PART 8：例題-計算 \(dy\) 與 \(\Delta y\)

若 \(y = {x^2} + 2x\) ，當 \(x = 3\) ， \(\Delta x = 0.1\) 時求 \(\Delta y\) 與 \(dy\)

SOL:

(1)  \(\Delta y = f(x + \Delta x) - f(x)\)  

\( = \left[ {{{(x + \Delta x)}^2} + 2(x + \Delta x)} \right] - \left[ {{x^2} + 2x} \right]\)  

\( = \left[ {{{(3 + 0.1)}^2} + 2(3 + 0.1)} \right] - \left[ {{3^2} + 6} \right]\)

\( = \left[ {{{(3.1)}^2} + 2(3.1)} \right] - \left[ {15} \right]\)  \( = 0.81\)

(2)  \(dy = (2x + 2)dx\)  \( \buildrel\textstyle.\over= 8\Delta x = 0.8\)