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In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. [1] It maps quadratic irrational numbers to rational numbers on the unit interval , via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the ...
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...
This last integral is , since (+) is the null function (because is a polynomial function of degree ). Since each function f ( k ) {\displaystyle f^{(k)}} (with 0 ≤ k ≤ 2 n {\displaystyle 0\leq k\leq 2n} ) takes integer values at 0 {\displaystyle 0} and π {\displaystyle \pi } and since the same thing happens with the sine and the cosine ...
Now any rational root p/q corresponds to a factor of degree 1 in Q[X] of the polynomial, and its primitive representative is then qx − p, assuming that p and q are coprime. But any multiple in Z [ X ] of qx − p has leading term divisible by q and constant term divisible by p , which proves the statement.
The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x) is denoted by f(x); for example, the value of f at x = 4 is denoted by f(4). Commonly, a specific function is defined by means of an expression depending on x, such as () = +; in this case, some computation, called function ...
satisfying respectively y(0) = 0, y ′ (0) = 1 and y(0) = 1, y ′ (0) = 0. It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that ...
For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The ...
Continuing to use Figure 1 as an example, price can be written as a function of quantity: = +, and be substituted into TR(Q) to get the TR function = +, which is a quadratic. In Figures 2 through 4, this function is shown graphically by using an example of demand for apples.