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Calculus is a very important branch of mathematics. The different branches of mathematics have been

enriched a lot by the application of calculus.

1)    Differential calculus

·         Real number

·         Function

·         Limit

·         Continuity

·         Differentiation

·         Second order derivative

2)    Integral calculus

·         Indefinite integral

·         Method of substitution

·         Integration by parts

·         Integrals of some special form of functions

·         Definite integral

3)    Differential equation

·         Order and degree of differential equations

·         Differential equations of the first order and of the first degree

·         Linear second order differential equations with constant coefficients

4)    Application of calculus

·         Significance of derivatives

·         Tangent and normal

·         Maxima and minima

·         Definite integral as an area

·         Velocity and acceleration

Few examples of advance calculus problem: -

Example 1: - Find the derivatives of y w.r.t. x when y = (x + 1) (x – 1)

Solution: - Differentiating both sides w.r.t. x,

dy / dx = (x + 1) d / dx ( x – 1) + (x – 1) d/dx (x + 1)

= (x + 1) [ d / dx (x) – d / dx (1)] + (x – 1) [d/dx (x) + d/dx (1)]

= (x + 1) (1 – 0) + (x – 1) (1 – 0)

= (x+1) + (x -1)

= x + 1+ x – 1

= 2x

Therefore dy /dx = 2x

Example 2: Evaluate ∫ x ^4 dx

Solution: -

∫ x ^4 dx = x^ (4 + 1) / (4+ 1)

= x^5 / 5

Therefore ∫ x ^4 dx = x^5 / 5