SOLUTION 4: Begin with the function
$$ f(x)= x^{3/4} $$
and choose
$$ x-values: 16 \rightarrow 14 $$
so that
$$ \Delta x = 14-16 = -2 $$
The derivative of $ \ y=f(x) \ $ is
$$ f'(x)= \displaystyle{ 3 \over 4 }x^{-1/4} = \displaystyle{ 3 \over 4 x^{1/4} } $$
The exact change of $y-$values is
$$ \Delta y = f(14) - f(16) $$
$$ = 14^{3/4} - 16^{3/4} $$
$$ = 14^{3/4} - (16^{1/4})^3 $$
$$ = 14^{3/4} - (2)^3 $$
$$ = 14^{3/4} - 8 $$
The Differential is
$$ dy = f'(16) \ \Delta x $$
$$ = \displaystyle{ 3 \over 4 (16)^{1/4} } \cdot (-2) $$
$$ = \displaystyle{ 3 \over 4 (2) } (-2) $$
$$ = \displaystyle{ 3 \over 8 } (-2) $$
$$ = \displaystyle{ -3 \over 4 } $$
$$ = \displaystyle{ -0.75 } $$
We will assume that
$$ \Delta y \approx dy \ \ \ \ \longrightarrow $$
$$ 14^{3/4} - 8 \approx -0.75 \ \ \ \ \longrightarrow $$
$$ 14^{3/4} \approx 8-0.75 \ \ \ \ \longrightarrow $$
$$ 14^{3/4} \approx 7.25 $$
NOTE: The number 16 was chosen for its proximity to 14 and for it's convenient fourth root. Check the accuracy of the final estimate using a CALCULATOR: $ 14^{3/4} \approx 7.2376 $
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