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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