SOLUTION 5: Begin with the function
$$f(x)= \sqrt{x}$$
a.) $\ \ \$ Choose
$$x-values: 64 \rightarrow 72$$
so that
$$\Delta x = 72-64 = 8$$
The derivative of $\ y=f(x) \$ is
$$f'(x)= \displaystyle{ 1 \over 2 }x^{-1/2} = \displaystyle{ 1 \over 2 \sqrt{x} }$$
The exact change of $y-$values is
$$\Delta y = f(72) - f(64)$$ $$= \sqrt{72} - \sqrt{64}$$ $$= \sqrt{72} - 8$$
The Differential is
$$dy = f'(64) \ \Delta x$$ $$= \displaystyle{ 1 \over 2 \sqrt{64} } \cdot (8)$$ $$= \displaystyle{ 1 \over 2 (8) } (8)$$ $$= \displaystyle{ 1 \over 16 } (8)$$ $$= \displaystyle{ 1 \over 2 }$$ $$= 0.5$$
We will assume that
$$\Delta y \approx dy \ \ \ \ \longrightarrow$$ $$\sqrt{72} - 8 \approx 0.5 \ \ \ \ \longrightarrow$$ $$\sqrt{72} \approx 8+0.2 \ \ \ \ \longrightarrow$$ $$\sqrt{72} \approx 8.2$$

NOTE: The number 64 was chosen for its proximity to 72 and for it's convenient square root. Check the accuracy of the final estimate using a CALCULATOR: $\sqrt{72} \approx 8.4853$

b.) $\ \ \$ Choose
$$x-values: 81 \rightarrow 72$$
so that
$$\Delta x = 72-81 = -9$$
The derivative of $\ y=f(x) \$ is
$$f'(x)= \displaystyle{ 1 \over 2 }x^{-1/2} = \displaystyle{ 1 \over 2 \sqrt{x} }$$
The exact change of $y-$values is
$$\Delta y = f(72) - f(81)$$ $$= \sqrt{72} - \sqrt{81}$$ $$= \sqrt{72} - 9$$
The Differential is
$$dy = f'(81) \ \Delta x$$ $$= \displaystyle{ 1 \over 2 \sqrt{81} } \cdot (-9)$$ $$= \displaystyle{ 1 \over 2 (9) } (-9)$$ $$= \displaystyle{ 1 \over 18 } (-9)$$ $$= \displaystyle{ -1 \over 2 }$$ $$= -0.5$$
We will assume that
$$\Delta y \approx dy \ \ \ \ \longrightarrow$$ $$\sqrt{72} - 9 \approx -0.5 \ \ \ \ \longrightarrow$$ $$\sqrt{72} \approx 9-0.5 \ \ \ \ \longrightarrow$$ $$\sqrt{72} \approx 8.5$$

NOTE: The number 81 was chosen for its proximity to 72 and for it's convenient square root. Check the accuracy of the final estimate using a CALCULATOR: $\sqrt{72} \approx 8.4853$