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QUESTION BANK mathematics ( CONTINUITY AND DIFFERENTIABILITY)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER CONTINUITY AND DIFFERENTIABILITY ALONG WITH THE ANSWER KEY AT THE END
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Continuity and DifferentiabilityQUICK RECAP
CONTINUITY X Discontinuity of a Function : A real
function f is said to be discontinuous at x = c,
8 A real valued function f is said to be continuous if it is not continuous at x = c.
at a point x = c, if the function is defined at x = c i.e., f is discontinuous if any of the following
and lim f (x ) = f (c) or we say f is continuous reasons arise:
x →c
(i) lim f (x ) or lim f (x ) or both does
at x = c iff lim f (x ) = lim f (x ) = f (c) x →c − x →c +
x → c− x →c + not exist.
, (ii) lim f (x ) ≠ lim f (x ) (iii) Polynomial function
x →c − x →c + (iv) Modulus function
(iii) lim f (x ) = lim f (x ) ≠ f (c) (v) Sine and cosine functions
x →c − x →c + (vi) Exponential function
X A function f is said to be continuous in an
interval (a, b) iff f is continuous at every
DIFFERENTIABILITY
point in the interval (a, b) ; and f is said to 8 Let f(x) be a real function and a be any real
be continuous in the interval [a, b] iff f is number. Then, we define
continuous in the interval (a, b) and it is (i) Right-hand derivative :
continuous at a from the right and at b from f (a + h) − f (a)
the left. lim , if it exists, is
X A function f is said to be discontinuous in the h →0 + h
interval (a, b) if it is not continuous at atleast called the right-hand derivative of f(x)
one point in the given interval. at x = a and is denoted by Rf ′(a).
X Algebra of Continuous Functions : If f and g (ii) Left-hand derivative :
be two real valued functions, continuous at f (a − h) − f (a)
x = c, then lim− , if it exists, is called
h →0 −h
(i) f + g is continuous at x = c.
(ii) f – g is continuous at x = c. the left-hand derivative of f(x) at x = a
(iii) f . g is continuous at x = c. and is denoted by Lf ′(a).
A function f(x) is said to be differentiable at
f x = a, if Rf ′(a) = Lf ′(a).
(iv) is continuous at x = c, (provided
g The common value of Rf ′(a) and Lf ′(a)
g(c) ≠ 0). is denoted by f ′(a) and is known as the
X Composition of two continuous functions is derivative of f(x) at x = a. If, however,
continuous i.e., if f and g are two real valued Rf ′(a) ≠ Lf ′(a) we say that f(x) is not
functions and g is continuous at c and f is differentiable at x = a.
continuous at g(c), then fog is continuous at c. X A function is said to be differentiable in
X The following functions are continuous (a, b), if it is differentiable at every point of
everywhere. (a, b).
(i) Constant function X Every differentiable function is continuous
(ii) Identity function but the converse is not necessarily true.
SOME GENERAL DERIVATIVES
Function Derivative Function Derivative Function Derivative
xn nxn–1 sin x cos x cos x – sin x
tan x sec2 x cot x – cosec2 x sec x sec x tan x
cosec x – cosec x cot x eax aeax ex ex
1 −1
sin–1 x ; x ∈(−1, 1) cos–1 x ; x ∈(−1, 1) tan–1 x 1
1− x 2 2 ;x∈R
1− x 1 + x2
1 1
cot–1 x sec–1 x ; x ∈ R – [–1, 1] cosec–1x 1
− ;x∈R 2 − ; x ∈ R –[–1,1]
1 + x2 x x −1
x x2 − 1
1
loge x 1 ax ax loge a; a > 0 loga x ; x > 0 and a > 0
;x>0 x log e a
x
, EXPONENTIAL FUNCTION is called natural logarithm.
X The function logax (a > 0, ≠1) has the
8 If a is any positive real number, then the
following properties :
function f defined by f(x) = ax is called the
exponential function. (i) loga(mn) = logam + loga n ; m, n > 0
LOGARITHMIC FUNCTION (ii) log a m = log a m − log a n ; m, n > 0
n
8 Let a > 1 be a real number. The logarithmic (iii) logamn = nlogam ; m > 0
function of x to the base a is the function
y = f(x)= logax i.e., loga x = b, if x = ab log x
(iv) log a x = ;x>0
X The logarithm function, with base a = 10, is log a
called common logarithm and with base a = e, (v) loga a = 1, loga1 = 0
SOME PROPERTIES OF DERIVATIVES
1. Sum or Difference (u ± v)′ = u′ ± v′
2. Product Rule (uv)′ = u′v + uv′
3. Quotient Rule u ′ u ′v − uv ′
= ,v≠0
v v2
4. Composite Function dy dy dt
(a) Let y = f(t) and t = g(x), then = ×
(Chain Rule) dx dt dx
dy dy dt du
(b) Let y = f(t), t = g(u) and u = m(x), then = × ×
dx dt du dx
5. Implicit Function Here, we differentiate the function of type f(x, y) = 0.
6. Logarithmic Function If y = uv, where u and v are the functions of x, then log y = v log u.
d v v du dv
Differentiating w.r.t. x, we get (u ) = uv + log u
dx u dx dx
7. Parametric Function dy dy / dt g ′(t )
If x = f(t) and y = g(t), then = = , f ′(t ) ≠ 0
dx dx / dt f ′(t )
8. Second Order Derivative dy
Let y = f (x), then = f ′(x )
dx
d dy d2 y
If f ′(x) is differentiable, then = f ′′( x ) or = f ′′(x )
dx dx dx 2
ROLLE’S THEOREM MEAN VALUE THEOREM
8 Let f : [a, b] → R be a continuous function on 8 Let f : [a, b] → R be a continuous function
[a, b] and differentiable on (a, b) such that on [a, b] and differentiable on (a, b), then
f(a) = f(b), then there exists some c ∈(a, b) there exists some c ∈(a, b) such that
such that f ′(c) = 0 f (b) − f (a)
f ′(c) =
b−a
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