3.1 Lösning 9c

Vi har:

\[ f(x) = \, x^3 \]

\[ f\,'(x) = 3\,x^2 \]

\[ c = 2,081\,666 \]

Tangenten: \( {\color{White} x} y \, = \, k\,x \, + \, m \)

Tangentens lutning = kurvans lutning i    \( x = 2,081\,666 \)   . Därför:

\[ k = f\,'(2,081\,666) = 3\cdot 2,081\,666^2 = 13\]

Tangentens ekvation: \( {\color{White} x} y \, = \, 13\,x \, + \, m \)

Beröringspunktens koordinater:

\[ x = 2,081\,666 \]

\[ y = f(2,081\,666) = 2,081\,666^3 = 9,020\,553 \]

Beröringspunkten ligger på tangenten:

\[\begin{array}{rcl} y & = & 13\,x \, + \, m \\ 9,020\,553 & = & 13 \cdot 2,081\,666 \, + \, m \\ 9,020\,553 & = & 27,061\,658 \, + \, m \\ -18,041\,105 & = & m \\ \end{array}\]

Tangentens ekvation:

\[ y \, = \, 13\,x \, - \, 18,041\,105 \]