tangent line approximation

In the previous post we have discussed about Newtons Law of Cooling Differential equation and In today's session we are going to discuss about tangent line approximation. The tangent line approximation in mathematics may be defined as approximating the value of any function that is hard to calculate with the help of the value of another function that is nearest to it.

To have more precise understanding of the tangent line approximation, let us take a very simple example. Consider a square root function f (x) that is f (x) = √x.

Now if we take x equal to 9 then f (9) = √9 which comes out as 3 and is very easily calculated. However if we consider x = 8.9 then f (8.9) = √8.9 and this cannot be calculated easily. Thus for f (8.9) = √(9 – 0.1), the value can be approximated with the help of the value which we got from f (9) = √9 = 3 ( since 0.1 is very small ).

Thus we can say that if the value of f (b + h) is hard to find whereas f ( b ) can be very easily calculated where | h | is very small (here the mod is taken as we need positive value of h) then the value of f (b+h) can be approximated with the help of the value of f (b). (know more about tangent line approximation, here)

Since in the example given above the value of h is equal to -0.1 thus the value of the function f can be approximated at the point (b+h) with the help of the tangent line to the graph of function f at the point b when | h | is very small.

In order to get more help on topics: tangent line approximation, Transcription and Translation and cbse paper for class 9, you can visit our next article.

Newtons Law of Cooling Differential equation

In the previous post we have discussed about Derivative of secx and In today's session we are going to discuss about Newton's Law of Cooling Differential equation. Hello friends I hope you have well underst0ood all the previous topic , today we are going to discuss a new topic Newton s Law Of Cooling Differential Equation. As we all know that Newton’s law of motion are very popular to all of us, but very few people are aware of Newton’s law of cooling, according to this law , if we put any body in air then its temperature decreases or increases with its surroundings. As we all know that the cooling rate of any body depend on its surroundings, on this same fact Newton’s law was given. As this law is a very old law, so people are not having any knowledge of heat, so people have to accept this law without knowing that weather Newton done this law practically or not. In the same time when Newton law was introduced, one more law was there called as Stefan’s law, according to this law the energy radiated by a body in atmosphere is directly proportional to the fourth power of its temperature. There is a condition called as condition of thermal equilibrium in which Newton Law and Stefan’s law are same, there is one difference between Newton Law and Stefan Law is Newton Law always Deals with the internal temperature while Stefan law deal with temperature of body. (know more about Differential equation, here)

We can get the Newton’s Law Of Cooling Equation by some simple calculation but before we need to have knowledge of some terms like Heat Conductivity, area of martial and thickness of wall.

We will denote heat conductivity by K, thickness by t and area by A, so required equation for cooling law is,

dQ / dt = KA(θ – θo) / t.

Here θ is the temperature of outer surface and θo is the temperature of inner surface.

If you have 10th Cbse Sample Paper. Go through Transition Metal topic, it is a important topic in mathematics.

Derivative of secx

In the previous post we have discussed about cosine law and In today's session we are going to discuss about Derivative of secx. Trigonometric can be defined as the relationship between the angles and sides of a triangle. In the trigonometric there are different types of function and derivative of functions. Here we will study the different types of derivatives. Derivatives of trigonometric  function are mention below:

d / da sin (a) = cos (a)	d / da cos (a) = - sin (a)
d / da tan (a) = sec2 (a)	d / da csc (a) = - csc (a) cot (a)
d / da sec (a) = sec (a) tan (a)	d / da cot (a) = - csc2 (a)

(know more about Derivative, here). These above mention are the different types of derivatives of trigonometric functions. Now we will find the Derivative of secx. We are earlier studies the derivative of sec x is sec (x) tan (x). Now see the prove of Derivative Of Secx. To find the derivative of sec x first we write the sec x in the derivative form:

Prove = d / dx sec = sec x tan x;

We know that sec x = 1 / cos x;

We can also write in place of sec x as:

= d / dx [1 / cos x]

We can solve it by u / v methods:

u / v = u d / dx (v) – v d / dx u / u2

So we can write the above expression as:

= [cos x d / dx (1) – 1 d / dx cos x] / cos2 x;

If we find the derivative of 1 and cos x we get:

= [cos x (0) – 1 (- sin x)] / cos2x;

If we solve we get:

= [0 + sin x] / cos2 x;

On further solving we get:

= Sec x tan x.

So the proof of sec x is sec x tan x.

