Differentials Solution 3

Processing math: 100%

SOLUTION 3: Begin with the function

$f(x)= \sqrt{x}$
and choose

$x-values: 100 \rightarrow 96$
so that

$\Delta x = 96-100 = -4$
The derivative of

$\ y=f(x) \$ is

$f'(x)= \displaystyle{ 1 \over 2 }x^{-1/2} = \displaystyle{ 1 \over 2 \sqrt{x} }$
The exact change of

$y-$ values is

$\Delta y = f(96) - f(100)$

$= \sqrt{96} - \sqrt{100}$

$= \sqrt{96} - 10$
The Differential is

$dy = f'(100) \ \Delta x$

$= \displaystyle{ 1 \over 2 \sqrt{100} } \cdot (-4)$

$= \displaystyle{ 1 \over 2 (10) } (-4)$

$= \displaystyle{ 1 \over 20 } (-4)$

$= \displaystyle{ -1 \over 5 }$

$= -0.2$
We will assume that

$\Delta y \approx dy \ \ \ \ \longrightarrow$

$\sqrt{96} - 10 \approx -0.2 \ \ \ \ \longrightarrow$

$\sqrt{96} \approx 10-0.2$

$\sqrt{96} \approx 9.8$

NOTE: The number 100 was chosen for its proximity to 96 and for it's convenient square root. Check the accuracy of the final estimate using a CALCULATOR:

$\sqrt{96} \approx 9.7979$

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