What is the significance of this equation?

I am studying pre-calc right now, and I bought a book called The Cartoon Guide To Calculus just to get started on the real thing. Despite the name, the book is really good and so far I've understood really well.

They introduced something that they called "The Fundamental Equation of Calculus". (Nobody else calls it that, though.)

$$\Delta f = h\frac{df}{dx} + FLEA$$ (where $x$ is a $FLEA$ if $\lim \limits_{h \to 0}\frac{x}{h} = 0$)

Despite giving this such a grandiose name, they didn't explain why it was so important. Can anyone explain what this equation means and why it's so important?

• The actual 'fundamental equation of calculus' is $f(x+ε)=f(x)+εf'(x)$; the most common name for it is the Taylor formula. Rearrange it for f'(x) if you want to calculate the gradient. Also note - it works for secants as well as tangents. – user117644 Mar 11 '16 at 5:45

I don't know what they meant by FLEA.

But $\Delta f$ means the change of the function in a short period of time. For example if the function is $f(x) = x^3 - 2x^2 + 3x + 1$ and we want to see the change if $f(x)$ from $x = 2$ and $x + h = 2.01$ That'd be $\Delta f(2.01) - f(2) = [(8.120601) - 2(4.0401) + 3(2.01) + 1] - [8 - 2*4 + 3*2 + 1] = .120601 - .0802 + .03 = .070401$ where $x = 2$ and $h = .01$.

Now the derivative of $f(x)$ is the formula for the rate of change of $f$ and the precise point $x$. So $f'(x) = 3x^2 - 4x + 3$ means at $x = 2$ the function is changing at a rate of $f'(2) = 3*4 - 4*2 + 3 =7$ units (in the "y-direction") for every unit (in the "x-direction").

Thus in $h$ units the function $f(x)$ should increase by $h*f'(x)$ units. So at $x = 2$ and $h = .01$ the function should increase by $0.01*7 = .07$. And indeed as we saw the function did increase by $0.070401$. That's close to $h*f'(x)$.

So ... now I think I do know what that mean be FLEA. $f'(x)$ is the rate of change at precisely $x$. $f(x)$ is constantly changing as is the rate of change. You can use the rate of change (the derivative) to approximate where the function will be at $x + h$ but as the function is changing there will be a slight error. This error is FLEA.

So $\Delta f = f(x + h) - f(x) = h*f'(x) + FLEA$. This is a fundamental formula for expressing changes in the value of $f$ in terms of the derivative/(intantaneous rat of change).

And in our example $\Delta f = f(2 + .01) - f(.01) = .070401 = (0.01)f'(2) + FLEA = 0.01*7 + FLEA = 0.07 + FLEA = 0.070401$. (FLEA = 0.000401).

• upvote because you have previously established your relationship to fleas. – Barry Smith Mar 11 '16 at 6:06
• And, because $\lim_\limits{h \to 0} \frac{FLEA}{h} = 0$, the $FLEA$ (the error margin) will get smaller and smaller as $h \to 0$. – naiveai Mar 11 '16 at 6:14
• The only reason I upvoted it is because you have flea in your name as well... Oh wait.... Someone already mentioned that :3 – Karan Singh Apr 7 '16 at 11:53

This is the canonical calculus picture. At the point $(x, f(x))$, on the graph of $y = f(x)$, draw a tangent line (the green line). The picture shows how well the tangent line approximates the curve $y = f(x)$ for small values of $h$. My guess would be that FLEA is the Fundamental Linear Error of the Approximation. This is sometimes taken as the definition of derivative:

$g(x)$ is the derivative of $f(x)$ if $f(x+h) =f(x)+h\,g(x)+o(h)$ as $h \to 0$.

Another way to write this is $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}-g(x) =0$.

You can also think of the derivative of $f(x)$ as the slope of the line through $(x, f(x))$ that best matches $f(x)$. The nice thing about this description is that if you generalize "best fitting line" to "best fitting polynomial of degree m", you end up with Taylor or Maclaurin series.