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.
Can you do 4,5 including II. of each number?
![ZOOM
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We have seen many things can contribute to the shape of a graph. There are symmetries and
periodicity. There are limits which contribute to vertical and horizontal asymptotes. There are local
maximums and minimums, places where tangent lines are horizontal and vertical and points of
inflection. In order to find an accurate graph there are many things you must look for:
A. The domain of the function
B. The x- and y-intercepts
C. Symmetry and Periodicity
D. Horizontal and vertical asymptotes
E. Local maximums and minimums
F. Points of Inflection.
I. For each of the following functions complete each of the steps above. Be sure to show all of your
work. You may use exact values or round to two decimal places where appropriate.
3
1) f(x) = 3 --+
3
1
2) f(x)
x3 + 1
3) f(x) = cos³(x)
4) f(x) = 6x1/3 + 3x4/3
5) f(x) = x – sin x
II. Graph each function by hand on graph paper. Be sure to use a large enough graph so that all
information can be found clearly. Label all points coinciding with relative extrema and all points of
inflection on your graph. All points should be found correct to one decimal place. Include dotted lines
to indicate any asymptotes the graph might have. If a function is symmetric or periodic, this behavior
should be evident from your graph. At each point where the graph has either a horizontal or vertical
tangent line, please sketch a brief tangent line to indicate this.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d2af6b8-35d3-4327-94c1-a05ff77beaf0%2Ffc6ccee8-3373-4d8a-bd19-6808920c6e2a%2Ff3ph5z_processed.jpeg&w=3840&q=75)
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