How to Learn Critical Point Problems

In the previous post we have discussed about  How to Define Inflection Point and In today's session we are going to discuss about Critical Point Problems, A critical point is a most important concept of calculus in mathematics. According to mathematical definition, a critical point can be define as a variable for any function with real variable where function has their derivative equal to zero or  function is non-differentiable. If any real value of a function exists in a critical point situation then that value consider as a critical value of function. The concept of critical point can be represented by using the graph of function f(). Here the graph should not be admit as tangent or graph’s tangent should be a line in horizontal or vertical form.

The concept of critical point can be defined as an expression if any function f (s) is in the critical point f' (s) = 0 and the value of the co domain of ‘f’ for which the derivative of the function is zero are known as the critical value that means value of ‘x’ on which differentiation of the function is not possible is known as the critical value.
We can define it by a simple example as if there is a function f (s) = s² then function gives the zero when s = 0 that means when s = 0 then differentiation of the function is not possible, so s = 0 define as the critical point of the function f (s).
It is also define as the points on the graph that is defined on the basis of the given function for which the value of the derivative is equal to zero or the derivation of the function is not possible. It is define by a simple graph plotted on the basis of defined function in which critical point is defined as when tangent is drawn on the critical point either it does not exist on the curve or the tangent line does not cross it.
Differentiation Strategy can be define as set of rules and concept to perform the differentiation on the function. indian science engineering eligibility test books is a syllabus that contains the detail description with topics of general ability subject which helps in the student to make their preparation in a better way.

How to Define Inflection Point

In the previous post we have discussed about Implicit Differentiation and In today's session we are going to discuss about Inflection point, It is a term which is generally used under the theory of the differential calculus in the subject of the math. We all must have heard about the point of the inflection but many of us do not know its exact meaning. So in this article we will discuss some of the facts about the inflection point.
We will now begin with the definition of the inflection point. The point of the inflection is generally defined as a point existing on the curve where the sign of the concavity or the sign of the curvature changes from the positive to the negative or from the negative to the positive. By the above definition of the point of the inflection we mean to say that at the point of the inflection the curve either changes from the concave upwards which is known as the positive curvature to the concave downwards which is known as the negative curvature or can change from the concave downwards that is the negative curvature to the concave upwards that is the positive curvature.
We have now discussed about the definition of the point of the inflection in the last paragraph so let us now take an easy, simple and a practical example in this paragraph to understand about the point of the inflection more precisely. Suppose a person is driving a car along a road which is of winding type then the point of the inflection there is that point where the wheel of the steering straight for a moment when the car is just turned from the right to the left or from the left to the right.
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Implicit Differentiation

In the previous post we have discussed about inverse trig functions and In today's session we are going to discuss about Implicit Differentiation. Hello friends today we will be discussing a new topic Implicit Differentiation Calculator, firstly we will talk about the normal differentiation then we will move to  implicit differentiation, if we talk about normal differentiation then in normal differentiation we differentiate with respect to one variable  as if have to differentiate  y = 7x+y  with respect to x then  the differentiation will be dy /dx = 7 ,  as we are differentiating with respect to x so all the variable which are not having x with them the differentiation  of them will be 0.  But in implicit differentiation we will differentiate with respect to both the function x and y.  As in the above case the differentiation of y was zero but if we do implicit differentiation then the differentiation of y will not be zero it will be dy/dx.
Step 1: in this step we have to separate the terms, means the term containing x one side and term containing y one side.  (know more about Implicit and explicit functions , here)
Step 2: the term which don’t have x and y will be treated as constant and we have to differentiate the other terms..
Step 3: now find the value of dy/dx and that will be the solution for the given problem.
Now we will see one example in which we will do the implicit differentiation of a function.
Example: find the implicit differentiation of
Y = 3x + 4y2
Solution
Step 1: we have to separate the terms as,
Y – 4y2 =3x
Step 2 :  dy /dx – 8y* dy/dx = 3
Take dy/dx as common
Dy/dx (1 – 8y) =3
Step 3:
Dy/dx = 3/(1 -8y)
This is the required solution for the given problem.
In this way we can find the implicit differentiation of any function.
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inverse trig functions

