Calculus/Basic Functions
- Some Basic Functions in Mathematica
Derivatives:
Calculates the Derivative of a function at a given point: using the limit definition of a derivative. The Derivative of a function is the slope of the tangent line to the function at a given point.
Parameters:
func
: A function that takes a singlef64
argument and returns anf64
. This is the function for which the derivative will be calculated.x
: The point at which the derivative will be calculated.
Returns:
The calculated derivative of the function at the given point.
Equation:
f′(x)=limh→0f(x+h)−f(x)h
Example:
use mathematica::Functions;
fn main() {
let function = |x: f64| x.powi(2);
let x = 2;
let derivative = Functions::derivative(function, x);
println!("The Derivative of x^2 as x=2 is: {}", derivative);
}
Execute as:
>>> The Derivative of x^2 as x=2 is: 4
Summation:
Calculates Summations of a function. The Summation of a function is the sum of all the values of the function from a given start to a given limit.
Parameters:
start
: The starting value of the summation.limit
: The ending value of the summation.func
: A function that takes a singlef64
argument and returns anf64
. This is the function for which the summation will be calculated.
Returns:
The summation of the function from the given start to the given limit.
Equation:
n∑i=mai=am+am+1+am+2+⋅⋅⋅+an−1+an
Example #1: Constant
use ferrate::special::Functions;
let start = 0;
let limit = 9;
let function = |x: f64| 3_f64;
let summation = Functions::summation(start, limit, function);
println!("The Summation of the constant, 3 from 0 to 9 is: {}", summation);
Execute as:
>>> The Summation of the constant, 3 from 0 to 9 is: 30
Example #2: Function
use ferrate::special::Functions;
let start = 4.5;
let limit = 100;
let function = |x: f64| 1_f64 / x;
let summation = Functions::summation(start, limit, function);
println!("The Summation of 1/x from 4.5 to 100 is: {}", summation);
Execute as:
>>> The Summation of 1/x from 4.5 to 100 is: 3.104044184318839
Product:
Calculates the Product (a.k.a. Capital Pi Notation) of a function. The Product of a function is the product of all the values of the function from a given start to a given limit.
Parameters:
start
: The starting value of the product.limit
: The ending value of the product.func
: A function that takes a singlef64
argument and returns anf64
. This is the function for which the product will be calculated.
Returns:
The product of the function from the given start to the given limit.
Equation:
n∏i=mai=am×am+1×am+2×⋅⋅⋅×an−1×an
Example #1: Constant
use ferrate::special::Functions;
fn main() {
let start = 2_f64;
let limit = 7_f64;
let f = |x: f64| 3_f64;
let product_series = Functions::product(start, limit, f);
println!("The Product of the constant, 3 from 2 to 7 is: {}", product_series);
}
Execute as:
>>> The Product of the constant, 3 from 2 to 7 is: 2187
Example #2: Function
use ferrate::special::Functions;
fn main() {
let start = 1_f64;
let limit = 6_f64;
let f = |x: f64| x.powi(2);
let product_series = Functions::product(start, limit, f);
println!("The Product of x^2 from 1 to 6 is: {}", product_series);
}
Execute as:
>>> The Product of x^2 from 1 to 6 is: 518400
Newton's Method:
Calculates the root of a function using Newton's Method. Newton's Method is an iterative method for finding the roots of a function. It is based on the idea that a continuous and differentiable function can be approximated by a straight line tangent to it.
Parameters:
x
: The initial guess for the root of the function.func
: A function that takes a singlef64
argument and returns anf64
. This is the function for which the root will be calculated.
Returns:
The root of the function.
Equation:
xn+1=xn−f(xn)f′(xn)
Example:
use ferrate::special::Functions;
fn main() {
let x = 1.5_f64;
let function = |x: f64| x.powi(2) - 2_f64;
let newton = Functions::newmet(x, function);
println!("The Root of x^2 - 2 is: {}", newton);
}
Execute as:
>>> The Root of x^2 - 2 is: 1.414213562373095