Suppose that 212 f(x) (A) Find all critical values of f, compute their average, and enter it below. Note: If there are no critical values, enter -1000 Average of critical values = (B) Use interval notation to indicate where f(x) is increasing. Note: Enter T' for, '-I' for -, and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x' Increasing: (C) Use interval notation to indicate where f(x) is decreasing Decreasing: (D) Find the x-coordinates of all local maxima of f, compute their average, and enter it below. Note: If there are no local maxima, enter -1000 Average of x values = (E) Find the x-coordinates of all local minima of f, compute their average, and enter it below Note: If there are no local minima, enter -1000. Average of x values = (F) Use interval notation to indicate where f(x) is concave up. Concave up: (G) Use interval notation to indicate where f(x) is concave down. Concave down (H) Find all inflection points off, compute their average, and enter it below Note: If there are no inflection points, enter -1000 Average of inflection points = (I) Find all horizontal asymptotes of f, compute the average of the y values, and enter it below Note: If there are no horizontal asymptotes, enter -1000 Average of horizontal asymptotes = | (J) Find all vertical asymptotes off, compute the average of the x values, and enter it below Note: If there are no vertical asymptotes, enter -1000 Average of vertical asymptotes = | (K) Use all of the preceding information to sketch a graph off. When you're finished, enter a "1" in the box below. Graph Complete:
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.
Section G,H,I
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