Question 715531
Your referring to a tangent line makes me think you are trying to use the tangent line to find an approximation to a function.  


Your function, f(x)=sqrt(x), has a value of 2 at x=4.  You also found that the slope of the tangent line at x=4 is {{{1/4}}}.  NOW, you want to actual equation of the line which is tangent to the f(x) at that f(4)=2 value.  Point-Slope form will be easiest but no matter, the form of the linear equation is up to you.  


The point of tangency is (4,2) and the slope is 1/4.  What is the line equation?
Using {{{y-2=(1/4)(x-4)}}}, you find {{{highlight(y=(1/4)x+1)}}}.  


(a) Approximately, {{{sqrt(4.4)}}}
ONLY an approximation, but it will be near {{{y=(1/4)4.4+1}}}, or near {{{2.1}}}.


(b) Approximately, what is {{{sqrt(9)}}}
Actually I'm not sure if you wanted that one or for 9.9?  I'll do for 9.
Only APPROXIMATELY, {{{sqrt(9)}}} ~ {{{(1/4)9+1=3.25}}}, actually not very good.  


You see that the farther away from the point of tangency, the worse is this approximation for the function.

{{{graph(300,300,-1,10,-1,10,sqrt(x),0.25*x+1)}}}