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Problem 19

Let $g(x)=\sqrt{x}$ for $x \geq 0$
a. Find the average rate of change of $g(x)$ with respect to $x$ owerthe intervals $[1,2],[1,1.5]$ and $[1,1+h]$.
b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$ and $0.000001 .$
c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$
d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$ .

a) See explanation for result.
b) See explanation for result.
c) 0.5
d) 0.5

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