Suppose that a population grows according to a logistic model with carrying capacity 6300 and k = 0.0012 per year. (a) Write the logistic differential equation for these data. dP/ dt = .0012P(1- 6300 P (b) Draw a direction field (either by hand or with a computer algebra system). Use the direction field to sketch the solution curves for initial populations of 1000, 2000, 4000, and 8000. P 8000 15000 6000- 10000 4000 5000 2000 500 1000 1500 2000 2500 3000 500 1000 1500 8000 14 000 12000 10000 8000 6000 4000 6000- 4000 2000 2000 500 1000 1500 500 1000 1500 2000 2500 What does the direction field tell you about the solution curves? All of the solution curves approach 6300 as t-. All of the solution curves approach 0 as t-. Some of the solution curves approach 0 as t- , and the others approach . All of the solution curves approach as t-. All of the solution curves approach 3150 as t ,
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.
which one of these graphs is correct from the 4 options and why
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