Legrange's Mean Value Theorem

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the right-hand side above is zero.
The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f : [a, b] → R is continuous on [a, b], and that for every x in (a, b) the limit

exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the real-valued cube root function mapping x to x^1/3, whose derivative tends to infinity at the origin.
Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f(x) = e^ix for all real x. Then

f(2π) − f(0) = 0 = 0(2π − 0)

while |f′(x)| = 1.

Torricelli's (Roger's) law

Torricelli's (Roger's) law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an opening to the height of fluid above the opening.

Torricelli's law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. , where g is the acceleration due to gravity (9.8 N/kg). This last expression comes from equating the kinetic energy gained, , with the potential energy lost, mgh , and solving for v.

Definition of the Day -Heat Of Combustion

The heat of combustion () is the energy released as heat when a compound undergoes complete combustion with oxygen under standard conditions. The chemical reaction is typically a hydrocarbon reacting with oxygen to form carbon dioxide, water and heat. It may be expressed with the quantities:

• energy/mole of fuel (kJ/mol)
• energy/mass of fuel
• energy/volume of fuel