Encyclopedia > Calculus with polynomials

  Article Content

Calculus with polynomials

Polynomials are perhaps the simplest functions to do calculus with. Their derivatives and integrals are given by the following rules:

<math>\frac{d}{dx} \sum^n_{r=0} a_r x^r = \sum^n_{r=0} ra_rx^{r-1}</math>
<math>\int \sum^n_{r=0} a_r x^r\,dx= \sum^n_{r=0} \frac{a_r}{r+1} x^{r+1} + c</math>

Hence the derivative of x100 is 100x99 and the integral of x100 is x101/101 + c.

Proof Because differentiation is linear, we have:

<math>\frac{d\left( \sum_{r=0}^n a_r x^r \right)}{dx} =
\sum_{r=0}^n \frac{d\left(a_r x^r\right)}{dx} = \sum_{r=0}^n a_r \frac{d\left(x^r\right)}{dx}</math>

So it remains to find <math>\frac{d\left(x^r\right)}{dx}</math> for any natural number r. The derivative of function f(x) is given by Newton's difference quotient

<math> f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} </math>

By the binomial theorem, and using the C-notation of combinations,

<math>\left(x+h\right)^r = \sum_{k=0}^r {}^rC_k h^k x^{r-k}</math>

and therefore

<math>\frac{\left(x+h\right)^r - x^r}{h} = \sum_{k=1}^{r} {}^rC_k h^{k-1} x^{r-k}</math>

The derivative is the limit of this as <math> h \rightarrow 0 </math>

<math>\frac{d}{dx}\left(x^r\right) = \lim_{h\rightarrow 0} \left(\sum_{k=1}^{r} {}^rC_k h^{k-1} x^{r-k}\right) = {}^rC_1 x^{r-1} = rx^{r-1}</math>

which gives the claimed result. Generalisation

<math>\frac{d}{dx} \left(ax^k\right) = akx^{k-1}</math>
is generally true for all values of k where xk is meaningful. In particular it holds for all rational k for values of x where xk is defined.

Similarly for integration, see Table of integrals.


All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Government of Israel

... a 5-year term. Since August 2000, this post has been filled by Moshe Katsav. The prime minister (head of government) exercises executive power and is selected by th ...