Differential Calculus

Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Differential helps in the study of the limit of a quotient, dealing with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The notations dy and dx are known as differentials. The process used to find the derivatives is called differentiation. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x).

Limits

Limit helps in calculating the degree of closeness to any value or the approaching term. A limit is normally expressed using the limit formula as,

lim_x→cf(x) = A

This expression is read as “the limit of f of x as x approaches c equals A”.

Derivatives

Derivatives represent the instantaneous rate of change of a quantity with respect to the other. The derivative of a function is represented as:

lim_x→h[f(x + h) − f(x)]/h = A

Continuity

A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied –

• f(a) is defined
• lim_x→af(x) exists
• lim_x→a⁻ f(x) = lim_x→a⁺ f(x) = f(a)

Continuity and Differentiability

A function is always continuous if it is differentiable at any point, whereas the vice-versa for this condition is not always true.

Integral Calculus

Integral calculus is the study of integrals and the properties associated to them. It is helpful in:

• calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval.
• calculating the area under a curve for any function.

Integration

Integration is the reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as a collection of small parts in order to form a whole. It is generally used for calculating areas.

Definite Integral

A definite integral has a specific boundary or limit for the calculation of the function. The upper and lower limits of the independent variable of a function are specified. A definite integral is given mathematically as,

∫_a^b f(x).dx = F(x)

Indefinite Integral

An indefinite integral does not have a specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:

∫ f(x).dx = F(x) + C