Continuity & Differentiability IIT JEE Super Revision

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Continuity

At a point At an interval Everywhere

❑ Addition, subtraction of two continuous function makes a continuous function.

Algebra of continuous function

❑ Product of two continuous functions is continuous and ratio is also continuous provided denominator is not equal to zero at the point where we are checking the continuity.

❑ Composition of two continuous functions is also a continuous function.

Polynomial functions are continuous everywhere

sinx, cosx, |x| are also continuous everywhere

log x is continuous in (0, ∞)

Points to Remember

tan x is continuous everywhere except odd multiple of π2

Differentiability

LHD RHD

Differentiation

Standard Formulae of Differentiation

SOME RULES FOR DIFFERENTIATION

1. The derivative of a constant function is zero i.e.

2. The derivative of constant times a function is constant times the derivative of the function, i.e.

3. The derivative of the sum or difference of two function is the sum or difference of their derivatives, i.e.,

4. PRODUCT RULE OF DIFFERENTIATION

The derivative of the product of two functions = (first function) x (derivative of second function) + (second function) x (derivative of first function) I.e.,

5. QUOTIENT RULE OF DIFFERENTIATION

The derivative of the quotient of two functions

6. DERIVATIVE OF A FUNCTION OF A FUNCTION (CHAIN RULE)

If y is a differentiable function of t and t is a differentiable function of x i.e.

y = f(t) and t = g(x), then

Similarly, if y = f(u) , where u = g(v) and v = h(x), then

10. DIFFERENTIATION BY TRIGONOMETRIC SUBSTITUTIONS

Function Substitution

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Function Substitution

(vii)

(viii)

(ix)

(x)

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Continuity & Differentiability 1 1 1 0 2 2

JEE MAIN 2020

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Continuity & Differentiability

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JEE MAIN 2019

JEE MAIN PAST YEARS CHAPTERWISE WEIGHTAGE

JEE MAIN 2019

Chapter name

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8th Apr(II)

9th April(I)

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Chapter name 2018 2017 2016 2015


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JEE MAIN PAST YEARS CHAPTERWISE WEIGHTAGE

A

Q1. Let f and g be differentiable functions on R such that fog is the identity function. If for some a,b R, g’(a) = 5 and g(a) = b, then f’(b) is equal to:

B

D

C 1

5

9th Jan 2020 - (Shift II)

JEE MAIN Continuity & Differentiability : Super JEE Revision

Solution :

A

Q1. Let f and g be differentiable functions on R such that fog is the identity function. If for some a,b R, g’(a) =5 and g(a) = b, then f’(b) is equal to:

B

D

C 1

5



A

Q2. Let [t] denote the greatest integer ≤ t and is discontinuous when x is equal to

B

D

C



Solution :


A

B

D

C

Q2. Let [t] denote the greatest integer ≤ t and is discontinuous when x is equal to


A

Q3. Let f be any function continuous on [a,b] and twice differentiable on (a,b). If for all x ϵ (a,b), f’(x) > 0 and f”(x) < 0, then for any Is greater than

B

D

C

9th Jan 2020 - (Shift I)


Solution :

A

Q3. Let f be any function continuous on [a,b] and twice differentiable on (a,b). If for all x ϵ (a,b), f,(x) > 0 and f,, (x) < 0, then for any is greater than

B

D

C



Q4. If is continuous at x = 0, then a + 2b is equal to :


A

B

D

C -2

-1

1

0


Solution :

Q4.If is continuous at x = 0, then a + 2b is equal to :

9th Jan 2020 - (Shift I)A

B

D

C -2

-1

1

0


A

Q5. The value of c in the Lagrange’s mean value theorem for the function f(x) = x3 - 4x2 + 8x +11, when x [0,1] is:

B

D

C

JEE-Main 2020, 7th Jan-II


Solution :

A

Q5. The value of c in the Lagrange’s mean value theorem for the function f(x) = x3 - 4x2 + 8x +11, when x [0,1] is:

B

D

C

JEE-Main 2020, 7th Jan-II


A

Q6. Let the function, f : [-7, 0] ⟶ R be continuous on [-7, 0] and differentiable on (-7, 0). If f(-7) = -3 and f,(x) ≤ 2, for all x (-7, 0), then for all such function f, f(-1) + f(0) lies in the interval:

B

D

C

JEE-Main 2020, 7th Jan-I


Solution:

A

Q6. Let the function, f : [-7, 0] ⟶ R be continuous on [-7, 0] and differentiable on (-7, 0). If f(-7) = -3 and f,(x) ≤ 2, for all x (-7, 0), then for all such function f, f(-1) + f(0) lies in the interval:

B

D

C

JEE-Main 2020, 7th Jan-I


A

Q7. Let f : [ - 1, 3 ] ⟶ R be defined as

where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:

B

D

C only two points

only three points

four or more points

only one point

JEE-Main 2019, 8th April-II


Solution:

A

Q7. Let f : [ - 1, 3 ] ⟶ R be defined as

where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:

B

D

C only two points

only three points

four or more points

only one point

JEE-Main 2019, 8th April-II


A

Q8. If the function f defined on is continuous, then k is

B

D

C

JEE-Main 2019, 9th April I


Solution:

A

Q8. If the function f defined on is continuous, then k is

B

D

C

JEE-Main 2019, 9th April I


Q9.

A

B

D

C


Solution:

Q9.

A

B

D

C


Q10.

A

B

D

C


Solution:

Q10.

A

B

D

C


Homework QuestionsQ1. It the function f defined on

is continuous, then k is equal to ______.

A

Q2. If c is a point at which Rolle’s theorem holds for the function. f(x) = loge in the interval [3,4], where α R, then f,,(c) is equal to :

B D

C

Homework Questions

A B

Q3. If

is continuous at x = 0, then the ordered pair (p, d) is equal to

DC

Q4. If f : R ➝ R is a function defined by where [x] denotes the greatest integer function, then f is

A

B D

C

discontinuous only at x = 0

continuous for every real x

continuous only at x = 0

discontinuous only at non-zero integral values of x

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Thank You

Q1. It the function f defined on

is continuous, then k is equal to ______.

Solution of Homework Questions

Solution:

A


B

D

C


Solution:

A


B

D

C


A

B

Q3. If


D

C


Solution:

A

B

Q3. If


D

C



A

B

D

C






Solution:


A

B

D

C






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