Remedial Mathematics
| Author’s Name | Sudhir Kumar Pundir |
| Editions | Latest Edition |
| Publishers | Asian Books Private Limited |
A Textbook of Remedial Mathematics ( PCI Books)
| Author’s Name | Dr P. K. Sharma |
| Editions | Fifth edition, 2019 |
| Publishers | NIRALI PRAKASHAN |
Remedial Mathematics Book Topic-wise Summary
✓ Partial fraction
• Splitting up a given fraction into simpler fractions is called a partial fraction. In this chapter, we study the methods of resolving a given fraction into a partial fraction.
• Any proper rational fraction f(x)/g(x) can be expressed as the rational fraction, each having a simple factor of g(x). Such types of fractions are called partial fractions and the process of obtaining them is called the resolution or decomposition of f(x)/g(x) into partial fractions.
✓ Logarithms
• Logarithms were invented in the 17th century as a calculation tool by John Napier. A logarithm is a mathematical operation that determines how many times a certain number, known as a base, is multiplied by itself to reach another number.
• Logarithm is the inverse operation of the exponentiation factors. The logarithm of any number y>0, to a given base a>0, and a#1 is the exponent (power) to which the base must be raised to equal the given number.
Thus, if ax=Y, then logaY=X.
✓ Function
• In this chapter we study a special type of relationship called functions. We can visualize a function as a rule that produces new elements out of some given elements.
• A relation f from a non-empty set A to a non-empty set B is said to be a function. If each element set A has one and only one image B. The function f from A to B is denoted by: A –> B
✓ Limits and continuity
• The limit of a function f(x) is said to be L at X=X0 if f(x) gets closer and closer to L as x moves closer and closer to X0.
• There are two ways X could approach several X0 either from the left or the right, i.e., all the values of X near X0 could to less or could be greater than X0. This naturally leads to two limits- the right-hand limit and the left-hand limit.
✓ Determinants
• Every square matrix can be associated with an expression or a number that is known as its determinant. Let, A=[aij] be a square matrix of order n, then the determinants of A are denoted by det (A).
• Applications of determinants to coordinate geometry are the Area of the triangle, Conditions for Collinearity of three points, and Equation of a line passing through two given points.
✓ Matrices
• Matrix is a rectangular array of numbers, functions, or symbols, arranged in columns and rows enclosed by a pair of brackets. The brackets sign is ( ), [ ], or || ||.
• The concept of the matrix was introduced by Arthur Carley in 1860. Today, it is an essential tool for the development of modern science and technology.
• Some applications of the matrices are chemical reactions kinetics, balancing of chemical equations, and chemical systems.
✓ Differentiation
• Differentiation is a process of finding the derivatives of a function at any point. The derivative is a concept that is at the root of the calculus.
• There are two ways of introducing this concept, one is the geometrical way (as the slope of a curve) and the second is the physical way (as the rate of change).
• Let y is a function of x i.e, y=f(x). Here x is the independent variable and y is the dependent variable. Then the rate of changes of dependent variable y concerning independent variable x is denoted by dy/dx.
✓ Analytical Geometry
• Analytical Geometry is the study of geometry use of coordinate systems, also known as Cartesian geometry or Coordinate geometry. In the study, to locate the position of a point on a plane, we square a coordinate system.
• Let X’OX and Y’OY be two perpendicular lines intersecting at O. Let the lines X’OX and Y’OY represent the coordinates axes say X-axis and Y-axis.
✓ Straight Line
• A straight line is a curve such that all points of the line segments joining any two points on it lie on it. In analytical geometry, a line in the plane is often defined as the set of points whose coordinates satisfy given linear equations.
✓ Differential Equations
• A differential equation is an equation that involves dependent variables, independent variables, and the derivatives of the dependent variables.
• A differential equation is said to be linear if the dependent variable and all of its derivatives occur only in the first degree & are multiplied together.
✓ Laplace Transform
• Laplace transformation is a powerful mathematical tool to solve ordinary and partial differential equations with the given initial conditions, without the findings of general solution, and then evaluating arbitrary constants.
• A transformation is a mathematical device that converts one function to another function. For Laplace transformation of a function f(t);
f(t) —(Laplace transform)—> F(s), where f(t) input functions and F(s) output functions.