• Concept of Anti Derivatives or Integration

    The inverse process of derivatives called anti–derivatives or integration. “A function being given and it is required to find a Second function whose derivative with respect to x, is, that is, Thus if Then The function , then, is called the anti-derivate or indefinite integral of . Symbol of Integration: The function is called integrand […]

  • Introduction to Differential Calculus

    In the seventeenth century, Sir Isaac Newton, an English mathematician (1642–1727) and Gottfried Wilhelm Leibniz, a German mathematician, (1646–1716) consider the problem of instantaneous rates of the change. They reached independently to the invention of differential calculus. After the development of calculus, mathematics becomes a powerful tool for dealing with rates of change and describing […]

  • Average and Instantaneous Rate of Change

    A variable which can assign any value independently is called independent variable and the variable which depends on some independent variable is called the dependent variable. For Example:             If etc, then We see that as behaves independently, so we call it the independent variable. But the behavior of or depends on the variable . […]

  • Examples of Average and Instantaneous Rate of Change

    Example: Let (a) Find the average rate of change of with respect to over the interval . (b) Find the instantaneous rate of change of with respect to at the point . Solution: (a) For Average Rate of Change:             We have                         Put                         Again Put                         The average rate of […]

  • Derivative of a Function

    Let be a given function of . Given to a small increment and let the corresponding increment of by , so that when becomes , then becomes and we have                                     Dividing both sides by , then                         Taking limit of both sides as             Thus, if be the function of , […]

  • Introduction to Functions

    In mathematics, the term Function is very famous, if we look in the historical background the term function was first used by very well known Mathematician Leibniz in 1676, who put the meaning of function in term of that dependence of one quantity on another quantity. Function is also known as the input and output […]

  • Concept of Functions

    Let A and B be any two non–empty sets. Then a function ‘’ is a rule or law which associates each element of ‘A’ to a unique element of set ‘B’. Notation: (i) Function is usually denoted by small letters i.e. etc and the Greek letter i.e. etc. (ii) If ‘’ is a function from […]

  • Examples of Functions

    Example: Find the range of the function . Solution: We have Put Thus, the domain is, Now for the range, we have For So, the range of the function is . Example: Let . Find the domain and range of . Solution: We have For For Thus, the domain is. Now for the Range, we […]

  • Types of Functions

    Constant Function: Let ‘A’ and ‘B’ be any two non–empty sets, then a function ‘’ from ‘A’ to ‘B’ is called Constant Function if and only if range of ‘’ is a singleton. Algebraic Function: The function defined by algebraic expression are called algebraic function. e.g. Polynomial Function: A function of the form Where ‘n’ […]

  • Nature of Functions

    One – One Function: Let ‘A’ and ‘B’ be any two non–empty sets then a function ‘’ from A to B is called one–one function, if and only if distinct elements of set A have distinct elements of set B. e.g. , In Mathematically, Let be a function then ‘’ is called one–one function if […]

  • Concept of Limit

    Meaning of the Phrase “Tend to Zero”: Suppose a variable ‘x’ assumes in succession a set of values. Clearly, ‘x’ is becoming smaller and smaller as n is increasing and can be made as small as we want by taking ‘n’ sufficiently large. This un-ending decrease of ‘x’ is symbolically expressed by “” and read […]