Exponential Function

So we threw down a challenge at the end of our discussion of Tangent Lines, lets solve it.

Recall we wanted to find a curve y=f(x) such that for any (x_{0},f(x_{0})) we have the line tangent to that point t_{0}(x) satisfy t_{0}(x_{0}-1)=0.

(Historical note: Debeaune asked Descartes this problem in a letter in 1638; Leibniz solved this problem in 1684 using “infinitesimals”. We will use modern calculus to solve the problem!)

So what is f(x)=? Spoiler below!

Well, we set up the equation describing the tangent line
(1)\displaystyle0=f(x_{0})+f'(x_{0})\cdot(x_{0}-1-x_{0})
We can rewrite this to be
(2)\displaystyle 0=f(x_{0})-f'(x_{0}).
Or equivalently f'(x_{0})=f(x_{0}) for any x_{0}.

Lets try using the tangent line as an approximation for this curve. So for small h we write
(3)\displaystyle f(x+h)\approx f(x)+f'(x)\cdot h.
But since f'(x)=f(x) we rewrite the right hand side as
(3′)\displaystyle f(x+h)\approx f(x)\left(1+\frac{f'(x)}{f(x)}\cdot h\right)=f(x)\cdot(1+h).
What to do?

Assumption: Lets write f(0)=1.

We can have a coarse approximation
\displaystyle f(x+h)=f(x)\cdot(1+h).
So for small h we have
\displaystyle f(h)=f(0)\cdot(1+h)=(1+h).
We could also break this down into two steps:
(4)\displaystyle f(h)=f(h/2)\cdot\left(1+\frac{h}{2}\right)=\left(1+\frac{h}{2}\right)^{2}.
Why can we say this? Well, because
\displaystyle f(h/2)=(1+h/2)
and plugging this into (4) gives the correct relation.

We could get a better approximation iterating this three times:
(4a)\displaystyle f(h)=\left(1+\frac{h}{3}\right)^{3}.
We could do this n times:
(4b)\displaystyle f(h)=\left(1+\frac{h}{n}\right)^{n}.
Taking the limit n\to\infty we obtain the definition
(5)\displaystyle f(h)=\lim_{n\to\infty}\left(1+\frac{h}{n}\right)^{n}.
For sufficiently large n, we have for any x the quantity x/n be really small. So Equation (5) works for any real number.

What is this f(x)? It is precisely the exponential function, usually written \exp(x)=\mathrm{e}^{x}.

Exercise 1. Prove \exp(a)\exp(b)=\exp(a+b).

Exercise 2. What if we drop our assumption and allow f(0) be some nonzero real number? What if we let it be zero?

Exercise 3. Prove \exp(-a)=1/\exp(a) for any positive real number a.

Exercise 4. Calculate \exp(1)=\mathrm{e} to 7 digits of precision. (This mathematical constant “\mathrm{e}” is called “Euler’s constant”.) [Hint: we need for k digits of precision to evaluate the limit with some n\geq 4\cdot10^{k}.]

Exercise 5. Lets prove some inequalities involving \exp(x)! Prove the following:
(a) \exp(x)>1+x
(b) \displaystyle \exp(x)<\frac{1}{1-x} for x<1
(c) \displaystyle \frac{x}{1+x}<1-\exp(-x)<x for -1<x
(d) \displaystyle x<\exp(x)-1<\frac{x}{1-x} for x\exp\bigl(x/[1+x]\bigr) for x>-1
(f) \exp(x)>x^{p}/p! for x>0 and p\in\mathbb{N} (i.e., p is a positive integer)
(g) \displaystyle \exp(x)>\left(1+\frac{x}{y}\right)^{y}>\exp\bigl(xy/[x+y]\bigr) for x>0, y>0
(h) \displaystyle \exp(x)<1-\frac{x}{2} for 0<x<1.

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Exponential Map, Tangent Line. Bookmark the permalink.

4 Responses to Exponential Function

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