Class 12 Mathematics Chapter 5 Notes
Mathematics can be easy or difficult, depending on how well a student understands the concepts. It is essential to have a strong foundation so that it becomes easier to grasp knowledge about high-level topics. This subject cannot be mastered with memorisation. To excel in Mathematics, students need to practice regularly. The Class 12 Mathematics chapter 5 notes- Continuity and Differentiability give a clear understanding of different concepts, such as the algebra of continuous functions. It also includes differentiability of different functions such as inverse trigonometric, implicit, composite and more. In this chapter, students will learn about second-order derivatives and the Mean Value theorem.
Extramarks, an online learning platform, provides one of the best academic materials to help students understand concepts and ace their exams. The Class 12 Mathematics notes chapter 5 help to understand the chapter. Differentiability and continuity are crucial concepts that help students learn other mathematical branches. The notes act as a helping guide and aid in quick revision.
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List Of The Topics Covered In Class 12 Mathematics Chapter 5 Notes
The list of topics covered in Extramarks class 12 Mathematics notes chapter 5 Continuity and Differentiability include the following.
- Introduction
- Continuity and Algebra of Continuous Function
- Differentiability
- Derivatives of Composite, Implicit and Inverse Trigonometric Function
- Logarithmic and Exponential Function
- Logarithmic Differentiation
- Parametric Form Differentiation
- Second-Order Derivative
- Mean Value Theorem and Rolle’s Theorem
Key Topics Covered In Class 12 Mathematics Chapter 5 Notes
The key topics covered under Class 12 Mathematics Chapter 5 notes include
Definition of Continuity at a point:
The continuity of a function (f) at any point x = c, where c belongs to the domain of the function, is given as, Left-hand limit of f(x = c) = Right-hand limit of f(x = c) = Value of f(x=c) , i.e.,
If x = c, then LHL = RHL = f(c).
Therefore, xa–f(x) = xa+f(x) = f (x = c) = f(c)
REMEMBER:
- To find the solution for LHL of f(x) at x=0, substitute x = a – h and for RHL, substitute x = a + h.
- If xa–f(x) and xa+f(x) exists but LHL RHL, then the function is not continuous.
- If xa–f(x) and xa+f(x) exists but LHL = RHL f(c), then the function is discontinuous.
- At least one limit of the f does not exist; then, the function is discontinuous.
Definition of Continuity in an Interval (INTERMEDIATE VALUE THEOREM):
The continuity of a function (f) in an interval [a, b] where a<b then, if the function is continuous at all points, including the end-points in the given interval.
Continuity at point ‘a’ is given as,
xaf(x) = f (a)
Continuity at point ‘b’ is given as,
xbf(x) = f (b)
Algebra of Continuous Functions
If f(x) and g(x) be two real functions. If they are continuous at any point c R, then
- f(x) + g(x) is continuous at c.
- f(x) – g(x) is also continuous at c.
- f(x) . g(x) is continuous at c.
- f(x) / g(x) is continuous at c, g(c) 0.
- The product c.f(x) is continuous, where c is a constant.
- Consider a composite function fog(x), if g (x) is continuous at x and f(x) is continuous at g(c), then fog(x) is continuous at point c.
- If f(x) is continuous, then f(x) is also continuous.
Definition and formula of differentiability:
If f(x) is a real function. It is said to be differentiable at a point c in its domain if the left-hand derivative is equal to the right-hand derivative of the function at point c, i.e.
LHD f(x=c) = RHD f(x=c).
The derivative of a function is given ash0f(c+h) – f(c)h
Therefore, RHD f’(c) = h0f(c+h) – f(c)h and LHD f’(c) = h0f(c-h) – f(c)-h
Types of Discontinuity:
REMOVABLE DISCONTINUITY:
If xaf(x) exists but is not equal to f(c), for c R, then function f(x) has removable discontinuity. In such a case, we can redefine the function and make it continuous.
1. Missing point discontinuity:
In this type of discontinuity, xaf(x) exists but f(c) does not exist.
