Consider the following figure. The x y-coordinate plane is given. A point, a vertical dashed line, and a function are on the graph. The point occurs at the point (3, 2). The vertical dashed line crosses the x-axis at x = 5. The curve enters the window in the second quadrant goes up and right, sharply turns at the approximate point (−5, 3.5), goes down and right, passes through the open point (−4, 2), crosses the x-axis at x = −3, ends at the closed point (−2, 1), restarts at the open point (−2, 0), goes down and right, exits the window almost vertically just left of the y-axis, reenters the window almost vertically just right of the y-axis, goes down and right, passes through the open point (1, 2), changes direction at the approximate point (2.2, 0.1), goes up and right, ends at the open point (3, 1), restarts at the open point (3, −1), goes down and right, exits the window almost vertically just left of the vertical dashed line at x = 5, reenters the window just right of the vertical dashed line at x = 5, goes up and right, crosses the x-axis at x = 7, and exits the window in the first quadrant. For each given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. If the function is discontinuous, state which of the three conditions that define continuity is not satisfied. (Select all that apply.) (a) x = −5 f is continuous. lim x→−5 f(x) does not exist. f(−5) does not exist lim x→−5 f(x) ≠ f(−5). (b) x = 0 f is continuous. lim x→0 f(x) does not exist. f(0) does not exist lim x→0 f(x) ≠ f(0). (c) x = 1 f is continuous. lim x→1 f(x) does not exist. f(1) does not exist lim x→1 f(x) ≠ f(1). (d) x = 3 f is continuous. lim x→3 f(x) does not exist. f(3) does not exist lim x→3 f(x) ≠ f(3).
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.
Consider the following figure.
- The point occurs at the point (3, 2).
- The vertical dashed line crosses the x-axis at x = 5.
- The curve enters the window in the second quadrant goes up and right, sharply turns at the approximate point (−5, 3.5), goes down and right, passes through the open point (−4, 2), crosses the x-axis at x = −3, ends at the closed point (−2, 1), restarts at the open point (−2, 0), goes down and right, exits the window almost vertically just left of the y-axis, reenters the window almost vertically just right of the y-axis, goes down and right, passes through the open point (1, 2), changes direction at the approximate point (2.2, 0.1), goes up and right, ends at the open point (3, 1), restarts at the open point (3, −1), goes down and right, exits the window almost vertically just left of the vertical dashed line at x = 5, reenters the window just right of the vertical dashed line at x = 5, goes up and right, crosses the x-axis at x = 7, and exits the window in the first quadrant.
For each given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. If the function is discontinuous, state which of the three conditions that define continuity is not satisfied. (Select all that apply.)
(b)
(c)
(d)
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