SOLUTION 7: Begin with the function
$$ f(x)= \ln x $$
and choose
$$ x-values: 1 \rightarrow 1.2 $$
so that
$$ \Delta x = 1.2-1 = 0.2 $$
The derivative of $ \ y=f(x) \ $ is
$$ f'(x)= \displaystyle{ 1 \over x } $$
The exact change of $y-$values is
$$ \Delta y = f(1.2) - f(1) $$
$$ = \ln 1.2 - \ln 1 $$
$$ = \ln 1.2 - 0 $$
$$ = \ln 1.2 $$
The Differential is
$$ dy = f'(1) \ \Delta x $$
$$ = \displaystyle{ 1 \over (1)} \cdot (0.2) $$
$$ = (1) (0.2) $$
$$ = 0.2 $$
We will assume that
$$ \Delta y \approx dy \ \ \ \ \longrightarrow $$
$$ \ln 1.2 \approx 0.2 $$
NOTE: The number 1 was chosen for its proximity to 1.2 and for it's convenient natural logarithm value. Check the accuracy of the final estimate using a CALCULATOR: $ \ln 1.2 \approx 0.1823 $
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