SOLUTION 10: Begin with the function
$$ f(x)= \sin x $$
and choose
$$ x-values: \displaystyle{ \pi \over 6 } \rightarrow \displaystyle{ \pi \over 6 } - 0.09 $$
so that
$$ \Delta x = (\displaystyle{ \pi \over 6 } - 0.09) - \displaystyle{ \pi \over 6 } = -0.09 $$
The derivative of $ \ y=f(x) \ $ is
$$ f'(x)= \cos x $$
The exact change of $y-$values is
$$ \Delta y = f(\displaystyle{ \pi \over 6 } - 0.09) - f(\displaystyle{ \pi \over 6 }) $$
$$ = \sin(\displaystyle{ \pi \over 6 } - 0.09) - \sin(\displaystyle{ \pi \over 6 }) $$
$$ = \sin(\displaystyle{ \pi \over 6 } - 0.09) - \displaystyle{ 1 \over 2 } $$
$$ = \sin(\displaystyle{ \pi \over 6 } - 0.09) - 0.5 $$
The Differential is
$$ dy = f'(\displaystyle{ \pi \over 6 }) \ \Delta x $$
$$ = \cos(\displaystyle{ \pi \over 6 }) \cdot (-0.09) $$
$$ = \displaystyle{ \sqrt{3} \over 2 } \cdot (-0.09) $$
$$ \approx \displaystyle{ 1.73205 \over 2 } \cdot (-0.09) $$
$$ \approx -0.0779 $$
We will assume that
$$ \Delta y \approx dy \ \ \ \ \longrightarrow $$
$$ \sin(\displaystyle{ \pi \over 6 } - 0.09) - 0.5 \approx -0.0779 \ \ \ \ \longrightarrow $$
$$ \sin(\displaystyle{ \pi \over 6 } - 0.09) \approx 0.5 - 0.0779 \ \ \ \ \longrightarrow $$
$$ \sin(\displaystyle{ \pi \over 6 } - 0.09) \approx 0.4221 \ \ \ \ \longrightarrow $$
NOTE: The number $ \displaystyle{ \pi \over 6 } $ was chosen for its proximity to $ \displaystyle{ \pi \over 6 }-0.09 $ and for it's convenient sine value. Check the accuracy of the final estimate using a CALCULATOR: $ \sin(\displaystyle{ \pi \over 6 }-0.09 ) \approx 0.4201 $
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