Derivatives of Cot functions for some special values

We can find the derivative of the cot function using the quotient rule of differentiation. The quotient rule can be states as follows: For a function f such that f(x) = u/v, where u and v are both function of x, the derivative of f(x) is given by the formula:
f’(x) = (v*u’ – u*v’)/v^2, where u’ is derivative of function u and v’ is derivative of function v.
We know that, cot x = cos x / sin x,
Therefore, (d/dx) cot x = (d/dx) (cos x / sin x),
= [sin x (d/dx) (cos x) – cos x (d/dx) (sin x)]/sin^2 (x)  … using the quotient rule stated above.
=[ sin x (-sin x) – cos x * cos x]/sin^2(x)
= [-sin^2(x) – cos^2(x)]/sin^2(x)
= -[sin^2(x) + cos^2(x)]/sin^2(x)
= -1/sin^2(x)
= -csc^2(x)
Therefore the derivative of cot x is –csc^2(x)

The above derivative can also be proved using the definition of derivative which is like this:
F’(x) = lim(t->x) [(f(t) – f(x)]/(t-x), for the function y = f(x) = cot (x) the definition would be like this:
(d/dx) cot x = lim(t->x) [cot t – cot x]/(t-x)
= lim(t->x) [1/tan t – 1/tan x]/(t-x)
  = lim(t->x) [tan x – tan t]/[(tan t * tan x)(t-x)]
= lim(t->x) {[tan x – tan t]/(t-x)} * 1/[tan t * tan x]
= lim(t->x) {-[tan t – tan x]/(t-x)} * lim(t->x) 1/[tan t * tan x]
= lim(t->x) {-[ tan(t-x) (1+tan t tan x)]/(t-x)} * 1/tan^2(x)
We know that lim(t->x) of tan(t-x)/(t-x) = 1. Therefore,
= -{1+tan^2(x)} *1/tan^2(x)
= - sec^2(x)/tan^2(x)
= [-1/cos^2(x)]/[sin^2(x)/cos^2(x)] … here the cos^2(x) cancel each other out.
= -1/sin^2(x)
= -csc^2(x)

Derivative of cot-x:
For finding the derivative of cot(-x) we use the chain rule.
Let –x = u, then du/dx = -1.
Cot (-x) = cot u
(d/du) cot u = - csc^2(u) …  from what we proved above.
Using the chain rule we have:
(d/dx) cot (-x) = (d/dx) cot u * du/dx = -csc^2(u) * (-1)
= -csc^2(-x) * (-1)
= csc^2(-x)
Thus we see that the derivative of cot(-x) is csc^2(-x)

Derivative of cot^-x (k):
Cot^(-x) (k) where k is some constant can be written like this also
= (cot(k))^(-x)
= 1/(cot k)^x
= (1/cot k)^x
Since k is a constant, cot k is also a constant and in turn, 1/cot k is also constant. So let 1/cot k = some constant say p. Therefore,
= p^x is the function.
(d/dx) p^x = p^x ln p
= (1/cot k)^x ln(1/cot k)
That is the derivative of the above function.

Triple integrals spherical co-ordinates

The concept of definite integration can be extended to two or three or multi dimensional system. For a bounded function f(x,y,z) defined on a rectangular box B (x0 = x = x1, y0 = y = y1, z0 = z = z1), the triple integral of f over B,
Integral (B) [f(x,y,z)] dV or  Integral  (B) [f(x,y,z)] dx dy dz,
can be defined as a suitable limit of Riemann sums corresponding to partitions of B into sub boxes by planes parallel to each of the co-ordinate planes. Triple integrals over more general domains are defined by extending the functions to be zero outside the domain and integrating over a rectangular box containing the domain.
All properties of double integrals hold good for triple integrals as well. In particular, a continuous function is integrable over a closed, bounded domain. If f(x,y,z) = 1 on the domain D, then the triple integral gives the volume of D.
Volume of D =  Integral (B)  dV
Triple integrals spherical coordinates: The spherical co-ordinates related to Cartesian co-ordinates x, y and z are given by the equations:

The triple integrals in spherical coordinates relationships are illustrated in the picture below

For the co-ordinate surfaces in the spherical co-ordinates, see picture below:

The volume in spherical co ordinates would be like this:

Spherical co ordinates are suited to problems involving  spherical symmetry and, in particular, to regions bounded by spheres centered at origin, circular cones with axes along the z axis, and vertical plains containing the z axis.

Infinite and removable discontinuity

Introduction To Discontinuity : Let us first define discontinuity  , a function f(x) is said to be continuous in an interval if it is continuous at every point in the domain of f(x). If the interval is closed, say [a, b], the function f(x) is continuous at every point of the interval including the end points a and b. If the interval is open, say ]a, b[, we exclude the end points of the interval. If the function is not continuous at a point, say c, we say that the function is discontinuous at c and c is called a point of discontinuity of f(x) .The graph of the function y = f(x)= 5 for x > 0 and 1 for x = 0, when drawn, we see that the function is discontinuous as there is a jump in the graph at x = 0. If we carefully observe that the function is continuous to the left of O and also to the right of O.

