Differentials Solution 7

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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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