Calculus: Derivative and Anti-derivative operators

In mosaicCalc: R-Language Based Calculus Operations for Teaching

Derivative and Anti-derivative operators

Description

Operators for computing derivatives and anti-derivatives as functions.

Usage

D(tilde, ...)

antiD(tilde, ..., lower.bound = 0, force.numeric = FALSE, .tol = 1e-04)

Arguments

tilde	A tilde expression. The right side of a formula specifies the variable(s) with which to carry out the integration or differentiation. On the left side should be an expression or a function that returns a numerical vector of the same length as its argument. The expression can contain unbound variables. Functions will be differentiated as if the formula f(x) ~ x were specified but with x replaced by the first argument of f.
...	Default values to be given to unbound variables in the expression expr. See examples.#' Note that in creating anti-derivative functions, default values of "from" and "to" can be assigned. They are to be written with the name of the variable as a prefix, e.g. y.from.
lower.bound	for numerical integration only, the lower bound used
force.numeric	If TRUE, a numerical integral is performed even when a symbolic integral is available.
.tol	Tolerance for numerical integration. Most users do not need this.

Details

D() attempts to find a symbolic derivative for simple expressions, but will provide a function that is a numerical derivative if the attempt at symbolic differentiation is unsuccessful. The symbolic derivative can be of any order (although the expression may become unmanageably complex). The numerical derivative is limited to first or second-order partial derivatives (including mixed partials). antiD() will attempt simple symbolic integration but if it fails it will return a numerically-based anti-derivative.

antiD() returns a function with the same arguments as the expression passed to it. The returned function is the anti-derivative of the expression, e.g., antiD(f(x)~x) -> F(x). To calculate the integral of f(x), use F(to) - F(from).

Value

For derivatives, the return value is a function of the variable(s) of differentiation, as well as any other symbols used in the expression. Thus, D(A*x^2 + B*y ~ x + y) will compute the mixed partial with respect to x then y (that is, \frac{d^2 f}{dy\;dx}). The returned value will be a function of x and y, as well as A and B. In evaluating the returned function, it's best to use the named form of arguments, to ensure the order is correct.

a function of the same arguments as the original expression with a constant of integration set to zero by default, named "C", "D", ... depending on the first such letter not otherwise in the argument list.

Examples

D(sin(t) ~ t)
D(A*sin(t) ~ t )
D(A*sin(2*pi*t/P) ~ t, A=2, P=10) # default values for parameters.
f <- D(A*x^3 ~ x + x, A=1) # 2nd order partial -- note, it's a function of x
f(x=2)
f(x=2,A=10) # override default value of parameter A
g <- D(f(x=t, A=1)^2 ~ t)  # note: it's a function of t
g(t=1)
gg <- D(f(x=t, A=B)^2 ~ t, B=10)  # note: it's a function of t and B
gg(t=1)
gg(t=1, B=100)
f <- makeFun(x^2~x)
D(f(cos(z))~z) #will look in user functions also
antiD( a*x^2 ~ x, a = 3)
G <- antiD( A/x~x ) # there will be an unbound parameter in G()
G(2, A=1) # Need to bound parameter. G(2) will produce an error.
F <- antiD( A*exp(-k*t^2 ) ~ t, A=1, k=0.1)
F(t=Inf)
one = makeFun(1 ~ x)
by.x = antiD(one(x) ~ x)
by.xy = antiD(by.x(sqrt(1-y^2)) ~ y)
4 * by.xy(y = 1) # area of quarter circle

mosaicCalc documentation built on Sept. 11, 2024, 9:10 p.m.