Difference quotient of a function: Tangent lines and their slopes

This section deal with the problem of finding a straight line L that is tangent to a curve C at a point P. Before we go to learning that, let us make a few assumptions to start with (so as to avoid mathematical errors before we come to the end result). C is the graph of a function y = f(x) and this function is continuous over the interval (a,b). The co-ordinates of the point P are (x0,y0) and the point P lies on the curve C so that f(x0) = y0. Also we assume that P is not the end point of C. That means x0 is not equal to a or b. Therefore C extends to some distance on either sides of P.

A reasonable definition of tangency can be stated in terms of limits. If Q is a point on C different from P, then the line through PQ is called a secant line to the curve. This line rotates around P as Q moves along the curve.  If L is a line through P, whose slope is the limit of the slopes of these secant lines PQ as Q approaches P along C (see picture below), then we say that L is tangent to C at P.

Since C is the graph of the function y = (x), then vertical lines can meet C only once. Since P = (x0,f(x0)), a different point Q on the graph must have a different x-co ordinate, say x0+h, where h ? 0. Thus the co-ordinates of Q would be (x0+h, f(x0+h)) and the slope of the line PQ would be:
= (f(x0+h) – f(x0))/h, based on the formula for slope of line joining two points with co-ordinates (x1,y1) and (x2,y2). Slope = m = (y2-y1)/(x2-x1)

So for our line PQ, that would be:
m = (f(x0+h) – f(x0))/(x+h – x) = (f(x0+h) – f(x0))/h

The above expression is called the difference quotient formula or simply the difference quotient of a function f. Note that h can be positive or negative based on whether Q is to the right or to the left of P.

This method of finding the difference quotient is further applied in finding the derivative of functions in calculus. The basic limit definition of derivatives stems from this difference quotient.

Given the function y = f(x) and a point x = x0, or given a table of values of f(x) for various values of x, to evaluate the difference quotient and subsequently to simplify the difference quotient, is simple.

Derivative of Cos II

This function is known as differentiation of trigonometric function with use of calculus and various trigonometry rules.

Derivative of Cos Squared X
Derivative of any trigonometric function that is we have to differentiate the function one time. Here we have to find the derivative of Cos squared X. for this first we know the basic rules of differentiation. And also rules of trigonometry.
Derivative of Cos Squared X means first we write the function in (cosX)^2 , now we have to use chain rule . Take external term derivative from the function and then differentiate the internal term we get (2cosxsinx) with negative sign because one time differentiation of cos function is negative sine function.
Now by using trigonometry rule change the result means -2cosxsinx is replaced with -2sin2x. so finally we get derivative of cos squared x is( -2sin2x).

Derivative of Cos -1
Derivative of Cos-1 means we have to differentiate only cos function and cos-1 is a cos function only. So by differentiating that is derivative of cos-1 is sin-1 with negative sign. We have to differentiate only one time.

Derivative of Inverse Cos
For finding the derivative of inverse Cos we use inverse trigonometric rule and calculus rules. Inverse trigonometric function also called sometime as cyclometric function. They are inverse of trigonometric function with proper domains. These are also used as arcos, arcsin and etc.
Inverse laws are very restricted and we can’t go out of the domains. These are proper subsets of domains. Like if function y=square root of x then we write this function as y^2=x also. Similarly function y=arccos then we write this function as cosy=x also. And then differentiate the function.
Derivative of inverse Cos by differentiating we get 1/square root of (1-x^2).
                                              d/dx (arcos) =1/square root of (1-x^2)

Derivative of Cos X Squared
Derivative of Cos X Squared, we write this function also as y=(Cosx)^2, now we differentiate one time this function with respect to x.  Differentiate the external term we get 2 cosx now differentiate the internal term cosx with respect to x then we get sinx with negative sign. Thus total term by differentiating we get 2sinxcosx with negative sign.
We have to write the result in more accurate form. For this we have to use trigonometric rules. By using these rules we get 2sinxcosx in another form is sin2x.so we use this form.
Finally derivative of Cos X Squared we get 2sinxcosx or sin2x with negative sign.
                                     d/dx(cosx)^2=(-sin2x)

Intoduction to Partial derivative

Generally we come across with functions of two or more variables. For example, the area of a rectangle, with sides of length x and y is given by A = xy, which obviously depends on the values of x and y, and so it is a function of two variables. Similarly the volume V of a rectangular parallelepiped having sides x, y and z is given by V = xyz and so it is a function of three variables x, y and z. Generally functions of two, three, …, n variables are denoted by f (x, y), f ( x, y, z),…., f(x1, x2,…,xn) respectively. If u = f(x, y) is a function of two variables x and y, then x and y are called independent variables and u is called the dependent variable.

Partial derivatives: Let f(x, y) be a function of two variables x and y. The partial derivatives of f(x, y) with respect to x is defined as Lim h -> 0 f(x + h, y) – f(x, y) / h. Provided that the limit exists and is denoted by del f/del x. Thus, the partial derivative of f(x, y) with respect to x is its ordinary derivative w.r.t. x when y is treated as a constant. Similarly, the partial differentiation of f(x, y) with respect to y is defined as Lim k -> 0  f(x, y + k) – f(x, y) / k .Provided that limit exists and is denoted by del f/del y.

