Boolean Algebra
Boolean Algebra Just like sets and logic, Boolean algebra is a modern concept. It is concerned with statements which are either true or false. It was George Boolen who developed this theory and its significance was realized when Claude Shanon introduced circuit algebra to deal with relay circuits in 1938. The digital computer which contains a large number of logic circuits in small space and works in switching them opened a wide field for this algebra and its recognition as significant part of modern mathematics. Definition A...
read moreLimits
A number ‘l’ is called limit of a function f(x) when i.e., if given , there exists such that |x –a| | f(x) – l | < . Right hand and left hand limits Let h be a small positive number. Left hand side limit of f(x) when , is denoted by f(a -0) and is defined as: Right hand side limit of f(x), when , is denoted by f(a + 0) and is defined as: exists if Indeterminate forms If a function f(x) takes the form , then say that f(x) is indeterminate at x=a. Other...
read moreContinuity and Differentiability
Definition of continuity at a point A function f(x) is said to be continuous at x=a if given , there exists such that | f(x) – f(a) | such that | x –a | < . Alternative definition: f(x) is said to be continuous at x =a if value of f(x) = limit of f(x) at x = a Or if lim f(x) = f(a) Of if lim f (a+h) = lim f(a –h) = f(a) …….Equation 1 If the condition (1) is not satisfied, then f(x) is said to be discontinuous at x =a. Note: In order to rest continuity of a function at a point, we verify the...
read moreMaxima and minima
It is obvious from the diagram that the function y=f(x) is maximum at P and is minimum at Q. At both these points tangent is parallel to x-axis, so that its slope is zero. for both maximum and minimum. Working Rule (1) Find out from the equation f(x,y) = c and put . Solve it for x. Let, on solving x=a, x=b. (2) If = negative for x=1, then y is maximum at x=a. If positive for x=b, then y is minimum at x=b. (3) If both are 0 for some value of x, we have to find . If it is 0 then f(x) is maximum...
read moreTangents and Normals
P(x,y) is any point on the curve f(x,y)=c. PT is tangent at p and PN is normal at P. Angle made by tangent PT with x-axis is denoted by in anticlockwise direction. is defined as slope of gradient of tangent PT. We also define = slope of tangent, Slope of normal = Equation of Tangent PT is Equation of normal PN is PM is perpendicular from P on x-axis. By , By We define: Sub tangent = TM = Sub normal = MN = Length of Tangent = PT = Length of normal = PN = Where is point P. The tangent is...
read morePartial Differentiation
Defination of Partial Differentiation If f is a function of several variables , then the derivative of f w.r.t. keeping other variables constant is called partial derivative of f w.r.t. and is denoted by or by and is defined as: provided the limit exists. 3. if f=f(x,y) and partial derivates are continuous then 4. if f=f(x,y), then . if g=g (x,y,z), then . if f=f(x,y) and , then: And so Homogenous Function If f is a homogenous function of x and y of...
read moreFormulas for Area of a Triangle.
There are a lot of formulas and techniques to find the area of a triangle. We can use many different formulas to calculate area of a triangle according to the given conditions. Here we shall derive some of the main formulas used to calculate area of a triangle. Formulas for Area of a Triangle: The area of a triangle is denoted by the symbol delta ( ) We shall appeal to the formula: And the half angle formula: etc. Where “s” is the semi circumference of the triangle or, We shall now derive different formula for the...
read moreHalf Angle formulas
Half Angle formulas?: The Half angle formulas are stated below: If ABC is a triangle , A , B and C are the three angles of the triangle and a , b , c are the sides opposite to the corresponding angles and “s” is the semi perimeter or , , Then: Proof of Half angle formula: First of all let’s prove the half angle formula for Using the cosine law: Now using the trigonometric sub-multiple angle formula: Now , let us prove the half angle formula for Using the cosine law: Lastly ,...
read moreProjection Law
Projection Law: Projection law states that in any triangle: Where , A , B , C are the three angled of the triangle and a , b , c are the corresponding opposite side of the angles. Projection law or the formula of projection law express the algebraic sum of the projection of any two side in term of the third side. Proof of Projection law: To prove the projection law we shall take the help of sine law which states that: Thus , using the above formula for sine law we can easily deduce the formula for projection law...
read moreSine Law
Sine Law: Sine law states that in any triangle ABC: And also with some mathematics we can also prove the following : Which is also closely related to the sine law. Where , a , b and c are sides of a triangle , A , B and C are angles opposite to sides a , b, and c correspondingly and R is the circum-radius of the triangle as shown in following figure: Proof of Sine Law: Let us consider a Triangle ABC , placed in the standard position with the vertex A at the origin and side AB along the positive x-axis as in the...
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