2.2 Lösning 8a

Definitionen till ändringskvot i intervallet mellan \( x \, \) och \( x+h \, \):

\[ {\Delta y \over \Delta x} \; = \; {f(x + h) \, - \, f(x) \over h} \]

Tillämpad på vårt exempel \( y = f(x) = 2\,x^2 - 5\,x + 32 \):

\[ f(x + h) = 2\,(x+h)^2 - 5\,(x+h) + 32 = 2\,(x^2 + 2\,x\,h + h^2) - 5\,x - 5\,h + 32 = \]

\[ = {\color{Red} {2\,x^2}} + 4\,x\,h + 2\,h^2 {\color{Red} {- 5\,x}} - 5\,h + {\color{Red} {32}} = 2\,h^2 + 4\,x\,h - 5\,h + ({\color{Red} {2\,x^2}} {\color{Red} {- 5\,x}} + {\color{Red} {32}}) \]

\[ \Delta y = f(x + h) \, - \, f(x) = 2\,h^2 + 4\,x\,h - 5\,h \]

\[ {\Delta y \over \Delta x} = {2\,h^2 + 4\,x\,h - 5\,h \over h} = {h\,(2\,h + 4\,x - 5) \over h} = 2\,h + 4\,x - 5 \]