Introduction to Elementary calculus

Introduction to Elementary calculus

Before going to understand the problems on continuous random variables, we need to know some fundamental knowledge about differentiation and integration, which are part of calculus in higher mathematics.

Hence, we introduce some simple concepts, techniques and formulae to calculate problems in statistics, which involve calculus.

Differentiation

We Studied about functions and functional values in earlier classes. Functional value is an exact value. For some function f(x), when x = a, we obtain the functional value as f (a) = k.

Another type of approximation gives the very nearest value to the functional value is known as limiting value. So the limiting value is an approximate value. This limiting value approaches the nearest to the exact value k.

Suppose the exact value is 4, the limiting value may be 4.00000001 or 3.999999994. Here we observe that the functional value and the limiting value are more or less the same and there is no significant difference between them.

Hence in many occasions we use the limiting values for some critical problems.

The limiting value of f (x) when x approaches a number 2 is denoted by lim_x→2 f(x) = f(2) = l (some existing value)

The special type of any existing limit,  is called the derivative of the function f with respect to x and is denoted by f’ (x) . If y is a function of x, and has a derivative, then the differential coefficient of y with respect to x is denoted by dy/dx. This process of finding the limiting value is known as differentiation.

Some rules on differentiation:

(i) Derivative of a constant function is zero.

i.e., f´(c) = 0, where c is some constant.

(ii) If u is a function of x and k is some constant and dash denotes the differentiation,

[k u]´ = k[u]´

(iii) (u ± v)´ = u´ ± v´

(iv) (u v)´ = u´v + u v´ (product rule)

(v)   (quotient rule)

Important formulae:

( i) (xⁿ) ´ = n x^n-1

(ii) (e^x)´ = e^x

(iii) ( log x) ´ = 1/x

Example 7.19

Evaluate the following limits:

Solution:

Example 7.20

Find the derivative of the following with respect to x:

Solution:

Repeated differentiation:

If the derivative of a function is again differentiated with respect to the same variable, we say that the differentiation is the second order differentiation and is denoted as

Integration

Integration is the reverse process of differentiation.  It is also called anti-derivative.

Suppose the derivative of x⁵ is 5x⁴. Then the integration of 5x⁴ with respect to x is x⁵. we use this in symbol as follows:

Therefore for the above examples, we usually write 

These integrals are also called improper integrals or indefinite integrals

Rules and some formulae on integration:

Example 7.21

Integrate the following with respect to x.

Solutions:

The above discussed integrals are known as improper integrals or indefinite integrals . For the proper or definite integrals we have the limiting points at both sides. These are called the lower limit and the upper limit of the integral.

This integral ∫(f)dx  is an indefinite integral. Integrating the same function with in the given limits a and b is known as the definite integral. We write this in symbol as

In a definite integral where a is known as the lower limit and b is known as the upper limit of the definite integral. To find the value of definite integral, we do as follows:

Example 7.22

Evaluate the following definite integrals:

Solutions:

Double integrals

A double integral is an integral of two variable function f(x,y) over a region R If R=[a, b] × [c, d] then the double integral can be done by iterated Integration( integrate first with respect to y and then with respect to x )

The notation used for double integral is  

Here the function f(x,y) is integrated with respect to y first and treat f(x) constant and then integrate with respect to x and apply limits of x and simplify

Example 7.23

Solutions:

Let us first integrate with respect to y and then with respect to x. Hence the double integral is written as

Example 7.24

Solution:

Example 7.25

Solution: