Answer
a. $0.414$, $0.449$, $\frac{\sqrt {1+h}-1}{h}$
b. See table.
c. $0.5$
d. $\frac{1}{2}$
Work Step by Step
a. Given $g(x)=\sqrt x$ ($x\geq0$), we have:
(i) the average rate change over $[1,2]$, $R_1=\frac{\sqrt 2-\sqrt 1}{2-1}=\sqrt 2-1\approx0.414$
(ii) the average rate change over $[1,1.5]$, $R_2=\frac{\sqrt {1.5}-\sqrt 1}{1.5-1}\approx0.449$
(iii) the average rate change over $[1,1+h]$, $R_h=\frac{\sqrt {1+h}-\sqrt 1}{1+h-1}=\frac{\sqrt {1+h}-1}{h}$
b. See table; the rate of change is given in the third column.
c. The table indicates that the rate of change of $g(x)$ with respect to $x$ at $x=1$ is $0.5$
d. $\lim_{h\to 0} \frac{\sqrt {1+h}-1}{h}=\lim_{h\to0}\frac{(\sqrt {1+h}-1)(\sqrt {1+h}+1)}{h(\sqrt {1+h}+1)}=\lim_{h\to0}\frac{1+h-1}{h(\sqrt {1+h}+1)}=\lim_{h\to0}\frac{1}{\sqrt {1+h}+1}=\frac{1}{\sqrt 1+1}=\frac{1}{2}$
