Continuity and Differentiability
Chapter 5 of the NCERT textbook starts with the definition of continuity. Students go on to learn about continuity, the algebra of continuous functions, the definition and meaning of differentiability, derivatives of composite functions, the derivative of implicit functions, the derivative of inverse trigonometric functions, derivative of exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second-order derivatives, mean value theorem via miscellaneous examples. Here, students can find the exercises explaining these concepts properly with solutions. Continuity and differentiability, the derivative of composite functions, chain rule, the derivative of inverse trigonometric functions, and the derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation is the derivative of functions expressed in parametric forms. Second-order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.
Continuity: Continuity of function at a point: Geometrically we say that a function y=f(x) is continuous at x=a if the graph of the function y=f(x) is continuous (without any break) at x = a.
A function f(x) is said to be continuous at a point x = a when:
f(a) exists i.e. f(a) is finite, definite and real.
[Tex]\lim_{x\to a}f(x) [/Tex] exists.
[Tex]\lim_{x\to a}f(x)=f(a) [/Tex]
A function f(x) is continuous at x = a if
[Tex]\lim_{h\to 0}f(a+h)=\lim_{h\to 0}f(a-h) = f(a) [/Tex]
where h→0 through positive values.
Continuity of a Function in a Closed Interval: A function f(x) is said to be continuous in the closed interval if it is continuous for every value of x lying between a and b continuous to the right of a and to the left of x = b i.e. [Tex]\lim_{x\to a-0}f(x)=f(a)\text{ and }\lim_{x\to b-0}f(x) = f(b) [/Tex]
Continuity of a Function in an Open Interval: A function f(x) is said to be continuous in an open interval (a,b) if it is continuous at every point in (a,b).
Discontinuity (Discontinuous function): A function f(x) is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.
The sum, difference, product, and quotient of continuous functions are continuous. i.e. if f and g are continuous functions, then
- (f ± g) (x) = f(x) ± g(x) is continuous
- (f . g) (x) = f(x) . g(x) is continuous
- {f/g)(x)=f(x)/g(x) (provided g (x) ≠ 0) is continuous)
Chain Rule: If f = v o u, t = u (x), and if both dt/dx and dv/dx exist, then: df/dx = dv/dt. dt/dx
Addition Rule: (u±v)′ = u′ ± v’
Product Rule: (uv)′ = u′v + uv’
Mean Value Theorem
Mean Value Theorem states that,
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) = (f(b)−f(a))/(b−a)
Rolle’s Theorem
Rolle’s Theorem states that,
If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) whereas f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0.
Lagrange’s Mean Value Theorem
Lagrange’s Mean Value states that,
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 [Tex]f'(c)=\dfrac{f(b)-f(a)}{b-a} [/Tex]
Learn more about, Rolle’s and Lagrange’s Mean Value Theorem
Derivatives of Some Standard Functions
The derivative of some standard functions are,
[Tex]\begin{aligned}\dfrac{\mathrm{d}}{\mathrm{d}x}(x^n)&=nx^{n-1}\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin x)&=\cos x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\cos x)&=-\sin x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\tan x)&=\sec^2 x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\cot x)&=-\cosec^2 x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\sec x)&=\sec x \tan x\\\dfrac{\mathrm{d}}{\mathrm{d}x}\cosec x&=-\cosec x \cot x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(a^x)&=a^x\log_e a\\\dfrac{\mathrm{d}}{\mathrm{d}x}(e^x)&=e^x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\log_e x)&=\dfrac{1}{x}\end{aligned} [/Tex]
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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