2.5 Lösning 6
Att bestämma \( C \, \):
\[ \begin{array}{rcl} f(x) & = & C \cdot e\,^{k\,x} \end{array}\]
\( \, f(0) = 50 \) innebär:
\[ \begin{array}{rcrcl} f(0) & = & C \cdot e\,^{k\,\cdot\, 0} & = & 50 \\ & & C \cdot e\,^{0} & = & 50 \\ & & C \cdot 1 & = & 50 \\ & & C & = & 50 \end{array}\]
Att bestämma \( k \, \):
\[ \begin{array}{rclcl} f(x) & = & 50 \cdot e\,^{k\,x} & & \\ f\,'(x) & = & 50 \cdot k \cdot e\,^{k\,x} & & \\ \end{array}\]
\( f\,'(0) = 5 \) innebär:
\[ \begin{array}{rcrcl} f\,'(0) & = & 50 \cdot k \cdot e\,^{k\,\cdot\, 0} & = & 5 \\ & & 50 \cdot k \cdot e\,^{0} & = & 5 \\ & & 50 \cdot k \cdot 1 & = & 5 \\ & & k & = & {5 \over 50} \\ & & k & = & 0,1 \end{array}\]
Att specificera \( f(x) \, \):
\[ f(x) \, = \, 50 \, e\,^{0,1\,x} \]
