a pulse (dirac's delta).
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The unit step function at t=0 is defined to have a value of 1.
The unit step function is also known as the Dirac delta function. It can be thought of as a function of the real line (x-axis) which is zero everywhere except at the origin (x=0) where the function is infinite in such a way that it's total integral is 1 - hence the use of the word 'unit'. The function is not a strict function by definition in that any function with the properties as stated (0 everywhere except the origin which by definition has a limit tending to 0), must therefore also have an integral of 0. The answer is therefore zero everywhere except at the origin where it is infinite.
Increase in cost: take the first derivative with respect to the unit produced of a cost function. Total cost: sub-in the new quantity into the cost function.
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It can be the basis of the trig functions because the hypotenuse, which is the radius, is 1. For related reasons, it can represent unit vectors in any direction.