2. Isolated point discontinuity:
In this type of discontinuity, xaf(x) exists and f(c) exist, but xaf(x) f(c).
NON-REMOVABLE DISCONTINUITY:
In this case, xaf(x) does not exist, and so the function cannot be redefined to make it continuous.
1. Finite Discontinuity:
This type of discontinuity exists when the LHL and RHL limits do not exist, but their values are both finite, yet LHL RHL.
2. Infinite Discontinuity:
This type of discontinuity exists when either LHL or RHL or both limits tend toward infinity.
3. Oscillatory Discontinuity:
This type of discontinuity exists when limits oscillate between two finite values.
Relation between continuity and differentiability:
- Every f(x) is differentiable; then it is also continuous. But, if f(x) is continuous, then it is not differentiable.
- A function that is not continuous (discontinuous) will not be differentiable.
- If f is function it is differentiable a a point x = c, then f is also continuous at x = c.
REMEMBER:
Differentiable Continuous
Not Differentiable Not Continuous
Not Continuous Not Differentiable
Derivatives of inverse functions:
If f and g are two real-valued functions such that y = f(x) and x= g(y) then
g(f(x)) = x ddx g(f(x)) = g’(f(x)). f’(x) = 1
Using this result, we can say that dydx . dxdy= 1 or dydx = 1dxdy or dxdy = 1dydx where dxdy0
Rules of Differentiation:
1. Sum and difference rule:
The derivative of y= f(x) g(x) is given as
dydx= ddyf(x) ddy g(x)
2. Product Rule:
The derivative of y= f(x). g(x) can be remembered with the help of the u-v rule. It is defined as
ddxu.v = (ddxu) v + (ddxv) u i.e.,
dydx= (ddyf(x)) g(x) + (ddyg(x)) f(x)
3. Quotient Rule:
The derivative of y= f(x) / g(x) can be remembered with the help of the the u-v rule. It is defined as
ddx(uv) = u (ddxv) + v (ddx u)v2 i.e.,
ddx(f(x)g(x)) = (ddxg(x)) f(x) + (ddx f(x)) g(x)g(x)2
4. Chain Rule:
i) for two functions:
Let y = f(u), u= f(x) and if dduy and ddxu exists then,
We can write ddxy = dduy . ddxu
ii) for three functions:
Let y = f(u), u= f(w), w= f(x) and if dduy, ddwu and ddxw exists then,
We can write ddxy = dduy . ddxu. ddxw
1. Differentiation in Parametric Form:
If x and y are expressed in the form x = f(t), y = g(t), where t is a parameter. It is given as
ddxy = ddtyddtx, provided ddtx 0.
2. Logarithmic Differentiation:
Consider y = f(x)g(x). This function can be treated in the following way:
Taking loge on both sides, we get loge y = g(x) loge f(x). The derivation can further be calculated by using the chain rule, given as
ddxy= f(x)g(x) g(x)f(x)f'(x) + g'(x) log f(x)
Second-Order Derivative:
It is denoted by y2 or y’’.
The derivative of the first-order derivative of a function is called the second-order derivative.
d2dx2y= ddx(ddxy)
Derivative of infinite series:
If y = f(x) +f(x) +f(x) +…. ∞
Then y can be treated as y= f(x) + y
By differentiating, we get, (2y-1) ddxy = f’(x)
Rolle’s Theorem:
If function f:[a, b] R is continuous of [a, b] and differentiable on (a, b) and f(a) = f(b), then at least one number c in the interval (a, b) such that f’(c) = 0, a,b,c R.
Lagrange’s Mean Value Theorem:
It is an expansion of Rolle’s Theorem. It states that if function f:[a, b] R is continuous of [a, b] and differentiable on (a, b) and f(a) = f(b), then at least one number c in the interval (a, b) such that
f’(c) = f(b) – f(a)b – a, where a,b,c R.
