Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
APPLICATION OF DERIVATIVES
Derivative as the rate of change:
If a quantity y varies with another quantity x, satisfying some rule y = f(x), then:
• dy/dx or f’(x) represents the rate of change of y w.r.t. x and
• dy/dx at x = a or f’(a) represents the rate of change of y w.r.t. x at x = a
Equations of Tangent and Normal:
Geometrically, f’(h) represents the slope of the tangent to the curve y = f(x) at point (h, k)
• Equation of the tangent at (h, k) will be y – k = f’(k) (x – h) and
• Equation of normal at (h, k) will be y – k = -1/f’(k) (x – h)
(using point-slope form)
o If f’(x) = 0, equation of tangent is y = k, and equation of normal is x = h.
o If f’(x) is not defined, equation of tangent is x = h, and equation of normal is y = k.
Increasing and Decreasing Functions:
A function f(x) is said to be strictly increasing in an interval (a, b) if
▪ x
1
< x
2
=> f(x
1
) < f(x
2
) for all x
1
, x
2
(a, b) OR
▪ f’(x) > 0 for each x (a, b)
A function f(x) is said to be strictly decreasing in an interval (a, b) if
▪ for x
1
< x
2
=> f(x
1
) > f(x
2
) for all x
1
, x
2
(a, b) OR
▪ f’(x) < 0 for each x (a, b)
A function f(x) is said to be increasing in an interval (a, b) if
▪ x
1
< x
2
=> f(x
1
) <= f(x
2
) for all x
1
, x
2
(a, b) OR
▪ f’(x) >= 0 for each x (a, b)
A function f(x) is said to be decreasing in an interval (a, b) if
▪ for x
1
< x
2
=> f(x
1
) >= f(x
2
) for all x
1
, x
2
(a, b) OR
▪ f’(x) <= 0 for each x (a, b)
A function f(x) is said to be neither decreasing nor decreasing (constant function) in an interval (a, b) if
▪ for x
1
< x
2
=> f(x
1
) = f(x
2
) for all x
1
, x
2
(a, b) OR
▪ f’(x) = 0 for each x (a, b)
Maxima and Minima:
Critical Point – A point c in the domain of f(x) is called a critical point if either
▪ f’(c) = 0 or
▪ f’(c) doesn’t exist
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
First Derivative Test:
If f(x) be a function defined on an open interval I, and f be continuous at a critical point c in I, then
▪ If f’(x) changes sign from +ve to -ve as x increases through c OR If f’(x) > 0 to the close-left of c and
f’(x) < 0 to the close right of c, then c is a point of local maxima and f(c) is a local maximum value.
▪ If f’(x) changes sign from -ve to +ve as x increases through c OR If f’(x) < 0 to the close-left of c and
f’(x) > 0 to the close right of c, then c is a point of local minima and f(c) is a local minimum value.
▪ If f’(x) doesn’t change sign as x increases through c, then c is neither a point of local maxima nor a point
of local minima, it is called a point of inflexion.
Second Derivative Test:
If f(x) be a function defined on an open interval I, and f be twice differentiable
at critical point c in I, then
▪ If f’( c) = 0 and f’’(c) < 0, the c is a point of local maxima and f(c) is a local maximum value.
▪ If f’( c) = 0 and f’’(c) > 0, the c is a point of local minima and f(c) is a local minimum value.
▪ If both f’( c) and f’’(c) = 0, then this test fails, and we revert to First Derivative test.
Finding absolute maxima and absolute minima:
➔ Find all critical points and take all the end points of the given interval
➔ Calculate values of f(x) at all these points
➔ The highest value obtained in step 2 will be the absolute maximum value and the lowest value
obtained in step 3 will be the absolute minimum value in the given interval.
