Subject: Calculus

Calculus With Polynomials

Any mathematical expression of variables and constants that is finite, uses only the four fundamental operations and positive integer exponents are called polynomials. We dealt with them a lot in algebra making our way to master basic algebraic operations with polynomials. In this article, we will not anymore deal with algebra with polynomials but instead, let us tackle the "calculus of polynomials".

As mentioned earlier, polynomials are any mathematical expression which can be of the form

2x^3+2x^2+x+5

or more generally,

\sum_{i=0}^{n}a_ix^{i}

as long as i is any real valued positive integer. In calculus, we are only concerned with derivative and integrals of functions and how these functions behave when applied with these "operations". So now, let me show you how to do basic calculus operations with polynomial functions.

Derivative of Polynomials

The formula for finding the derivative for any polynomial function is not difficult. For any polynomial function (such as those stated above),

\sum_{i=0}^{n}a_ix^{i}

the derivative is given by

\sum_{i=0}^{n}a_iix^{i-1}

Since a is just a constant, the derivative can also be written as

a_i\sum_{i=0}^{n}x^{i}

Say we have a polynomial function where n=3, a_0=5, a_1=1, a_2=2 and a_3=2. Then

\sum_{i=0}^{n}a_ix^{i} = a_3x^3+a_2x^2+a_1x+a_0

would become

2x^3+2x^2+x+5
.

If we get its derivative, then we have

\sum_{i=0}^{n}a_iix^{i-1} = 3 \cdot a_3x^{3-1}+ 2\cdot a_2x^{2-1}+1\cdot a_1x^{1-1}+0

or

6x^2+4x+1

Note that if we were to differentiate directly the polynomial function 2x^3+2x^2+x+5 using the linearity of differentiation rule, we would still get 6x^2+4x+1.

Derivatives of polynomial functions are important in mathematics and sciences. Derivatives determine the "degree of steepness" or slope and so with calculating the "rate of change" of a polynomial function.

Anti-derivative of Polynomials

Integrals of polynomial functions is achieved by the formula below. For any polynomial function

\sum_{i=0}^{n}a_ix^{i}

the anti-derivative is then given by

\sum_{i=0}^{n}\frac{a_i}{i+1}x^{i+1} + C

where C is a constant commonly called as the "constant of integration". We can easily verify the validity of this formula by taking into principle the fact that the inverse of differentiation is integration. So if we apply this formula to the derivative of the polynomial function we just had solved above, we should get back to the original function.

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