The Case of Multiple Cycles. Handles a variety of curves. Each family of curves line segments, poly-lines, circular arcs, conic arcs, etc.

The Decomposition Method Available decomposition methods: Optimal. The small-side angle-bisector approximation scheme. Uses the same infrastructure for multi-way polygon union, used by the decomposition method. The same algorithm also works for the case of closed convolution cycles. The Convolution Method.

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5. The package can treat them as follows: discard them output the regularized sum , or give access to them through the underlying arrangement. Low-Dimensional Features. Rational approximation of vertical segments Rational approximation of vertical segments Rational approximation of vertical segments Rational approximation of vertical segments Rational approximation of vertical segments

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