Description Usage Arguments Details Value Note Examples

Operators for computing derivatives and anti-derivatives as functions.

1 2 3 4 5 6 7 |

`formula` |
A formula. 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 |

`...` |
Default values to be given to unbound variables in the expression |

`.hstep` |
horizontal distance between points used for secant slope calculation in numerical derivatives. |

`add.h.control` |
logical indicating whether the returned derivative function should have an additional parameter for setting .hstep. Meaningful only for numerical derivatives. |

`lower.bound` |
for numerical integration only, the lower bound used |

`force.numeric` |
If |

`.function` |
function to be integrated |

`.wrt` |
character string naming the variable of integration |

`from` |
default value for the lower bound of the integral region |

`.tol` |
Numerical tolerance. See |

`f` |
A function. |

`wrt` |
Character string naming a variable: the var. of integration. |

`av` |
A list of the arguments passed to the function calling this. |

`args` |
Default values (if any) for parameters. |

`vi.from` |
The the lower bound of the interval of integration. |

`ciName` |
Character string giving the name of the symbol for the constant of integration. |

`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).

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, *d2f/dydx*). 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.

`numerical_integration`

is not intended for direct use. It packages
up the numerical anti-differentiation process so that the contents
of functions produced by `antiD`

look nicer to human readers.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
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)
antiD( A/x~x ) # This gives a warning about no default value for A
F <- antiD( A*exp(-k*t^2 ) ~ t, A=1, k=0.1)
F(t=Inf)
one = makeFun(1 ~ x + y)
by.x = antiD(one(x=x, y=y) ~ x, y=1)
by.xy = antiD(by.x(x = sqrt(1-y^2), y = y) ~ y)
4 * by.xy(y = 1) # area of quarter circle
``` |

```
Loading required package: mosaicCore
Attaching package: 'mosaicCalc'
The following object is masked from 'package:stats':
D
function (t)
cos(t)
function (t, A)
A * cos(t)
Warning message:
In makeFun.formula(formula, ...) :
Implicit variables without default values (dangerous!): A
function (t, A = 2, P = 10)
A * (cos(2 * pi * t/P) * (2 * pi/P))
[1] 12
[1] 120
[1] 72
[1] 7200
[1] 720000
function (z)
numerical.first.partial(.function, .wrt, .hstep, match.call())
<environment: 0x6d8b728>
function (x, a = 3, C = 0)
a * 1/3 * x^3 + C
function (x, C = 0, A)
A * log((x)) + C
Warning message:
In makeFun.formula(A * log((x)) + C ~ x, C = 0) :
Implicit variables without default values (dangerous!): A
[1] 2.802496
[1] 3.141593
```

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