Thus, the partial differentiation of f(x, y) with respect to y is its ordinary derivative w.r.t. y when x is treated as a constant. The process of finding partial differentiation of a function is known as partial differentiation.

Let us see some Partial derivative examples: If f(x, y) = x^3 + y^3 – 3 axy, then Del f/del x = 3 x^2 + 0 – 3 ay = 3 ( x^2 – ay) [because y is treated as a constant].And del f/del y = 0 + 3 y^2 – 3 ax = 3 (y^2 – ax) [because x is treated as a constant].

Partial derivative symbols: Let f(x, y) be a function of two variables such that its partial differentiation (del f/del x) and del f/del y both exists. Then del f/del x and del f/del y are functions of x and y, so we may further differentiate them partial with respect to x or y. The partial differentiation  of del f/del x with respect to x and y are denoted by del^2 f/del x^2 and del^2 f/(del y del x). Similarly, the partial differentiation of del f/del y with respect to x and y are denoted by del^2 f/(del x del y) and del^2 f/del y^2.

A short note on inverse trig functions

Introduction:
We know that not every function as an inverse. A function has an inverse if and only if it is one to one and onto. As all trigonometric functions are periodic, they are all many to one type of functions. So technically, inverse of trig functions do not exist. But if we can suitably restrict the domain of the trigonometric function , then it becomes one to one and onto. Therefore with this modified domain the trigonometric function can have an inverse. Let us look at the following examples:

Inverse trig functions examples:

1. Inverse of sine function:
The sine function defined as follows: sin = {(x,y) | y = sin(x), x belongs to R, y belongs to [-1,1]} is an onto function. It is a many to one function. It is a periodic function with at period of 2belongs tobelongs to. But instead of R, if the domain is restricted to say, [-pi/2,pi/2], or [pi/2,3*pi/2], or [3pi/2, 5pi/2] etc, then it becomes one to one and still remains onto. Thus now we can define the inverse of sin function using any one of the above domains as follows: sin^(-1) = [(y,x) | y = sin(x), x belongs to [-pi/2,pi/2], y belongs to [-1,1]}, where “sin^(-1)” is the symbol for inverse sine function.

2.  Inverse of cosine function:
Just like how we defined the inverse of sine function, we can define the inverse of cosine function by restricting its domain as well. The domain restrictions can be made to suite our purpose. Therefore, inverse cosine function can be defined as follows: cos^(-1) = [(y,x) | y = cos(x), x belongs to [0,pi], y belongs to [-1,1]}

The other trigonometric function inverses can also be defined similarly.

Interrelations between inverse trigonometric functions:
Sin^(-1) x = cosec^(-1)(1/x), or cosec^(-1)(x) = sin^(-1)(1/x). The inverse functions of cos and sec  and the inverse function of tan and cot are related the same way.

Integral of inverse trig functions:
To find integral of inverse trigonometric functions, we use the method of integration by parts. We know that we follow the order LIATE (Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential function) for integration by parts. When we integrate inverse trigonometric functions, the inverse function becomes the u of the integration by parts and since there is no other function it is multiplied to, we take v = 1. Thus for example, if we were to integrate the function like tan^(-1)x using the integration by parts rule, then here u = tan^(-1)x and v = 1 and then integrate using integration by parts.

Trigonometric Identities

A function is a mathematical statement relating the different variables. When we assign the values for a set of independent variables, we get the definite value for the dependent variable.
Identity is a mathematical expression, which is valid for all the values of the variables. When the identity has terms involving the algebraic expression we call it as algebraic identity and the function involving the trigonometric expressions are called trigonometric identities.
Verifying Trigonometric identities:
Identities are mathematical statements that are valid for all the values of the variables. For example
(a +b)^2 = a^2 +2ab +b^2

The above expression is valid for all the values of ‘a’ and ‘b’. Trigonometry is the field of study involving triangles’ sides and angles. If the identity has any of the trigonometric functions, then it is called trigonometric identity. As the trigonometric functions are related to the angles, we substitute the angles in the functions of identities. We adopt the following steps to verify an identity.
Step 1: Choose an angle from the defined domain.
Step 2: Replace the variable by the chosen value
Step 3: Evaluate the function and simplify
Example: Sin^2 (x) + cos 2(x) =1
Step 1: For the value of x = 90
Step 2: Sin^2 (90) + cos^2 (90)
Step 3: 1+ 0 =1. Hence verified

Fundamental Trigonometric Identities
Trigonometry, a field of study involving the sides and angles of a triangle has a set of fundamental definition of trigonometric functions. Using the theorem of Pythagoras, we have three fundamental trigonometric identities called as Pythagorean identities, which are as follows.

