2.3 Lösning 6a

\[ \begin{array}{rcl} f(25+h) & = & 4\,(25+h)^2 - 380\,(25+h) + 9000 & = \\ & = & 4\,(625+50\,h+h^2) - 9500 - 380\,h + 9000 & = \\ & = & 2500 + 200\,h + 4\,h^2 - 500 - 380\,h & = \\ & = & 4\,h^2 - 180\,h + 2000 \end{array}\]

\( f(25) = 4\cdot 25^2 - 380\cdot 25 + 9\,000 = 4\cdot 625 - 9500 + 9000 = 2000 \)

\[ \begin{array}{rcl} {\Delta y \over \Delta x} & = & {f(25+h) - f(25) \over h} & = & {4\,h^2 - 180\,h + 2000 -2000 \over h} & = \\ & = & {4\,h^2 - 180\,h \over h} & = & {h\cdot (4\,h - 180) \over h} = 4\,h - 180 \end{array}\]

\[ f\,'(25) \; = \; \lim_{h \to 0} \; (4\,h - 180) \; = \; - 180 \]

Dvs vid tiden \( x = 25\, \) sjunker oljans volym med \( 180\, \) liter per minut.