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The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable, (1) It is sometimes known as ...

Let the values of a function f(x) be tabulated at points x_i equally spaced by h=x_(i+1)-x_i, so f_1=f(x_1), f_2=f(x_2), ..., f_4=f(x_4). Then Simpson's 3/8 rule ...

The 2-point Newton-Cotes formula int_(x_1)^(x_2)f(x)dx=1/2h(f_1+f_2)-1/(12)h^3f^('')(xi), where f_i=f(x_i), h is the separation between the points, and xi is a point ...

The derivative of the power x^n is given by d/(dx)(x^n)=nx^(n-1).

A recurrence relation between the function Q arising in quota systems, Q(n,r)=Q(n-1,r-1)+Q(n-1,r).

The divisibility test that an integer is divisible by 9 iff the sum of its digits is divisible by 9.

A negative number multiplied by another negative number gives a positive number.

On an algebraic curve, the sum of the number of coincidences at a noncuspidal point C is the sum of the orders of the infinitesimal distances from a nearby point P to the ...

Let lim stand for the limit lim_(x->c), lim_(x->c^-), lim_(x->c^+), lim_(x->infty), or lim_(x->-infty), and suppose that lim f(x) and lim g(x) are both zero or are both ...

Given a set of linear equations {a_1x+b_1y+c_1z=d_1; a_2x+b_2y+c_2z=d_2; a_3x+b_3y+c_3z=d_3, (1) consider the determinant D=|a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|. (2) Now ...