By what factor does the reaction rate change if [b] is doubled (and the other reactant concentrations are held constant)?

In kinetics, the reaction follows this general equation: 

aA + bB ---> cR

wherein A and B are the reactants while R is the product of reaction. The terms a, b and c are the stoichiometric coefficients of the two reactants and the product, respectively. For elementary reactions, the reaction rate has the following equation:

-r = k[A]^a[B]^b

where -r denotes for the reaction rate. The negative sign indicates that the reactant concentration decreases with time.

For example, let's assume A to be 0.1 M and B to be 0.2 M. The reaction is A + 2B --> R (example). Then,
-r = k(0.1)(0.2)^2 = 0.004

If reactant B is doubled:
-r = k(0.1)(0.2x2)^2 = 0.016

The reaction rate increased by a factor of 4. This is how the concentrations of reactants affect the reaction rate.