Calculus with R

Calculus, computing, and notation

A dominant style of notation in teaching calculus is the analytic style, where functions are written in the form \(f = 1 / x\), derivatives are referred to in any of several notations such as \(f'\) or \(\frac{d}{dx} f\). Integrals are denoted \(\int f(x) dx\). In this notation, \(dx\) is a formal device that serves to identify the variable with respect to which the operation of integration or differentiation is being performed.

This contrasts with mainstream computer languages where functions are identified by names and there is no universal notation for “with respect to ….”. The calculus operations in mosaicCalc exploit the tilde-formula notation in R, e.g. 1 / x ~ x. The symbol on the right-hand side of tilde identifies the variable with respect to which the operation is to be performed. To the left of tilde is a statement in standard computer notation of the transformation carried out by the function.

Traditionally, a typographic distinction is made between “variables” and “parameters.” For instance, it’s implicit that the straight-line function \(mx + b\) involves the variable \(x\) and the parameters \(m\) and \(b\). Usually the variables are letters toward the end of the alphabet such as \(t\), \(x\), and \(y\) whereas parameters are written using earlier letters in the alphabet (e.g., \(a\), \(b\), \(c\), \(k\)) sometimes using upper-case letters, Greek letters, and/or subscripts.

In mosaicCalc, no formal distinction is made between variables and parameters except as provided by the “with respect to …” component of a formula. In mosaicCalc you would write m * x + b ~ x as a formula for a straight-line function.

A formula like m * x + b ~ x stands for a function, but it is not an R function. Formulas in mosaicCalc are routinely turned into actual R functions. This happens automatically in operations such as differentiation and anti-differentiation, which return a function rather than a formula.

The operation makeFun() in mosaicCalc performs this action. For instance, here is the definition of a function called f:

f <- makeFun(m * x + b ~ x)
f

## function (x, m, b) 
## m * x + b

Note that f is a function of three arguments. There’s no formal distinction being made between the “variable” and the “parameters.” (makeFun() can also be used to turn linear and logistic regression reports into functions.)

Contrast this with the way an R programmer might naturally define the straight-line function:

# NOT the style in mosaicCalc
straight_line <- function(x) m * x + b

straight_line is a legitimate function in R, but the parameters \(m\) and \(b\) are not represented by arguments to the function. Instead, they are set by the scoping rules of the R language. In particular, given this definition of straight_line, at the time that straight_line() is applied to an input, the values of m and b will be looked up in the R environment in which the function was defined.

The mosaicCalc approach is to avoid as much as possible the need to understand the lexical scoping used by R. And so every quantity in a function made by the likes of makeFun() will become an argument.

In makeFun() the “with respect to …” component of the formula is used only to determine the order of the arguments: symbols used on the right-hand side of the tilde are put first in the argument list.

Differentiation and anti-differentiation

The mosaicCalc operations for differentiation and anti-differentiation are called D() and antiD() respectively. Like makeFun(), D() and antiD() take as a primary argument a formula where the right-hand side identifies the “with respect to …” variable. And, all three of those operations are similar in that they return a function (not a formula) whose arguments include every symbol used in the body of the function.

D(a / x ~ x)

## function (x, a) 
## -(a/x^2)

In all three operations, you can declare a default value for the parameter, which enables you have the parameter symbolically identified while not having to assign a value to the parameter every time it is evaluated. For instance,

makeFun(a / x ~ x, a = 3)

## function (x, a = 3) 
## a/x

D( a / x ~ x, a = 3)

## function (x, a = 3) 
## -(a/x^2)

Famously, anti-differentiation introduces an additive quantity usually denoted C, so \(\int 1/x\ dx \rightarrow \ln x + C\). This \(C\) is treated as a parameter in the output of mosaicCalc::antiD():

antiD(a / x ~ x)

## function (x, C = 0, a) 
## a * log((x)) + C

Analytic or numeric

The symbolic capabilities of mosaicCalc are weak. Only a few, simple forms of functions can be handled symbolically by antiD(). The D() operation is similarly limited, although it handles composition and linear combinations of functions well. For instance:

D(a * x + b * x^2 + c * sin(x^2) ~ x)

## function (x, a, b, c) 
## a + b * (2 * x) + c * (cos(x^2) * (2 * x))

while antiD() will generate a numeric-integration result:

F <- antiD(a * x + b * x^2 + c * sin(x^2) ~ x)
F

## function (x, a, b, c, C = 0) 
## {
##     numerical_integration(.newf, .wrt, as.list(match.call())[-1], 
##         formals(), from, ciName = intC, .tol)
## }
## <environment: 0x7fe9bd712438>

Note that even though the function F() involves applying numerical methods, it is still a function and can itself be differentiated or integrated using D() and antiD() in mosaicCalc.

Graphics

The meaning of “with respect to …” when making a graphic is to set the coordinate axes. For instance, the mosaic package includes a function plotFun()

mosaic::plotFun(1 / x ~ x, xlim=c(0,5))

plotFun() can be thought of as a legacy function. It uses the lattice graphics system. (We plan soon to be adding ggplot2 compatible calculus graphics functions.)