This is how we can solve the derivative value. Centripetal Acceleration Formula is used to find the speed along to a given circular path and its radius is directed along to the center. icse sample papers 2013 is very helpful for exam point of view.

cosine law

Hello friends, In the previous post we have discussed about Derivative of cot and today we are going to discuss a very important topic cosine law, as we are now familiar with trigonometric parameters. All the parameters have their specific role in trigonometry and we use all the parameters according to our need. As we well know that the six parameters are sin, cos, tan, sec, cot and cosec. As we well know that we can apply all these law to a triangle and we also know that any triangle consists of three angle and three sides, now if we talk about cosine law then  if we are having the value of  two sides of a triangle and we are also having the value of the angle opposite to the side of triangle which we want to measure, suppose if a triangle consists of three sides as a ,b and c and if the value of  b and c are given and we are asked to find side a by cosine law then we must have the value of angle A.  We can write cosine law as,

a2 = b2 + c2 – 2bccosA

Here a, b and c are sides of the triangle and A is the angle between them.

Now we will see an example, in which we will apply cosine law

Example:  A triangle is having sides as b= 2 cm and c = 3cm and angle A = 60 degree find the side a by cosine law?

Solution: as we know by cosine law,

a2 = b2 + c2 – 2bccosA

Now we will put our values in this,

a2 = 4 + 6 – 2 *2*3 *1/2

a2 = 10 -6

a2 = 4

a = 2

This is the required solution of the given problem by cosine law.

 If you are having cbse class 10 sample papers then go through First Derivative Test, it is an important topic in mathematics.

Derivative of cot

A branch of mathematics, which represents the relationship among the angles and sides of a triangle, is said to be trigonometry. Different types of function are explained in the trigonometry; here we will discuss the different types of derivatives. Derivatives of trigonometric function are as follows:

⇨ d / da sin (a) = cos (a);

⇨ d / da cos (a) = - sin (a);

⇨ d / da tan (a) = sec2 (a);

⇨ d / ds csc (a) = - csc (a) cot (a);

⇨ d / da sec (a) = sec (a) tan (a);

⇨ d / da cot (a) = - csc2 (a);

These are all different types of derivatives of trigonometric functions. Here we will see the Derivative of cot a. Let’s discuss the prove of derivative of Cot a. First we write the cot a in the derivative form:

Proof = d / da cot a = - csc2 a; we can also write the cot a as:

⇨ Cot a = 1 / tan a;

We can also write in place of cot a as:

= d / da [1 / tan a]

We can solve it by u / v methods:

u / v = [u d / dx (v) – v d] / dx u / u2;                         

Put the expression in this method so that we can easily find the solution.

Now, we can write the above expression as:

= [tan a d / da (1) – 1 d / da tan a] / tan2 a;

 If we find the derivative of 1 and tan a, we get:

= [tan a (0) – 1 (sec2 a)] / tan2 a;

If we solve we get:

= [0 – sec2 a] / tan2 a;

On further solving we get:

= - csc2 a.

There are different types of methods to Solving Multistep Equations. To get more information about the multistep equation then prefer icse board syllabus and In the next session we will discuss about cosine law.

Ellipse Equation

An ellipse can be defined as a closed curve on a plane which can be obtained from intersection of a cone onto a plane. In the standard definition we can say that an ellipse is a curved line that creates closed shape in which summation of distances from two focus points to all other points on the line are constant. We can describe the circle as an example of ellipse, in the case when it is squashed into an oval. In the simple mean we can define the ellipse by the distance of two focus point. In the case select any on ellipse then sum of the distance to the focus point is constant.
Ellipse is a part of geometrical mathematics which also has the properties like other geometrical shape. Like ellipse has center, major and minor axis, focus point, circumference, area, chord, tangent and secant property. Here we are going to discussing about the ellipse equation.
Ellipse equation means to say to represent the concept of ellipse in the form of equation. In the form of equation ellipse can be define as a 2-demensional closed curve which can be represented as:
                                   (p – r)² / a² + ( q – s)² / b² = 1
In the above given ellipse notation r, s, a and b can be consider as a real numbers, in which a and b are the category of positive numbers.  (know more about Ellipse, here)
In the general form an ellipse equation can be represented as below given format:
  Pa² + Qb² + Ra + Sb + e = 0
There are some of the terms are describe below  which is related to an ellipse equation:
focus :  sometime this can be refer as foci point which ha sthe nature of fix points that shows the distances to any point of an ellipse that are represented in linear relation.
Area:  It can be consider as a mathematical term which express a particular measurement or region which is associated to a surface.
Directrix:  In the concpet of Ellipse equation the term directrix can be define as a fixed line that demonstrates a curve or surface.

The Box Plot is a concept of descriptive mathematics that performs the task of representing the data in the graphical form. CBSE Board Syllabus is design and issued by the Central Board Of Secondary Education to guide the teachers and students for board examination preparation.