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 \( y = f(x) = 2\,x^2 - 5\,x + 32 \):

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

\[ = 2\,x^2+4\,x\,h +2\,h^2 - 5\,x-5\,h + 32 = \]

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

\[ \Delta x \, = \, x + h \, - \, x \, = \, h \]

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