Class 12 Maths Revision Notes for Application of Derivatives of Chapter 6

CBSE Class 12 Maths Revision Notes Chapter 6 Application of Derivatives

• If a quantity y varies with another quantity x, satisfying some rule y = f(x), then represents the rate of change of y with respect to x and represents the rate of change of y with respect to x at x = x_o.
• If two variables x and y are varying with respect to another variable t, i.e., if   then by Chain Rule
  
• A function f is said to be increasing on an interval (a,  b) if  in  for all   Alternatively, if   for each  in, then f(x) is an increasing funciton on (a, b).
• A function f is said to be decreasing on an interval (a,  b) if  in  for all   Alternatively, if   for each  in, then f(x) is an decreasing funciton on (a, b).
• The equation of the tangent at    to the curve y =  f (x) is given by
  
• If   does not exist at the point , then the tangent at this point is parallel to the y-axis and its equation is.
• If tangent to a curve  is parallel to x-axis, then = 0
• Equation of the normal to the curve y = f (x) at a point is given by 
• If  at the point is zero, then equation of the normal is .
• If  at the point  does not exist, then the normal is parallel to x-axis and its equation is.
• 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  given by  or  is a good appApplication of Derivatives Class 12 Notes Mathematicsroximation of ∆y when  dx x = ∆  is relatively small and we denote it by dy ≈ ∆y.
• 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.
• 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,

(i) 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.
(ii) 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.
(iii) 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. In fact, such a point is called point of inflexion.

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

(i) 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.
(ii) 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.
(iii) 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.

• Working rule for finding absolute maxima and/or absolute minima

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