Continuity And Differentiability Class 12 Notes- Chapter 5

CBSE Class 12 Maths Chapter 5 Notes : Continuity and Differentiability

List of Content
What is a continuous Function ?
Chain Rule
Logarithmic Differentiation
What is Rolle’s Theorem?
What is Mean Value Theorem?

What is a Continuous Function?

When in a function, the real value at a point is said to be continuous when at that point, the function of that point is equal to the limit of the function at that point. The continuity exists when all of the domain is continuous.

The difference, product, and quotient are continuous when it comes to continuous function. Let’s understand with the help of an example.

(f\(\pm\)g)(x)=f(x)\(\pm\)g(x) is said to be continuous.

(f.g)(x)=f(x).g(x) is again said to be continous.

(f/g)(x)=f(x)/g(x) and g(x) \(\neq\) 0 and is said to be continous.

All the functions which are differential are said to be continuous but the vice versa is not true.

Chain Rule

The composite of the functions can be differentiated with the help of chain rule. If f=v, t=u(x)
Then the existence of \(\frac{\partial t}{\partial x}\) and \(\frac{\partial v}{\partial x}\) can be witnessed, then, \(\frac{\partial f}{\partial x}=\frac{\partial v}{\partial t}.\frac{\partial t}{\partial x}\)

Logarithmic Differentiation

When the differential equation is in the form \(f(x)=[u(x)]^{v(x)}\). Here, the positive values of f(x) and u(x) is considered.

What is Rolle’s Theorem?

Let us consider, a continuous function f:[a,b]\(\rightarrow\) R which is continous on the point [a,b] and differentiable on the point (a,b) then , f(a)=f(b) and some external point exists such as c in (a,b) such that f'(c)=0.

What is Mean Value Theorem?

Let us consider, a continuous function f:[a,b]\(\rightarrow\) R which is continuous on the point [a,b] and differentiable on the point (a,b), some external point exists such as c in (a,b) such that

\(f'(c)=f(b)-f(a)/b-a\)

 

Important Questions

  1. If \(\cos y=x\cos (a+y), with \cos a\neq \pm 1, prove \; that \frac{\partial y}{\partial x}=\cos ^{2}(a+y)/\sin a\)
  2. If \(x=a(\cos t+t\sin t) and, y=a(\sin t-t\cos t), find \frac{\partial^2 y}{\partial x^2}\)
  3. If \(f(x)=\left | x \right |^{3}\) show that f’’(x) is available for the value of x and find it.
  4. With the help of mathematical induction, prove that \(\frac{\partial }{\partial x}(x^{n})=nx^{n-1}\) for integers of n anf positive.
  5. Obtain the same formula as cosines when \(\sin (A+B)=\sin A\cos B+\cos A\sin B\)

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