The Fundamental Theorem of Calculus
The beginner’s guide to proving the Fundamental Theorem of Calculus, with both a visual approach for those less keen on algebra, and an algebraic, slightly more rigorous approach, for those keen on exactness.
By the end, I hope you feel a bit more like a mathematician 🙂
Introduction, motivation and ‘hello!’
Hello! We are going to understand one of the most historically important and brilliant proofs in mathematics. Important and brilliant because it reduces previously impossible problems — that of integrating functions — into the art of spotting a derivative. But more on that soon.
What is wonderful about this proof is that there are two approaches, both of which complement each other, but also can be understood independently. To begin with, we will see an informal statement of the theorem, and an informal statement of the proof. This will give the intuition and ‘essence’ of what we are doing. This proof will be visual in nature, and not require excessive or complicated algebra. This part will convey some key ideas without algebra, but at the cost of being less exact. Next will come a formal statement and proof. This is optional. Why do I nevertheless encourage you to try and understand it, even if you aren’t very comfortable with ‘algebra’ proofs compared to visual proofs?
- The visual proof captures the key ideas, but the formal proof shows how mathematicians turn those ideas into mathematical objects and then prove things about the mathematical object
- Having seen the visual proof, you will have some idea what is going on in the algebra proof even if you don’t follow all the details
- Ideas in mathematics sometimes take a while to sink in. Taking time to think about something is never time wasted. At some later point the ideas will click, or be handy elsewhere. Time spent thinking about mathematics is fundamentally time well spent. Although, I am somewhat opinionated 🙂
- You’ll never know if you never try 🙂
A (very) short introduction to Derivatives (for those who haven’t encountered derivatives before)
Derivatives are about approximating functions with straight lines. The idea is that, near a point, the tangent line provides a pretty good approximation to how the function is changing.
The derivative of a line at a point can be viewed as the slope of the ‘best’ linear approximation at that point.
The derivative as the ‘best’ linear approximation near a point. Attribution: derivative work: Pbroks13 (talk)Tangent-calculus.png: Rhythm / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)
The idea is that, for many functions, a lot of the information about the function is contained in using a linear function to approximate it. Obviously, this approximation isn’t perfect, but if such an approximation holds everywhere, we learn a lot about the function: in fact we can recreate the function upto a constant term.
At the end of the article are some resources on understanding derivatives and other aspects of calculus, if you want to go into greater detail. We will also define the derivative at bit more precisely later.
The Fundamental Theorem of Calculus then tells us that, if we define F(x) to be the area under the graph of f(t) between 0 and x, then the derivative of F(x) is f(x).
Let’s digest what this means. Below is a red line — this is our function f. We want to find out the area between 0 and x — x is marked red on the x-axis. Our function F tells us, for each point on the x axis, what the area is under the curve at that point. [Please excuse my poorly drawn ‘x’]
source: Desmos.com
We want to determine what derivative of our function F is — at x. We can get a graphing calculator (I used desmos, but geogebra is also good and free) to plot F(x), which I have done below:
Graph of the function F
So this function looks like it should have a derivative. But what is it?
Imagine we look at the best line approximation to F(x) close to x. What might this look like? Well, how about we make a good guess.
Let’s suppose F(x+dx) is roughly equal to F(x) + dx*f(x)
For instance, at x = 8, we might say that F(8.00001) is well approximated by F(8) +0.00001*f(8). What is the ‘visual’ proof of this?
Let’s look at the area under the graph again.
When we use the approximation F(x+dx) roughly equals F(x) + dx*f(x) we see the following. dx*f(x) is represented by the area of the red rectangle, which has height f(x) and width dx. Is this a good linear approximation? Yes! Rewriting, our approximating function at x = 8 is F(8) + h*f(8). We also see that the rectangle contains ‘nearly’ all the area F(x) would have gained by going to F(x+dx). This can be seen below, where the area we ‘missed’ is just the small blue shaded region, which is much much smaller than the rectangular region.
However, there are several improvements we can make on this proof. Yes, it certainly looks like a good approximation on this graph, but does it work on all graphs? After all, graphs can look very different. Also, how are we defining our ‘best linear approximation’? This leads to formulating the problem using some algebra.
Part II: A Statement using Algebra
First, we want to define our derivative. This is done as follows:
[‘lim’ stands for ‘limit’]
The limit just means to look at what happens to the expression as dx gets arbitrarily close to 0. So, you might compute the following sequence and see where it ‘tends’ towards
For the functions we’re interested in, it won’t matter which sequence you pick for ‘dx’ , provided it tends to 0.
We get a nice visual feel for it in the following diagram:
My own creation!
We then see that, as dx tends to 0, the limit of the gradient of the straight line connecting F(x) with F(x+dx) is defined to be our derivative. This can be seen below.
Tangent animation.gifFrom Wikimedia Commons, the free media repositorycommons.wikimedia.org
We use a limit because, while x = 0.01 or 0.00001 may seem small to us, for a function like x¹⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰ a 0.01 difference suddenly results in a huge change in output. The limit means that dx can be made arbitrarily small so that we can always zoom in enough that our function can be approximated by a straight line.
***Note: there are some functions which cannot be approximated nicely by a straight line locally to a point no matter how far you zoom in, but these are dragons to slay on another day, with different techniques!***
Next, we want some notation to represent the area under the curve between 0 and x. We write:
***what does the f(t)dt mean? One way to look at it is f is a function of some variable t, so we integrate over t. The variable which denotes how far we integrate is x, so the upper bound of integration is x but we write f(t) as a function of t. It doesn’t matter really which variable name we use for f apart from avoiding using ‘x’ twice, because then we would have given the symbol two different meanings.***
Now, our task is to prove that:
Where, in the second line, we have just plugged in our definition of F(x) as the area under the curve, using the notation introduced above.
Part IIIa: A Proof using Algebra
We now prove
First we observe that
This is because we are only interested in the area between x and x+dx. This is seen in the diagram below, where we are really interested in the red area.
So, the problem then is to work out what the following limit is:
Here we assume the f(t) is continuous at t = x. (We can actually use a weaker assumption, but it requires more effort, as we will see in the final section)
What is the definition of continuity at x? This will take a bit of time to wrap your head around! (Read through the definition twice, then continue, as I will explain it in more informal language)
What does this mean? It means for any (small) number, we can find a small band around x where f(t) is less than that small number away from f(x). For instance, you might set the ‘small’ number to be 0.001. Then, I might find that if t is less than 0.00001 away from x, then we are guaranteed that|f(t) — f(x)| < 0.001. In this case, suppose x = 8, then |f(8) — f(8.000001)|