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 $
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