Question

How do you evaluate the function `f(x)=e^x` at the value of `x=3.2`?

Answer 1

`f(3.2) approx 24.53253`

Explanation:

`f(x) = e^x`

`e^x` is a transcendental function meaning that is both irrational and cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Hence, `e^x` can never (except in the trivial case `x=0`) be expressed as a fraction, the root of any polynomial with rational coeffients or the sum of any finite series. Thus, it can only ever be approximated by a number of any base.

Several definitions of `e^x` exist. Two of the most well known of these are:

`e^x = lim_(n->oo) (1+x/n)^n` (limit known to exist `forall x in RR`)

`e^x = sum_(n=0) ^oo (x^n)/(n!)` (sum known to converge `forall x in RR`)

From the second definition above we can approximate `e^3.2` as a decimal as follows:

`e^3.2 = 1 + 3.2 + 3.2^2/(2!)+ 3.2^3/(3!) + 3.2^4/(4!) + .......`

`approx 24.53253`