This chapter read the signs of calculus one by one. The guide below gathers the questions that help read them together.

Five questions for calculus

  1. What varies, and with respect to what? In $\frac{dy}{dx}$, both letters are written. In $f'$, the variable is implicit; in $\dot x$, it is time. If it is not clear, write it down.
  2. Is it a finite difference or a limit? $\Delta$ and $\Sigma$ are calculations with numbers; $d$ and $\int$ are what remains when the calculation shrinks without end. The shape looks alike; the meaning changes.
  3. Does cancelling hold here? With one variable and one order of differentiation, $\frac{dy}{dx}$ behaves like a fraction. With second derivatives or partial derivatives, check before cancelling.
  4. Which letter is a dummy? In the integral, the letter of the $d$ is bound; in the limit, the letter of the arrow is too. Neither may appear outside.
  5. Is the $\infty$ a direction or a quantity? In $x\to\infty$ and $\sum^\infty$, it is a direction, and calculations with it need a limit. Only Cantor’s $\aleph$s are quantities.

A complete reading of an equation

$$\frac{d}{dx}\int_0^{x}e^{-t^2}\,dt=e^{-x^2}.$$
  1. The integral $\int_0^xe^{-t^2}\,dt$ is the area under the bell-shaped curve, from $0$ to $x$. It is a function of $x$: each $x$ gives an area.
  2. The $t$ is a dummy letter. Replacing it with $s$ changes nothing. Replacing it with $x$ would be a mistake, because $x$ is already in the limit.
  3. The $\frac{d}{dx}$ is an operator: it asks for the rate at which that area grows as $x$ grows.
  4. The $=$ states an identity, valid for every $x$. It is the first part of the fundamental theorem.
  5. The remarkable detail: the integral $\int_0^xe^{-t^2}\,dt$ has no formula in elementary functions. Even so, its derivative is written in one line. The notation lets you talk about a function without knowing how to write it.

Glossary

  • Difference ($\Delta$). Finite change in a quantity: $\Delta y=f(x+\Delta x)-f(x)$.
  • Differential ($dy$). Change along the tangent line: $dy=f'(x)\,dx$. Approximates $\Delta y$ when the step is small.
  • Derivative. Limit of $\frac{\Delta y}{\Delta x}$ as $\Delta x\to0$. Written $\frac{dy}{dx}$, $f'(x)$, $\dot y$ or $Dy$.
  • Operator. Rule that turns one function into another, such as $D$, which differentiates. It can be added, multiplied and, with care, factorised.
  • Characteristic equation. Polynomial obtained by replacing $D$ with $r$ in a linear differential equation with constant coefficients.
  • Riemann sum. Sum $\sum f(x_i)\,\Delta x$ of areas of rectangles that approximate an integral.
  • Definite integral. Limit of the Riemann sums: $\int_a^b f(x)\,dx$, a number.
  • Antiderivative (indefinite integral). Function $F$ with $F'=f$. Determined up to a constant: $\int f(x)\,dx=F(x)+C$.
  • Fundamental theorem of calculus. Differentiating and integrating are inverse operations: $\frac{d}{dx}\int_a^xf=f(x)$ and $\int_a^bF'=F(b)-F(a)$.
  • Indeterminate form. Expression such as $\infty-\infty$ or $\frac00$ whose value depends on the route, not only on the form.
  • Series. Sum with infinitely many terms. It converges when the partial sums approach a number, which is called the sum of the series.
  • Conditional convergence. Series that converges, but whose sum changes when the terms are rearranged.
Exercise: apply the guide

Read $\displaystyle\lim_{h\to0}\frac{\sin(x+h)-\sin x}{h}=\cos x$.

What varies is $h$, bound by the limit; $x$ is free. The fraction is a finite difference divided by the step, $\frac{\Delta y}{\Delta x}$, and the limit turns it into the derivative. The $=$ is an identity, valid for every $x$. In Lagrange’s notation, it is the sentence $\sin'=\cos$; in Arbogast’s, $D\sin=\cos$.

Exercise: find the captured letter

A student wrote $F(x)=\int_1^x\frac{1}{x}\,dx=\ln x$. The result is right. What is wrong with the writing?

$x$ appears as the upper limit, which is free, and as the variable of integration, which is bound. The correct writing is $F(x)=\int_1^x\frac1t\,dt=\ln x$. The result comes out right by luck; with integrands that depend on both roles at once, it would come out wrong.

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