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 such that for any we have the line tangent to that point satisfy .
(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 ? Spoiler below!
Well, we set up the equation describing the tangent line
We can rewrite this to be
Or equivalently for any .
Lets try using the tangent line as an approximation for this curve. So for small we write
But since we rewrite the right hand side as
What to do?
Assumption: Lets write .
We can have a coarse approximation
So for small we have
We could also break this down into two steps:
Why can we say this? Well, because
and plugging this into (4) gives the correct relation.
We could get a better approximation iterating this three times:
We could do this times:
Taking the limit we obtain the definition
For sufficiently large , we have for any the quantity be really small. So Equation (5) works for any real number.
What is this ? It is precisely the exponential function, usually written .
Exercise 1. Prove .
Exercise 2. What if we drop our assumption and allow be some nonzero real number? What if we let it be zero?
Exercise 3. Prove for any positive real number .
Exercise 4. Calculate to 7 digits of precision. (This mathematical constant “” is called “Euler’s constant”.) [Hint: we need for digits of precision to evaluate the limit with some .]
Exercise 5. Lets prove some inequalities involving ! Prove the following:
(d) for for
(f) for and (i.e., is a positive integer)
(g) for ,
(h) for .