Question

Viteza unui automobil este descrisa de formula v = 9t^4+ 7t + 2. Gasiti formula care exprima acceleratia acestui automobil.

Answer 1

$a\equiv\dfrac{\Delta v}{\Delta t}$

Pentru un interval foarte mic de timp $\Delta t$:

$a=\dfrac{v(t+\Delta t)-v(t)}{\Delta t}=\dfrac{9(t+\Delta t)^4+7(t+\Delta t)+2-9t^4-7t-2}{\Delta t}$

$a=\dfrac{\Delta t(36 t^3 + 54 t^2 \Delta t+ 36 t (\Delta t)^2 + 9 (\Delta t)^3 + 7)}{\Delta t}\\a= 36 t^3 + 54 t^2 \Delta t+ 36 t (\Delta t)^2 + 9 (\Delta t)^3 + 7\approx 36t^3+7\:(\Delta t\to 0)$

Poti verifica imediat acest lucru prin a deriva functia v(t). In rezolvare am folosit formula $(a+b)^4= a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4$ si faptul ca $\Delta t$ e ales neglijabil fata de $t$.