In other words, we say that left hand limit is existing and also the right hand limit is existing but they are not equal. ).Now let us more understand what is discontinuity? The function f is said to be discontinuous at a point a of its domain D if it is not continuous thereat. The point a is then called the point of discontinuity of the function. The discontinuity may arise due to any of the following situations: (i) lim x -> a+ f(x) or lim x -> a- f(x) of both may not exist. (ii) lim x -> a+ f(x) as well as lim x -> a- f(x) may exist, but are unequal. (iii) lim x -> a+ f(x) as well as lim x -> a- f(x) both may exist, but either of the two or both may not be equal to f(a).

We classify the points of discontinuity according to the various situations discussed above:
1. Removable discontinuity: Now let us understand What is a Removable Discontinuity ? A function f is said to have removable discontinuity at x = a if lim x -> a- f(x) = lim x -> a+ f(x) but their common value is not equal to f(a). Such a discontinuity can be removed by assigning a suitable value to the function f at x = a.

2. Discontinuity of the first kind: What is infinite discontinuity ? A function f is said to have a discontinuity of the first kind at x = a if lim x -> a- f(x) and lim x -> a+ f(x) both exist but are not equal. f is said to have a discontinuity of the first kind from the left at x = a if lim x -> a- f(x) exists but not equal to f(a). Discontinuity of the first kind from the right is similarly defined.

3. Discontinuity of the second kind: A function f is said to have a discontinuity of the second kind at x = a if neither lim x -> a- f(x) nor lim x-> a+ f(x) exists. F is said to have discontinuity of the second kind from the left at x = a if lim x -> a- f(x) does not exist. Similarly, if lim x -> a+ f(x) does not exist, then f is said to have discontinuity of the second kind from the right at x = a.

Know more about Free calculus help.
Let I be an open interval let c belongs to I, and let f be a function defined on I except possibly at the point c. The function f has a jump discontinuity at c if the one sided limits of f exists at c but are not equal. The function f is said to have a removable discontinuity at c if lim x -> c f(x) exists, but f(x) is neither not defined or has a value different from the limit. As a specific example, the function g defined by g(x) = x/|x| has a jump discontinuity at 0 since the left hand limit is -1 and the right hand limit is 1. A removable discontinuity can be removed by giving the function the value of the limit at the given point. The function h defined by h(x) = x sin (1/x) has a removable discontinuity at 0; assigning h(0) = 0 makes h a continuous function at 0.

Instantly calculate vertical asymptotes

An asymptote is a line that runs very close to a curve but the curve never meets the line. At infinity the curve coincides with the asymptote.  In other words, an asymptote of a curve is a line such that the distance between the curve and the line tends to zero. The asymptote gets very very close to the curve but never touches it.

Vertical asymptote:
If the above defined asymptote is vertical, that means, that if the asymptote is parallel to the y axis it is called the vertical asymptote. The general equation of the vertical asymptote is y = a, where a is any real number.

How to find vertical asymptotes:
The simplest way to find vertical asymptotes of functions is to graph them.  Consider for example that we need to find the vertical asymptote of the function: f(x) = 1/x. Let us graph this function and see what we get. To be able to graph the function we first need the table of values. So let us make a table of value of x and the corresponding values of f(x).

Now let us plot those values on a graph.

Vertical asymptotes can also be found algebraically. In case of rational functions, there is a denominator. If we equate the denominator to zero and then solve for x, the values of x that we thus get would be the vertical asymptote. Let us try to understand that with an example:

Find the vertical asymptotes of the function f(x) = (2x^2-4x+5)/(x^2-2x+1)
Solution: In such a function, we shall work only on the denominator. Here the denominator is x^2 - 2x + 1. Therefore we equate that to zero. So we have:
x^2 - 2x + 1 = 0 now we factor the left hand side
(x-1)(x-1) = 0 now use the zero product rule
(x-1) = 0, (x-1) = 0 now solve for x
X = 1. That is the equation of the vertical asymptote

Quadrilaterals: an introduction

What are quadrilaterals?
A quadrilateral is a plane figure bounded by four line segments such that:
(a) No two line segments cross each other, and
(b) No two line segments are collinear.
In simple words, a four sided closed two dimensional figure is called a quadrilateral.

Types of quadrilaterals:

The next question that comes to our mind is what is a quadrilateral shape? There are various types of quadrilaterals whose description is given below:
(i) A quadrilateral with both pairs of opposite sides parallel is called a parallelogram.
(ii) A quadrilateral with exactly one pair of parallel sides is called a trapezium or a trapezoid. An isosceles trapezium  is a trapezium in which two non parallel sides are equal.
(iii) A kite is a quadrilateral in which two pairs of adjacent sides are equal.
(iv) A parallelogram in which all angles are right angles is called a rectangle.
(v) A parallelogram in which all sides are equal is called a rhombus
(vi) A rectangle with all sides equal, or a rhombus in which all angles are right angles is called a square.