In the previous post we have discussed about Trigonometric Equations and In today's session we are going to discuss about inverse trig functions.  In trigonometry we also use trigonometrically function, so here we are going to discuss the inverse trig functions. They are sometimes called Cyclometric function. Limited opposite functions for trigonometric functions are the inverse trig function.
Let’s see how the inverse trigonometric functions are represented:
Function Inverse
Sin arcsin
Cos arccos
Tan arctan
Sec arcsec
Cosec arccosec
Cot arccot
The notation sin ^-1, cos ^-1, etc are generally used for the arcsin, arccos etc. but sometimes they are not correct (know more about inverse trig functions, here)
The relationship between the inverse trigonometric function for the various arguments:
Complimentary angles:
Arccos y = ⊼ / 2 – arcsin y;
Arccot y = ⊼ / 2 – arctan y;
Arccsc y = ⊼ / 2 – arcsec y;
Negative arguments:
Arcsin (-y) = - arcsin y;
Arccos (-y) = ⊼ - arccos y;
Arctan (-y) = - arctan y;
Arccot (-y) = ⊼ - arccot y;
Arcsec (-y) = ⊼ - arcsec y;
Arccsc (-y) = - arccsc y;
Reciprocal arguments
Arccos (1 / y) = arcsec y;
Arcsin (1 / y) = arccsc y;
Arctan (1 / y) = ½ ⊼- arctan y = arccot y, if the value of ‘y’ is greater than zero;
Arctan (1 / y) = - ½ ⊼ - arctan y = - ⊼+ arccot y, if the value of ‘y’ is less than zero;
Arccot (1 / y) = ½ ⊼- arccot y = arctan y, if the value of ‘y’ is greater than zero;
Arccot (1 / y) = - 3 / 2 ⊼ - arccot y = ⊼+ arctan y, if the value of ‘y’ is less than zero;
Arcsec (1 / y) = arccos y;
Arccosec (1 / y) = arcsin y;

Arccos y = arcsin √ (1 – y2), this condition is satisfy if o ≤ y ≤ 1;
Arctan y = arcsiny
√ (y2+ 1)
Arcsin y = 2arcsin y
1 + √ (y2+ 1),

Arccos y = arcsin √ (1 + y2,
1 + y
This function will be satisfied if -1 ≤ y ≤ +1.
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Trigonometric Equations

In the previous post we have discussed about How to Solve Median Worksheets and In today's session we are going to discuss about Trigonometric Equations. Hello students, in this section we are going to discuss the Trigonometric equations. An equation that denotes trigonometric functions is a trigonometric equation.
For example: Cos q = ½.
When we solve trigonometric equations using different relations then the equation is converted to form through which value of other variable can be obtained. The roots of trigonometric function are obtained by the inverse trigonometric functions.
Suppose we have sin q + sin 2q + sin 3q = 0;
This trigonometric equation can be reduced to form 2 sin 2q cos q + sin 2q = 0;
Or we can also write as:
Sin 2q (2 cos q + 1) = 0;
When we solve this equation we get the value of sin 2q as 0;
Sin 2q (2 cos q + 1) = 0;
Sin 2q = 0;
And the value of cos is:
2 cos q + 1 = 0;
2 cos q = -1;
Cos q = -1/2;
So the value of cos q is -1/2.This equation result gives the result of the trigonometric equations.
So q = ½ arcsin 0 = n ⊼/2;
q = arcos (-1/2);
q = 2/3 ⊼(3n +½);
Here, the value of ‘n’ may be positive or negative integer. (know more about Trigonometric Equations, here).

Now we will see how to solve the trigonometric equations.
Assumes we have the trigonometric equation 4 tan³ q – tan q = 0, which lies in the interval [0, 2⊼]. Then we can solve this as shown below:
So the given trigonometric equation is: 4 tan³ q – tan q = 0;
We can write the equation as:
=> 4 tan³ q – tan q = 0;
=> Tan q (4 tan² q – 1) = 0;
So the value of tan ‘q’ is 0;
Or tan q = +1/√3. For every value of q ∈[0, 2⊼],
Tan q = 0;
It means the value of ‘q’ is 0, ⊼or 2⊼.
While
Tan q = 1/√3,
q = ⊼/ 6 or 7⊼/ 6,
Tan q = -1/√3,
q = 5⊼/ 6 or 11⊼/ 6.
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How to Solve Median Worksheets

