SOLUTION 6: Begin with the function
$$ f(x)= e^{-x} $$
and choose
$$ x-values: 0 \rightarrow 0.3 $$
so that
$$ \Delta x = 0.3-0 = 0.3 $$
The derivative of $ \ y=f(x) \ $ is
$$ f'(x)= e^{-x} (-1) = -e^{-x} $$
The exact change of $y-$values is
$$ \Delta y = f(0.3) - f(0) $$
$$ = e^{-0.3} - e^{0} $$
$$ = e^{-0.3} - 1 $$
The Differential is
$$ dy = f'(0) \ \Delta x $$
$$ = -e^{0} \cdot (0.3) $$
$$ = (-1) (0.3) $$
$$ = -0.3 $$
We will assume that
$$ \Delta y \approx dy \ \ \ \ \longrightarrow $$
$$ e^{-0.3} - 1 \approx -0.3 \ \ \ \ \longrightarrow $$
$$ e^{-0.3} \approx 1 - 0.3 \ \ \ \ \longrightarrow $$
$$ e^{-0.3} \approx 0.7 $$
NOTE: The number 0 was chosen for its proximity to 0.3 and for it's convenient exponential value. Check the accuracy of the final estimate using a CALCULATOR: $ e^{-0.3} \approx 0.7408 $
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