Interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points. The Vandermonde method is the most straight-forward method which may be used to find an interpolating polynomial. The substitution of the points into the desired polynomial yields a system of linear equations in the coefficients of the polynomial. Suppose we have 3 points and we want to fit a quadratic interpolating polynomial through these points, We can find an interpolating polynomial that passes through 3 points (21, Yı), (72, 42), (x3, Y3) by solving the following matrix system. To + r121 + 3r2 = y1 To +ri12 + 3r2 = y2 To +ri23 + 3r2 = y3 a) The coefficient matrix for this system is a Vandermonde matrix. Compute the determinant of the Vandermonde matrix. b) Suppose entries of x are (0, 1, 2), and entries of y are the last 3 digits of your student number (If last 3 digits are y1y2y3, the first entry of vector y is y1, the second entry is y2, the third entr is y3). Solve the matrix system with Cramer's Rule and obtain the interpolating polynomial.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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