In the previous session we discuss about Rate of Change Formula and now today we will discuss about How to Solve Median Worksheets. We use median to solve the mid value of the given data.  We say that the medial is the middle item of the series of the given raw data. In order to find the median of the given data, we need to first arrange the data in the ascending or descending order. In order to find the median of the given values, the values of all the items of the series are not taken into consideration as it happens in the case of mean. Let us take an example. If we have the series of data 23, 34, 36, 54, 76. Here we have 5 data items. So we say that the data  has 5 observation and thus the middle term is 3. Thus we say that the third data 36 is the median. Let us take another series of data 12, 20, 36, 89, 98. Here again we have 5 data items. Thus we say that the middle term will be 3^rd term. So we say that the median will be  36.  Though it is not the same situation as in the case of mean. In both the series we observe that the mean will be different.  To learn about median we take 2 steps A) to find the location of the middle term and B) to find the value of middle term.
We have median worksheets online, which is useful to understand the real data handling and solving the problems related to solve the median of the given data. Rate of Change can be learned online by using math tutor. It helps us to focus on the main content and to relate it with the real world problem. We can take the help of CBSE previous years question papers in order to learn about the different patterns of the question paper in the past years and thus helps to get the guidance  to prepare for the exams.

Rate of Change Formula

In today's session we are going to discuss about Rate of Change Formula. Hello students, in this session we are going to discuss the rate of change formula. But before discuss the rate of change formula we should know first the means of rate of change. Rate of change means a ratio between changes in a one variable that is comparative changes in another variable. The rate of change is expressed by the slope of line.

Let’s take an latest example the rate of change of the petrol price is now so high because in 2009 the petrol price was 40 rupees per liter approx and now(2012) the petrol price is 80 rupees per liter approx.
So rate of change definition also states that a speed on which a particular variable alters over a definite interval of time. The formula of rate of change is similar to the slope of the equation. So it will be: -
Slope y₂ - y₁ / x_{2 –} x₁
Rate of change x with correspond to y is dx / dy
Rate of change y is dx / dy
Let’s take an example to clear it more.
Example 1: Determine the rate of change of the following points
(5, 6) and (2, -3)
Solution: -
We have points x₁ = 5, x₂ = 2 and y₁ = 6, y₂ = -3
Formula:-
Rate of change = y₂ - y₁ / x_{2 –} x₁
= -3 - 6 / 2 – 5
= -9 / -3
=3
So the rate of change is 3.
We can also find the rate of change for algebraic equations also.
Decimal to fraction calculator coverts the decimal value into the fraction for example, 15.5 is the decimal value and 155/10 and 31/2 is the fraction value.
CBSE class 11 previous year question papers, if student will read these sample papers, they will increase their confidence level and in next session we will discuss about How to Solve Median Worksheets.

Triangle

In the previous post we have discussed about  and In today's session we are going to discuss about Triangle, A triangle is a closed figure bounded by 3 line segments or edges, which is also called as polygon. Before understanding the concept of properties of triangles understand about its types. Triangles are categorized on the basis of the side length and the angles of the triangle.
Types of triangles with relative to the length of the sides are:
1. Scalene triangle: A triangle having all sides of unequal length and whose angles are also different is called as scalene triangle.
2. Isosceles triangle: In this type of triangle two sides are of equal length and two angles are also same.
3. Equilateral Angle: Where all the three sides of the triangle are of same length and all angles are equal with angle of 60 degree is defined as equilateral angle.
Types of triangle in relation to internal angles:
1. Right angle triangle: It is that type of triangle in which one interior angle is of 90 degrees. The side of the triangle which is opposite to the 90 degree angle is the longest side and is called as hypotenuse and the other two are called as legs. Pythagorean Theorem is applicable on right angle triangle.
2. Oblique triangles are those triangles which do not measure any angle equal to 90 degree.
3. Acute triangle are those triangles whose angles are lesser than 90 degree.
4. Obtuse triangles are those whose angle measure greater than 90 degree.
Area of a Triangle is given as: Area=1/2 * base * height
Commonly we consider the bottom side as a base and height is a line which is perpendicular on the base it means the side which makes a 90 degree angle with the base of the triangle.
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