Successive Differentiation(theory)

This post deals with the concepts of Successive Differentiation. Successive Differentiation means differentiating a given function successively or one after the other(like its 1st derivative,2nd derivative and so on) in order to obtain its nth derivative.The pdf attached below covers every concept important from examination point of view.

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Successive Differentiation

Tangents and Normals(theory)

This post deals with the chapter Tangents and Normals. This chapter forms the foundation for the next chapter Curvature wherein formulas of this chapter are extensively used in proving the formulas for radius of curvature and also while solving its questions. So,it becomes indispensible to study this chapter. Also,its questions would be dealt with in the subsequent post.

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Tangents and Normals

Infinite series theory(part 1)

This post deals with the tests of convergence of infinite series. The tests are mentioned in a concise and structured manner which makes it easier for students to study for their exams. It includes all the concepts pertaining to infinite series. The concept of Alternating series is included in the next post.

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Infinite series theory

Expansion of Functions

This post deals with the expansion of functions. The application of this concept would be dealt with in the subsequent posts.

Curvature(theory)

This post deals with the theory portion of Curvature in a very concise and easy to learn format.
SHORT NOTES:-
•Definitions:-
(a)Radius of Curvature:- Radius of curvature is defined as the radius of a circle made at a point on a curve.
(b)Curvature:- Curvature is the reciprocal of radius of curvature and gives a measure of the bending of the graph of a function about a point.
More is the bending of a graph about a point,less is the radius of curvature as a smaller circle would be formed in this case.

Click below for viewing the concepts of curvature
Curvature(theory)

Asymptotes

This pdf attached here covers the all important concepts of Asymptotes. It has a working rule which could be applied everywhere to find asymptotes with ease. It covers all concepts important from the examination point of view and illustrations which uses all the concepts so as to clear the application part of the concept.

Click below to access the pdf:-

Asymptotes Engineering Mathematics

Shifting of origin

Let us suppose we have a point P in the plane. We have a cartesian coordinate system XY such that the coordinates of this point P with reference to this frame is (x,y). Now,let us shift the coordinate axis system to a point (h,k) and let us call this new coordinate system X'Y' and the x and y coordinates of a point be represented by X and Y. So,according to the diagram below,X=(x-h), Y=(y-k). Hence,the coordinates of a point change to (x-h,y-k) whenever the coordinate axis system gets shifted to a point (h,k). This concept is useful in determining the equation of curve in a plane in the new coordinate system as shown below.

Interesting Application of De Moivre's Theorem

This is an interesting application of De Moivre's theorem wherein we get the series expansions of sin nx,cos nx, tan nx using this theorem. Note that here n is a positive integer.

Rotation Matrices

This post deals with the concept of rotation matrices which rotates a given vector by an angle. And in fact,this rotation matrix is nothing but e^i(theta) which we used to rotate a complex number. I will throw more light upon it in subsequent posts.

Rotation of Coordinate Axes

This post deals with one of the most beautiful concepts of mathematics. It tells us about the coordinates of a point relative to new coordinate axis system generated by rotating the original coordinate system by an angle.Like for example,we are given the equation of parabola in a coordinate system,and we rotate the axis by an angle then this concepts helps us in finding the equation of the same parabola in the new coordinate system.

Graph theory

General tricks for sketching any curve:-
1. Suppose,we have a function y=f(x) and we replace x by (x-a),then the graph will shift horizontally by a units towards the right(assuming that a is positive here which is implicitly assumed everywhere) and this is called horizontal shift.
2. If we replace x by (x+a), then graph will shift horizontally a units towards the left.
3. If we have a function y=f(x) and we replace y by (y-a),then graph will shift vertically up by a units.This is called vertical shift.
4. If we replace y by (y+a),then graph will shift vertically down by a units.
5. If we replace x by (-x) in the given function,then the graph will rotate 180° about the y axis(or in simple terms mirror image about y axis)
6. Similarly,if we replace y by (-y) in the given function,then the curve will rotate 180° about the x axis.
7. If we interchange x and y(ie replace x by y and y by x),then the graph will be a mirror image about the line y=x. This is actually how the graphs of inverse of a function is drawn.
8. If we replace x by -y and y by -x,then graph is a mirror image about the line y=-x.

●These tricks are useful if there is slight variation in the given function from a standard function whose graphs are already known to us.For ex. we have to plot a function y=|x-2|.So,we already know the graph of y=|x|. It is just that x is replaced by (x-2) in the given function. So by using trick 1,the graph of y=|x| will shift 2 units toward the right and hence this gives the graph of the given function easily.
9. Checking if the curve passes through origin gives us an idea about our curve.So,this could be done by plugging x=0 and y=0 and checking if it is satisfied by the equation.
10. Finding the points of intersection of the curve with x and y axis also helps to trace the curve.
11. Find the value of the function as x approaches infinity. It tells us about the nature of the function(whether it tends to a point or tends to infinity as x approaches infinity).
12. Find dy/dx and check for what values of x the graph is increasing or decreasing and find points of maxima and minima.
13. We could even find double derivative of a f
13. Find the domain and range of a function to get the area of plane where the curve is confined.

Rotation of a vector by an angle using complex numbers

This post deals with the concept of rotation of a given vector in a plane using complex numbers(although there are many other ways of doing so too). In next post,I will be discussing the rotation of vector using matrix.

Cramer's rule for Quadratic equations(condition of common roots)

This post deals with the condition for having common roots and the formula to find that common root for two given Quadratic equations

LU Decomposition (Crout Decomposition)

This post deals with a method of solving a system of linear equations called LU decomposition(and specifically Crout decomposition)

Trigonometric formulas using Geometry

This post deals with finding out the formulas of trigonometry like compound angle formulas(and thereafter, all the other formulas could be easily derived from it like for example putting B as 90 in the equations would give us that sin(90+A)=cos A etc.) using geometry.

Tangent to a polar curve

This post deals with how to find the tangent to a polar curve at any general point.

Trigonometry using complex numbers

In class 11, we spent countless hours and days trying to memorise the formulas of trigonometry. We tried our level best learning that Sin(A+B)=sinAcosB+cosAsinB, Cos(A+B)=cosAcosB-sinAsinB and what not. But we never tried to question why?Where suddenly these formulas landed from was the question which kept haunting me. But now I have come to a level where I can figure out why these formulas hold good. So here attached is the logic behind those formulas via the complex number method.

Projection Formulas

This post deals with the Projection Formulas used in SOT.

Solution of Triangles

Solutions of Triangle

In this post,I have discussed the logic behind the frequently used Herons formula to find out the area of a triangle. In the coming posts,I will discuss more about triangles and its properties.

Eigen Value and Eigen Vector

Have you ever thought why is it significant to study eigen values and eigen vectors of a square matrix and what do they mean really?Generally,most of the teachers tend to give formulas for finding them but fail to explain what really are we doing. So what students do is cram those formulas and properties and just blind-foldedly apply them in the exams without knowing the actual concepts which is the real reason why students tend to struggle with maths. This image attached explains the full concept behind eigen values,eigen vectors and characteristic equation. After going through this, I am sure that you won't have to cram a single formula and concept will be crystal clear.