Class 12 Maths Chapter 2 Notes

Handwritten notes of Class 12 Maths Chapter 2 Notes written by Professor Rana Azhar. These notes are very helpful in the preparation of Class 12 Maths Chapter 2 Notes for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.

Summary and Contents:    
Topics which are discussed in the notes are given below:
  • Important ch 2 maths class 12 important questions for Intermediate part-II students.
  • Introduction: The ancient Greeks knew the concepts of area, volume and centroids etc. which are related to integral calculus. Later on, in the seventeenth century, Sir Isaac Newton, an English mathematician (1642 - 1727) and Gottfried Whilhelm Leibniz, a German mathematician, (1646 - 1716) considered the problem of instaneous rates of change. They reached independently to the invention of diffrential calculus. After the development of calculus, mathematics became a powerful tool for dealing with rates of change and describing the physical universe.
  • Important ch 2 maths class 12 for Intermediate part-II students.
  • Dependent and Independent Variables: In differential calculus, we mainly deal with the rate of change of a dependent variable with respect to one or more independent variables. Now, we first explain the terms dependent and independent variables. 
  • Important class 12 maths 2.1 for Intermediate part-II students.
  • We usually write y=f(x) where f(x) is the value of f at x belongs to Df (the domain of the function f). Leter us consider the functional relation y=f(x)=x^2 +1.
  • Important class 12 maths ch 2 for Intermediate part-II students.
  • It is obvious that the change in the value of expression x^2 +1 depends upon the change in the value of the variable x. As x behaves independently, so we call it the independent variable. But the behaviour of y or f(x) depends on the variable x, so we call it the dependent variable.
  • Important class 12 maths ch 2 ex 2.1 for Intermediate part-II students.
  • The change in the value of x ( positive or negative) is called the increment of x and is denoted by the symbol delta x. The correspondence change in the dependent variable y or f(x) for the change delta x in the value of x is denoted by delta y. Usually the small changes in the values of the variables are taken as increments of variables.
  • Important class 12 maths ch 2 ex 2.2 for Intermediate part-II students.
  • Note: In this chapter we shall discuss functions of the form y = f(x) where x belongs to Df and is called an independent variable while y is called the dependent variable.
  • Important class 12 maths ch 2 ex 2.3 for Intermediate part-II students.
  • Average Rate of Change: Suppose a particle (or an object) is moving in a straight line and its positions (from some fixed point) after times t and t1 are given by s(t) and s(t1), then the distance traveled in the time interval t1 - t where t1 > t is s(t1) - s(t). and the difference quotient s(t) and s(t1) / t1 - t represents the average rate of change of distance over the time interval t1 - t. If t1 - t is not small, then the average rate of change does not represent an accurate rate of change near t. We can elaborate this idea by a moving particle in a straight line whose position in metres after t seconds is given by s(t) = t^2 +t.
  • Important ex 2.1 class 12 maths solutions for Intermediate part-II students.
  • THE CHAIN RULE: The composition fog of functions f and g is the function whose values f [g(x)] are found for each x in the domain of g for which g(x) is in the domain of f. (f[g(x)]) is read as f of g of x).
  • Important ex 2.2 class 12 maths solutions for Intermediate part-II students.
  • DERIVATIVE OF A FUNCTION GIVEN IN THE FORM OF PARAMETRIC EQUATIONS: The equations x = ar and y = 2at express x and y as function of t. Here, the variable t is called a parameter and the equations of x and y in terms of t are called the parametric equations. Now, we explain the method of finding derivatives of functions given in the form of parametric equations by the following examples:
  • Important exercise 2.3 class 12 maths solutions for Intermediate part-II students.
  • Divide 20 into two parts so that the sum of their squares will be minimum
  • Find two positive integers whose sum is 9 and the product of one with the other's square is a maximum.
  • Examples and Exercise problems with their Solutions.
  • Important class 12 maths chapter 2 exercise 2.1 for Intermediate part-II students.
  • Important class 12 maths exercise 2.1 for Intermediate part-II students.
  • Important class 12 maths exercise 2.1 solutions for Intermediate part-II students.
  • Important exercise 2.1 class 12 maths for Intermediate part-II students.

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