Chapter 6

Class 12 PCMB Mathematics Chapter 6

Application of derivatives

Application of derivatives

3. A function f is said to be 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, then f(x) is an increasing function on (a, b).

4. A function f is said to be 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, then f(x) is an decreasing function on (a, b).

5. The equation of the tangent at (x0, y0) to the curve y = f (x) is given by

6. 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 = x0.

Application of derivatives

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

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

11.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 dy = f’(x) dx or dy=dydxdx is a good of ∆y when dx x = ∆ is relatively small and we denote it by dy ≈ ∆y.

12. 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.

13. 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. In fact, such a point is called point of inflexion.

14. 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.

15. 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 Step3.

This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f.

Application of derivatives

Also Read :-NCERT Solutions Class 12th PCMB Chemistry Chapter 13 AMINES

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