Engineering Mathematics: Handwritten notes of Differentiation

Engineering Mathematics: Handwritten notes of Differentiation

Differentiation notes of Engineering

Rolle' Theorem

If a function f is 

• Continuous in [a,b]
• Differentiable in (a,b)
• f(a) = f(b)

Then there exists a number c belongs to (a,b) such that f'(c) = 0

Mean Value Theorem

It's of two types:

1. Lagrange's Mean Value Theorem
2. Cauchy's Mean Value Theorem

1. Lagrange's Mean Value Theorem

    If a function f is

• Continuous in [a,b]
• Differentiable in (a,b)

Then there exist a number c belongs to (a,b) such that

           f(b) - f(a)/b - a = f'(c)

2. Cauchy's Mean Value Theorem

     If f(x) and g(x) are two function

• Continuous in [a,b]
• Differentiable in (a,b)
• g'(x) doesn't vanish anywhere inside the interval

Then a point c in (a,b) such that

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

Intermediate form and L'Hospital Rule

L'Hospital Rule:

 Suppose f(x) and g(x) are two function

• Continuous in same interval [a,b]
• Differentiable in (a,b)
• g'(x) doesn't vanish anywhere inside the interval

Then, if f(a) = 0 = g(a)

        

Provided 

 exists

Intermediate form:

It may appear in different forms:

• 0/0, ∞/∞, 0×∞
• ∞-∞
• 0⁰, ∞⁰

Taylor's and Maclaurin's Theorem

Taylor's Formula:

f(x) = f(xo) + f'(xo)(x-xo) + f"(xo)(x-xo)²/2! + .......+ Rn(x)

Maclaurin's Formula:

If we set xo = 0 in the Taylor's Formula of the function f(x), then it is called as Maclaurin's Formula.