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
CONTINUITY AND DIFFERENTIABILITY
Continuity of a function:
• A real valued function f(x) is said to be continuous at x = a, if
= f(a).
• If the graph of a function has no breaks or jumps then the function is continuous.
• Point of discontinuity: If f(x) is not continuous at x = c, then c is called a point of discontinuity of f.
• A function is continuous if it is continuous in whole of its domain.
• Sum, difference, product and quotient of two continuous functions is also continuous.
• If g is continuous at x = c and f is continuous at g(c) then (fog) is also continuous at x = c.
Examples of continuous functions:
• All trigonometric, polynomial, exponential, logarithmic and modulus functions are continuous in
their respective domains.
Examples of dis-continuous functions:
• The “greatest integer” function y = [x] and
• The “fractional part of x” function y = {x}
are discontinuous at all integral values of x i.e., for all x Є Z.
Differentiability of a function:
• A function is said to be differentiable at x = a if it is continuous at x = a and the limit
exists i.e., both the limits
are finite and
equal.
• If a function is differentiable at x = a, then it is also continuous at x = a, as continuity is a
prerequisite of differentiability.
• Derivative of a function y = f(x) is denoted by y’ or
or f’(x) or
• If the graph of a function has a corner or a kink at x = a, then it is not differentiable at x = a.
Cases of non-differentiability:
Discontinuity
Vertical tangent
Corner or a kink
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
Derivatives of standard functions:
1.
(sin x) = cos x
2.
(cos x) = - sin x
3.
(tan x) = sec
2
x
4.
(cot x) = - cosec
2
x
5.
(cosec x) = - cosec x cot x
6.
(sec x) = sec x tan x
7.
(sin
-1
x) =
8.
(cos
-1
x) =
9.
(tan
-1
x) =
10.
(cot
-1
x) =
11.
(sec
-1
x) =
12.
(cosec
-1
x) =
13.
(a
x
) = a
x
ln a
14.
(e
x
) = e
x
15.
(ln x) =
16.
(
) =
17.
(x
n
) = nx
n-1
18.
(c) = 0, where c is a constant.
Techniques of Differentiation:
a) Using first principles by evaluating the limit:
b) Using direct formulae (given above)
c) Using product and quotient rules.
(uv) = u’v + uv’ and
(u/v) = (u’v – uv’) / v
2
d) For composite functions, we use chain rule. i.e., for y = f(g(x)); y’ = f’(u) * g’(x) where u = g(x).
Explanation: Let u = g(x) so y = f(u).
So,
= g’(x) and
= f’(u); Now,
=
*
= f’(u) * g’(x)
e) For implicit functions, i.e., for a given equation in y and x, just differentiate both sides of the
equation w.r.t. x and solve for y’.
f) For functions of the form y = [u(x)]
v(x)
, we take log on both sides and differentiate. This process is
called Logarithmic Differentiation.
g) For parametric functions of the form y = f(t) and x = g(t),
=
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.
Mean Value Theorem: If f : [a, b] → R is continuous on [a, b]
and differentiable on (a, b), then there exists some c in (a, b)
such that f’(c) =
• Rolle’s Theorem is a special case of Mean Value Theorem when
f(a) = f(b).
