Answer
The initial concentrations of the reactants and the initial reaction rate are recorded when calculating the method of initial rates, and $k$ is calculated by manipulating the rate law and solving for the orders of recants. The units for $k$ are not the same for every rate law because it varies with the orders of reactants.
If a reaction is first order in $A$, the rate will triple when $[A]$ is tripled.
If the initial reaction rate increases 16 times when $[A]$ is quadrupled, then the reaction is second order.
If a reaction is third order in $A$ and $[A]$ is doubled, the initial rate will increase 8 times.
If a reaction is 0 order, any change in $[A]$ will not affect the reaction rate.
Work Step by Step
Procedure given in textbook.
If a reaction is first order in $A$, then the rate is directly proportional to $A^1$, so the rate will increase by a factor of $3^1$ when $[A]$ is tripled.
If the initial reaction rate increases 16 times when $[A]$ is quadrupled, then $4^n=16$. Rewriting this equation gives $n=log_4{16}$, so $n$ must be 2, and the equation is second order.
If a reaction is third order in $A$, then the rate is directly proportional to $A^3$, so the rate will increase by a factor of $2^3$ or 8 when $[A]$ is tripled.
If a reaction is 0 order, any change in $[A]$ will not affect the reaction rate because the exponential function $A^0$ will give the same output for the reaction rate no matter the value of $[A]$.
