Ch 2 Maths Class 12 Solutions
Handwritten notes of Ch 2 Maths Class 12 Solutions Class 12 written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Ch 2 Maths Class 12 Solutions Class 12 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 ex 2.1 class 12 for Intermediate part-II students.
- Important Multiple Choice Questions (MCQs) of Chapter No. 2: Differentiation of 2nd Year Mathematics.
- Introduction and Applications of Derivatives with their examples of Chapter No. 2: Differentiation of 2nd Year Mathematics.
- Average rate of change: Let f be a real valued function the (difference quotient) f(x1)-f(x) is called average rate of x-x change.
- Important ex 2.2 class 12 for Intermediate part-II students.
- Derivative:- Let f(x) be afunction, then its derivative is denoted by f'(x) or a f(x) and defined as; f'(x) = Lim f(x+Sx)-f(x) Sx/dx. The process of finding f' is called differen - tiation.
- Important ex 2.3 class 12 for Intermediate part-II students.
- Notation for Derivative For the function y = f(x) +y+by=f(x+6x) by = f(x+6x)-f(x) (i) where by is the increment of y (change in the value of y corresponding to 8x (increment of x Dividing is by Ex on both sides, we get f(x+8x)-f(x) Taking limit on both sides as 6xo, we get Lim 89 Lim Sy as Lim f(x+8x)-f(x) is denoted by dy dx dy = f'(x) dx so (ii) is written
- Important ex 2.4 class 12 for Intermediate part-II students.
- Note: The symbol dy/dx is used for the derivative of y with respect to x and here it is nota quotient of dy and dx/dy is also denoted by y'.
- Important ex 2.5 class 12 for Intermediate part-II students.
- Derivative of exponential functions of f(x)= a where a>0 then f(x) is called general exponential function. Here the base a is constant and exponent x is variable. If f(x) = e where e22.71 then f(x) x is called natural exponential function Here base e is constant and exponent x is variable.
- Important ex 2.6 class 12 for Intermediate part-II students.
- Derivative of the logarithmic function: Logarithmic function:- If aso, az 1 and ५ x=a, then the function defined by y=logx (x>0) is called the logarithm of x to the base a. The logarithmic functions loge and logox. are called natural and common logarithms resp. y= loge is written as y= Inx
- Important ex 2.7 class 12 for Intermediate part-II students.
- Note: A function f can be expanded in Machlaurin series if the function is defined in the interval containing O and its derivatives exist at x = 0 The expansion is only valid if it is convergent.
- Important ex 2.8 class 12 for Intermediate part-II students.
- Increasing Function: A function f is said to be increasing function on an interval (a,b) if for every x1, x2 belongs to (a,b) than f(x1) < f(x2) whenever x1 < x2
- Important ex 2.9 class 12 for Intermediate part-II students.
- Constant function:- A function f is said to be constant function on an interval (a,b) if for every x1, x2∈(a,b) then f(x1) = f(x2) whenever x1 < x2
- Alternative definitions: Increasing function: A differentiable function f is said to be increasing function if f'(x)>0 for all x ∈ (a,b)
- Decreasing function: A differentiable function f is said to be decreasing function if f'(x) <0 for all x ∈ (a,b)
- Constant function:- A differentiable function f is said to be constant function if f'(x)=0 for all x ∈ (a,b)
- Important ex 2.10 class 12 for Intermediate part-II students.
- Example 1. Determine the values of x for which f(x) = x²+2x-3 is (is increasing (ii) decreasing (iii) find the point where the function is neither increasing nor decreasing.
- Important class 12 ex 2.1 for Intermediate part-II students.
- Stationary Point: Any point where f is neither increasing nor decreasing is called a stationay point. Provided that f'(x) = 0 at that point.
- Relative Maxima: A function f(x) has relative maxima f(c) at x = c belongs to (a,b) if (i) f'(c - Sx)>0 (ii) f'(c) = 0 (iii) f'(c+Sx)<0 where Sx is small +ve number.
- Important class 12 ex 2.2 for Intermediate part-II students.
- Relative Minima: A function f(x) has relative minima f(c) at x = c belongs to (a,b) if f'(c-Sx) <0 (ii) f'(c) = 0 where Sx is small +ve number.
- Relative Maxima: A function f(x) has relative maxima f(c) if f'(x) changes sign from positive from negative as x increases through c.
- Relative Extrema: Both relative maxima and relative minima are called in general " relative extrema".
- Critical values and critical points: If c belongs to Df and f'(c) =0 or f'(c) does not exist, then the number c is called a critical value for f while the point (c,f(c) ) on the graph of f is named as a critical point.