Let $f(x)=4\sqrt[\phantom{2}]{x}+1.$

a$.$ What is the average rate of change of $f$ over the interval from $x=2$ to $x=3\mathrm{.}5?$

b$.$ What is the average rate of change of $f$ over the interval from $x=3\mathrm{.}5$ to $x=5\mathrm{.}9?$

c$.f(a)=60\mathrm{.}46$ and the average rate of change of $f$ over the interval from $x=a$ to $x=a+206$ is $0\mathrm{.}113.$ What is the value of $f(a+206)?($Hint: think of the average rate of change as a constant rate of change.$)$

$f(a+206)=$

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