NCERT MATHS CLASS-12

  1. Home
  2. NCERT MATHS CLASS-12

CONTINUITY AND DIFFERENTIABILITY

  1. A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.

  2. Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then

  3. (f ± g) (x) = f (x) ± g(x) is continuous.

  4. (f . g) (x) = f (x) . g (x) is continuous.

  5. Every differentiable function is continuous, but the converse is not true.
  6. Following are some of the standard derivatives (in appropriate domains):
math diagram

7.Logarithmic differentiation is a powerful technique to differentiate functions of the form f (x) = [u (x)]v(x) . Here both f (x) and u(x) need to be positive for this technique to make sense. v (x)

8.Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ′(c) = 0.

APPLICATION OF DERIVATIVES

1.If a quantity y varies with another quantity x, satisfying some rule y = f(x) the dy/dx (or f(x)) represents the rate of change of y with respect to x

2.If two variables x and y are varying with respect to another variable t, i.e., if x = f(t) and y = g(t) , then by Chain Rule.

3.A function f is said to be
(a) increasing on an interval (a, b) if x1 < x2 in (a,b) ⇒ f(x1) ≤ f(x2) for all x1 , x2 ∈ (a,b)
Alternatively, if f ′(x) ≥ 0 for each x in (a, b)
(b) decreasing on (a,b) if x1 < x2 in (a, b) ⇒ f (x 1 ) ≥ f (x 2 ) for all x 1 , x 2 ∈ (a,b)

4.If dy/dx does not exist at the point (x0,y0) then the tangent at this point is parallel to the y-axis and its equation is x = xo.

  1. If dy/dx at the point (x0,y0) is zero, then equation of the normal is x = xo.

  2. If dy/dx at the point (x0,y0) does not exist , then then the normal is parallel to x-axis and its equation is y =yo

  3. Let y = f (x), △ x be a small increment in x and △y be the increment in y corresponding to the increment in x, i.e., △y = f (x + △x) – f (x). Then dy given by below equaion and is a good approximation of △y when dx =△x is relatively small and we denote it by dy ≈ △y

    1. A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f.

    2. First Derivative Test Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then

      1. If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

      2. If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

      3. If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Infact, such a point is called point of inflexion.

    3. Second Derivative Test Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then

      1. x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0 The values f (c) is local maximum value of f .

      2. x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0 In this case, f (c) is local minimum value of f .

      3. The test fails if f ′(c) = 0 and f ″(c) = 0. In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.

    4. Working rule for finding absolute maxima and/or absolute minima

      1. Step 1: Find all critical points of f in the interval, i.e., find points x where either f ′(x) = 0 or f is not differentiable.

      2. Step 2:Take the end points of the interval

      3. Step 3: At all these points (listed in Step 1 and 2), calculate the values of f .

      4. Step 4:Identify the maximum and minimum values of f out of the values calculated in Step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f .

        INTEGRALS

        Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation

        1. From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.

        2. From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.

        3. Some standard integrals

math questions

  1. Integration by partial fractions

     
math questions

  1. Integration by substitution

    1.  tan x dx = log |sec x| + C

    2.  cot x dx = log |sin x| + C

    3.  sec x dx = log |sec x + tan x| + C

    4.  cosec x dx = log |cosec x – cot x| + C