Properties of quadrilaterals:
1. All quadrilaterals have four sides. The word quadrilateral itself means ‘four sided’.
2. Since all quadrilaterals have four sides, it is obvious that it would have four angles as well.
3. All sides are straight.
4. The figure has to be two dimensional.
5. The sum of interior angles of a quadrilateral is always 360 degrees.

Area of a quadrilateral:

The area of quadrilateral formula depends on the type of quadrilateral we have. Let us look at the formulae for area of various types of quadrilaterals:

1. Parallelogram: Area = length of base * perpendicular distance between the two bases
2. Trapezoid: Area = (1/2) * sum of parallel sides * perpendicular distance between the parallel sides
3. Kite: Area = area of upper triangle + area of lower triangle, where area of triangle = (1/2) * base * height.
4. Rectangle: Area = length * width
5. Rhombus: Area = (1/2) * product of the diagonals
6. Square: Area = (side)^2.
If a quadrilateral is none of the above, then to find the area we split the quadrilateral into two triangles. Find the area of both the triangles and add them up, to give the area of the quadrilateral. Alternatively if we know the co-ordinates of the vertices of a quadrilateral also we can find the area using co-ordinate geometry.

Introduction of Statistics

The modern development in the field of not only management, commerce, finance, social sciences, mathematics and so on but also in our life like public services, defence, banking, insurance sector, tourism and hospitality, police and military etc. are dependent on statistics. Learning statistics plays a vital role in enriching a specific domain by collecting data in that field, analyzing the data by applying various statistical techniques and finally making statistical inferences. Government applies statistics to make the economic planning in an effective way. A business man applies statistics to make his economic plan. Researchers learn statistics and apply it to present their research papers in an authoritative manner. Thus we see that statistics is applied in all walks of life.

Statistics history:

The origin of the word ‘statistics ‘ remains a mystery even though it was used in the ancient and medieval period. In those days, statistics was analogous to state or the data that are collected and maintained for the welfare of the people belonging to the state. Ancient Indian rulers, kept records of births and deaths. In the 16th century, also the kings and monarchs kept records of agriculture. In Egypt, the first census was conducted by the Pharaoh  during 300 B.C. to 2000 B.C.

What is statistics?

We may define statistics either in a singular sense or in a plural sense Statistics, when used as a plural noun, may be defined as data qualitative as well as quantitative, that are collected usually with a view of having statistical analysis.

Statistics, when used as a singular noun, may be defined, as the scientific method that is employed for collecting, analyzing and presenting data, leading finally to drawing statistical inferences about some important characteristics it means it is ‘science of counting’ or ‘science of averages’.

Variables statistics:

A variable is statistics is the parameter or attribute of which data is gathered for purpose of study. For any given study there can be one or more variables. The variable essential describes a thing, place, person or an idea all collectively called entities. The value of variable of each entity is different.

Types of variables Statistics:

Statistical variables are mainly classified as qualitative or quantitative variables. Qualitative variables refer to some adjective of the entity where as quantitative variables would have a number as its value always.
Quantitative variable can be of two types:  discrete or continuous variable.  A discrete variable will always assume an integral value where as a continuous variable can assume any real number value.

Percentile

In this article we shall talk about percentiles or centiles that divide a given set of observations into 100 equal parts. The points of subdivisions being P1,P2, ….. P99. P1 is the value for which one hundredth of the observations are less than or equal to P1 and the remaining ninety nine hundredths observations are greater than or equal to P1 once the observations are arranged in an ascending order of magnitude.

Percentile Formula:

For unclassified data, the pth percentile is given by the (n+1)pth value, where n denotes the total number of observations. So p = 1/100,2/100,… 99/100 for first percentile, second percentile, third percentile Ninety ninth percentile, respectively. In general let’s call thatP_i. For calculating percentile of ungrouped data the following formula can be used:
n  = P_i. N/100 + 0.5, where n = the observation of the ith percentile, P_i = ith percentile (in percent) and N = total number of observations. Here n is also called the rank of the observation. So the answer for the ith percentile would be the nth numbered observation. Another formula used for calculating this rank is
n = (N+1)* P_i/100

Percentile chart:

Percentile charts have various applications in the field of national census, health care, competitive exams etc. These charts are made using a very large number of data and a merit list is prepared based on the percentile chart. For example, a pediatrician would use a height and weight percentile chart to check on the growth of a particular baby based on the national averages. Percentile charts also have their applicability in deciding the ranks in case of competitive exams. For example, GRE score is given not as absolute score out of 200 but as a percentile. That means the person with highest score is treated as 100 percentile and other scores are calculated based on that, so a candidate with 90 percentile would be 10th rank and so on.

Know more about the College algebra help, Online Math Homework. This article give basic information about percentile. Next article will cover more concept on algebra tutoring and its advantages and many more. Please share your comments.