MTH403 Assignment No.1 Solution



   MTH403 Assignment Solution

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1. Introduction
Graphs are beneficial because they summarize and display information in a manner that is easy for most people to comprehend. Graphs are used in many academic disciplines, including math, hard sciences and social sciences. They make appearances in corporate settings, serving as useful tools to convey financial information and facilitate data analysis.
Different graphs are used depending on the information that individuals wish to convey. Many graphs are used to concisely and clearly summarize data; the best type of graph to use depends on the type of data being conveyed (such as nominal, scale-discrete, scalecontinuous and ordinal). Data summary graphs are generally nominal or contain data that can be reduced in some way; pie charts and bar charts are common and popular examples.
In the coming lectures, we discuss the graphs in a rectangular coordinate system. Equations can be graphed on a set of coordinate axes. The location of every point on a graph can be determined by two coordinates, written as an ordered pair, (x,y). These are also known as Cartesian coordinates, after the French mathematician Rene Descartes, who is credited with their invention.
Although the use of rectangular coordinates in such geometric applications as surveying and planning has been practiced since ancient times, it was not until the seventeenth century that geometry and algebra were joined to form the branch of mathematics called analytic geometry. French mathematician and philosopher Rene Descartes (1596-1650) devised a simple plan whereby two number lines were intersected at right angles with the position of a point in a plane determined by its distance from each of the lines. This system is called the rectangular coordinate system (or Cartesian coordinate system).
2. Outlines
    The Coordinate Plane.
    Asymptotes.
    .
2.1. The Coordinate Plane. The coordinate plane is a plane determined by two perpendicular lines, the x-axis and the y-axis. The x-axis is the horizontal axis, and the y-axis is the vertical axis. Every point in the plane can be stated by a pair of coordinates that express the location of the point in terms of the two axes. The intersection of the x- and y-axes is designated as the origin, and its point is (0,0).
As you can see from the figure, each of the points on the coordinate plane is expressed by a pair of coordinates: (x, y). The first coordinate in a coordinate pair is called the xcoordinate. The x-coordinate is the point’s location along the x-axis and can be determined by the point’s distance from the y-axis (where x = 0). If the point is to the right of the y-axis, its x-coordinate is positive, and if the point is to the left of the y-axis, its x-coordinate is negative. The second coordinate in a coordinate pair is the y-coordinate. The y-coordinate of a point is its location along the y-axis and can be calculated as the distance from that point to the x-axis. If the point is above the x-axis, its y-coordinate is positive, and if the point is below the x-axis, its y-coordinate is negative.
Definition 2.1. An asymptote is a line to which the graph gets arbitrarily close. That means that for any distance named, no matter how small, the graph will get within that distance and stay within that distance for some section of the graph with infinite length.
More precisely

Definition 2.2. The graph of y = f(x) has a horizontal asymptote of y = b, if and only if, either
(2.1)    lim f(x) = b or lim f(x) = b.
x→∞              x→−∞
As x →±∞, the curve approaches some constant value b.
Definition 2.3. The graph of y = f(x) has a vertical asymptote of x = c, if and only if, either
(2.2)    .
As x →±c, then the curve goes towards ±∞.

Definition 2.4. Let r(x) be a rational function with polynomial p(x) = anxn + ... + a0 of degree n in the numerator and polynomial q(x) = bmxm + ... + b0 of degree m in the denominator
    If n < m, then r(x) has a horizontal asymptote of y = 0.
    If n > m, then r(x) becomes unbounded for large values of x (positive or negative).
    If n = m, then r(x) has a horizontal asymptote of .
Example 2.5. Discuss the Asymptotes of the following rational function
(2.3)                                                                                                                   ,
this function is defined ∀x 6= 0. Consider the sequence of numbers  for k = 1,2,··· ,n. These numbers are getting closer and closer to zero. Since 
. So, r(xn) = 2,3,4, ··· ,k,··· for n = 2,3,3,··· ,k,···, which is getting larger and larger, so approaching the vertical line x = o. Thus there is a vertical asymptote at x = 0.

Example 2.6. Discuss the Asymptotes of the following rational function
(2.4)                                                                                                                   ,
defined ∀x 6= −2. The numerator and denominator are linear functions (degree of polynomials are the same). Alternatively we can see that as x get large then 2 in the denominator becomes insignificant. Thus= 10. Thus horizontal asymptote occurs at x = 10.
Example 2.7. Discuss the Asymptotes of the following rational function
(2.5)                                                                                                                    ,
defined ∀x 6= ±2. Clearly the function passes through the origin, so the x and y-intercept is (x,y) = (0,0). Note that this function is an even function, the edge of the domain is x = 2 , so we see there are vertical asymptotes at x = 2.
The numerator and denominator are quadratic functions (degree of polynomials are the same). Thus, for x large4. Thus, a horizontal asymptote occurs at y = −4.
Definition 2.8. We have seen that a rational function will have a horizontal asymptote if the degree of the numerator p(x) is less than or equal to the degree of the denominator q(x): In particular, if the degree of p(x) is strictly less than that of q(x); then the x-axis will be the horizontal asymptote-a geometrical condition that can be expressed analytically by saying r(x) → 0 as x →∞ and as x →−∞.


If the degree of p(x) is greater than or equal to the degree of q(x); then long division can be used to obtain more accurate information about the large scale behavior of the rational function. Recall that p(x) divided by q(x) gives a quotient f(x) and a remainder g(x) ; provided that p(x) = q(x) × f(x) + g(x) and provided that the degree of G(X) is strictly less than the degree of g(x)
In terms of rational functions, we have
(2.6)    
Because of the degree condition on g(x); it is clear that  so that g(x) and q(x) are close to each other when |x| is large. Thus the graph of the rational function r(x) is asymptotic to the graph of the polynomial f(x) as x →±∞.In other words, the two graphs are close to each other as x →±∞and as x →−∞: In the special case where the degree of p(x) is one more than the degree of q(x); the quotient is a linear function, whose graph is a non-horizontal line in the plane. That line is called an oblique or slant asymptote to the graph of the particular rational function.

Example 2.9. Discuss the oblique asymptote of the following rational function












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