# Calculus

Calculus is used to study the mathematical changes. It has two branches in major namely differential calculus and integral calculus.

Differential calculus deals with the studies of rates of change, slopes of various curves while integral calculus deals with the studies of accumulation of various quantities, the areas between curves, the areas under curves. Both differential and integral calculus make use of the concept of convergence of infinite series and sequences to a properly well-defined limit. Also both of these branches that is differential and integral are related to each other with the fundamental theorem stated in calculus.

## Fundamental Theorem

The fundamental theorem of calculus has two parts:

The first part states that an indefinite integral of a given function an easily be reversed by differentiation. The second part of the theorem states that the definite integral of a given function can easily be computed by using any of the anti-derivatives out of the infinitely many that it has.

In general calculus is allowing us to go from the non-constant rates of change towards the total change and vice versa. Due to its varied use calculus is used widely in almost all branches of physical sciences, computer science, engineering, actuarial science, statistics, business, economics, demography, medicine and many more.

Differential Calculus:

It is the study of the derivative of a given function. The derivative of a function is defined as the measure of sensitivity to the change of a quantity. The process in which we find derivative is termed as differentiation.

The first derivative of a function at a point gives the slope of the line that is tangent the given function at that particular point. It is denoted by $\frac{d}{dx}$ $(f(x)) = f’(x)$. For finding derivative of a function some rules are set to make it easy to differentiate any function.

Integral Calculus:

The integral calculus is the study of two related concepts the indefinite and the definite integral along with their definitions, properties and also the applications. The process in which we find integral of any given function is termed as integration. Indefinite integral is also the anti-derivative, that is, the inverse operation being applied to a derivative.

The definite integral inputs a function and simply results outputting a number which also gives the algebraic sum of the areas lying between the graph of the input function and the x- axis. Again for finding integrals of the function given, some rules are set that makes it easy to find definite or indefinite integrals as required.

## Problems

Let us see some problems on differential or integral calculus.

Example 1: Find the derivative of sin x. cos x.

Solution: By using rules of differentiation, we have,

$\frac{d}{dx}$ $(sin x\ cos x)$ = $cos x (cos x)$ + $sin x (-sin x)$

= $cos^2 x$ - $sin^2 x$

Example 2: Find the integral of $x^3$ + 6$x$ + 7

Solution: By using laws of integration, we have,

$\int(x^3 + 6 x + 7) dx$ = $\frac{x^4}{4}$+ 6 $\frac{x^2}{2}$ + 7$x$ + C

where C is the constant of integration.

$\int(x^3 + 6 x + 7) dx$ = $\frac{x^4}{4}$ + 3 $x^2$ + 7$x$ + C

Example 3: The marginal cost function $g'(C)$ = 1 + 10$x$ + 32$x^{3}$ where $x$ is the output. Find the total cost, average cost, variable cost and average variable cost, given that fixed cost is 100.

Solution: Fixed cost = 100

Total cost = $\int$ (Marginal cost) dx

= $\int$ (1 + 10x + 32x$^{3}$)$dx$

= x + $\frac{10x^{2}}{2}$ + $\frac{32x^{4}}{4}$ + $C$

= x + 5$x^2$ + 8$x^4$ + C

Total cost = x + 5$x^2$ + 8$x^4$ + 100

Total variable cost = x + 5$x^2$ + 8$x^4$

Average cost = $\frac{Total\ \ cost}{x}$

= 1 + 5x + 8$x^{3}$ + $\frac{100}{x}$

Average variable cost = 1 + 5x + 8$x^{3}$