Useful substitutions for finding derivatives:
Expression: |
Substitution: |
1)a2 + x2 |
x = a tan or x = a cot |
2)a2 – x2 |
x = a sin or x = a cos |
3) x2 – a2 |
x = a sec or x = a cosec |
4) a – xa + x or a + xa – x |
x = a cos 2 |
5) a2 – x2a2 + x2 or a2 + x2a2 – x2 |
x2= a2 cos 2 |
Derivatives of Trigonometric functions:
Expression: |
Derivatives: |
Expression: |
Derivatives: |
ddxsin x |
cos x |
ddxsec x |
sec x . tan x |
ddxcos x |
– sin x |
ddxcosec x |
– cosec x . cot x |
ddxtan x |
sec2x |
ddxcot x |
– cosec2x |
Derivatives of Inverse Trigonometric functions:
ddxsin-1x |
11 – x2 |
ddxcosec-1x |
1x x2 – 1 |
ddxcos-1x |
-11 – x2 |
ddxsec-1x |
-1x x2 – 1 |
ddxtan-1x |
11 + x2 |
ddxcot-1x |
-11 + x2 |
Derivatives of standard functions:
ddxxn |
n.xn-1 |
ddx (constant) |
0 |
ddxex |
ex |
ddxLog x |
1x, where x 0 |
ddxax |
ax log a, where a > 0 |
|
|
List of Continuous Functions:
Function f(x) |
Interval in which f is continuous |
Constant c |
(-∞, ∞) |
xn, n is an integer, n 0 |
(-∞, ∞) |
x-n, x is a positive integer |
(-∞, ∞) – {0} |
x-a |
(-∞, ∞) |
P (x) = a0xn+ a1xn-1+ a2xn-2+…..+ an |
(-∞, ∞) |
p(x)q(x), where p(x) and q(x) are polynomial in x |
(-∞, ∞), provided q(x) 0 |
sin x |
(-∞, ∞) |
cos x |
(-∞, ∞) |
tan x |
(-∞, ∞) – {(2n +1)2: n I |
cot x |
(-∞, ∞) – {n: n I} |
sec x |
(-∞, ∞) – {(2n +1)2: n I |
cosex x |
(-∞, ∞) – {n : n I} |
ex |
(-∞, ∞) |
Log x |
(0, ∞) |
In addition to Class 12 Mathematics chapter 5 notes, students may also access various other study materials on Extramarks pertaining to Continuity and Differentiability.
Class 12 Mathematics chapter 5 notes Exercises & Answer Solutions.
Chapter 5 Mathematics class 12 notes begin with a recall of all concepts students learnt in Class 11. To revise the previse topics, students can refer to CBSE Solutions for class 11. With the help of the notes, students can learn in detail about Continuity and Differentiation. Topics such as second-order derivatives, Rolle’s theorem, and MVT theorem are explained in detail in the Class 12 Mathematics chapter 5 notes. Students will also gain knowledge about Derivatives of standard functions and Continuous Functions.
To help students gain knowledge of the chapter, Extramarks provides detailed and well-structured Class 12 Mathematics chapter 5 notes. All important definitions, formulas, properties, and theorems are included in the notes. With the help of the CBSE solutions, it will become easier for students to tackle the most difficult questions in no time. The step-by-step solutions in the Class 12 Mathematics chapter 5 notes help students to understand each step and how to solve the problems ahead. It empowers students to get closer to pursuing their dream careers.
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Key Features of Class 12 Mathematics chapter 5 notes
The key features of Extramarks Class 12 Mathematics Chapter 5 notes include
- Class 12 Mathematics chapter 5 notes are focused on the fundamental basic concepts.
- The notes strictly follow the CBSE Syllabus and the latest guidelines given by the board.
- It is curated by the in-house team of experts at Extramarks.
- Extensive research is carried out to provide apt knowledge and authentic information to every student in an easy language and detailed manner.
- It enables a thorough understanding and aims to clear all the doubts.
- The students get an idea of the type of questions that will be asked in the CLass 12 board exams.
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