In a triangle ABC:
(1) 1 = sin^2 (A) +cos^2 (A)
(2) 1+ tan^2(A) = sec^2(A)
(3) 1+ cot^2(A) = cosec^2(A)

Simplifying Trigonometric Identities
To simplify a given trigonometric identity, we always rely on the algebraic methods. Especially, we adopt PEMDAS/ BOADMAS in simplifying the identities.  In addition to this, we use the above three identities in simplifying the given trigonometric identities. At the same time, we ensure that the defined trigonometric identity is a valid one in the defined domain of variables.

Table of Trigonometric Identities:
The fundamental trigonometric identities are as follows
Reciprocal Identities:
Sec(A) = 1/cos(A) Cosec (A) = 1/ Sin(A) tan(A) = sin(A)/cos(A),   cot(A) = 1/ Tan(A)= cos(A)/sin(A)

Pythagorean Identities
(1) 1 = sin^2 (A) +cos^2 (A) (2) 1+ tan^2(A) = sec^2(A) (3) 1+ cot^2(A) = cosec^2(A)

Even-Odd Identities
(1) Sin (-A) = -sin(A) (2) cos(-A) = cos (A) (3) tan(-A) = -tan (A)
(4) Cosec (-A)= -cosec(A) (5) sec(-A)= -sec A (6) cot(-A) = -cot(A)

Standard deviation is a Measure of Dispersion

When we are dealing with data sets experimenter is interested to know two things about data set. Those are Measures of Central Tendency and Measures of Dispersion. Measures of Central Tendency are Mean, Median and Mode. Measures of Dispersion are Inter Quartile Range, Range, Mean deviation, Standard deviation and Variance. Measures of Central Tendency give the measures for center of the data. Measures of Dispersion give the measure for the variation in the data.

That is, it explains how much the data spread. Range is one type of measure of dispersion and it is the difference between maximum and minimum value in the data. It depends only on the extreme values and it neglects remaining values this is the drawback for this method. Inter Quartile Range is the difference between first and third quartiles. It gives the percentage of observations in between first and third quartile. In this case also it depends only on the first and third quartile values and neglecting remaining values. To come over from this drawback we can use Mean Deviation which includes each and every observation.

Main drawback of Mean Deviation is dealing with mathematical operations like differentiations and integrations are quite difficult. To come over from this drawback we can use Standard Deviation and Variance. Variance is the average of squared differences of the observations from their mean. Standard Deviation is the square root of the Variance. When we do not know standard Deviation for the population (It is a collection of huge similar type of items) then we have to estimate it by sample standard deviation. Symbol for Standard Deviation is’ ’. Symbol for Sample Standard Deviation is‘s’ Sample Standard Deviation Formula is .

Where ‘n’ is the sample size. Standard Deviation of the constant variable is zero. Constant variable means a variable which takes a fixed and single value. For example, a variable which takes only value 5 is called as constant variable and standard deviation of such variable is zero. For clear understanding of standard deviation, Standard Deviation Examples are discussed below. When we are Adding Standard deviations we cannot add them simply. First we have to square each individual standard deviation to make them variances. Now add these variances and take square root of this to add standard deviations. For subtracting standard deviations also we have to do the same thing. Standard Deviation Problems are not solvable directly first we have to find out the variance and then by taking square root of it we can get standard deviation. Standard Deviation Example, for the data 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Variance is 9.1667 and standard deviation is 3.02765. Mean of this data is 4.5 here the standard deviation value explains the average distance of the observations from its mean 4.5.


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Mode definition

Mode (statistics) is the value of the variable corresponding to the maximum of the ideal curve which gives the closest possible fit to the actual distribution of the frequency. It represents the value which is the most frequent or typical, the value which is, in fact, the fashion. The mode is sometimes denoted by writing the sign ? over the variants symbol, for example X? denotes the mode of the values X1,X2, …. Xn.

Mode formula:

It is evident that, mode is to be determined by inspection only. There is no stereotyped method listed for determination of the mode for a data set. It purely depends on the intuitions of the statistician or researcher. However there is an empirical relation between the mean, median and mode.
Mode = Mean – 3*(Mean – Median).
The above relation holds good with surprising closeness for moderately asymmetrical distributions.  Putting that in words, we say that the median lies one third of the distance mean to mode from the mean towards the mode.

Usually mode represents a single humped distribution unless specifically stated otherwise. When the distribution is of a complicated form, there may be more than one mode. Such distributions are therefore sometimes called multimodal. The mean and the median are still unique for such distributions.

What is mode for grouped frequency distribution:

Based on how we define mode, it is in fact difficult to determine the mode for grouped frequency distributions that are more common in practice. At max we can find the class with the maximum frequency. But beyond that it’s no use giving merely the mid value of the class interval into which the greatest frequency falls, for this is entirely dependent on the choice of the scale of the class intervals. It is again no use making the class interval very small to avoid error on that account, for the class frequencies will them become small and the distribution irregular. What we actually want to arrive is at the mid value of an interval for which the frequency would be a maximum, if the intervals could be made indefinitely small and at the same time the number of observations be so increased that the class frequencies should run smoothly. As the observations cannot, in a practical case, be indefinitely increased, it is evident that some process of smoothing out the irregularities that occur in the actual distribution must be adopted, in order to ascertain the approximate value of the mode.

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