=for timestamp
Mi Feb 23 16:09:32 CET 2005

=head3 40. Hausaufgabe

=head4 Selbstgestellte Aufgabe

M<f(x) = ax^3 + bx^2 + cx + d> berührt die Gerade M<g(x) = 12x + 13> in M<A(-1,
1)> und hat in M<B(1, 5)> eine waagrechte Tangente.

M<<
{} f(-1) = 1 = -a + b - c + d; \\
{} \qquad \Rightarrow d = 1 + a - b + c; \\
{} f(1)  = 5 = a + b + c + d = a + b + c + 1 + a - b + c; \\
{} \qquad \Rightarrow 4 = 2a + 2c; \\
{} \qquad \Rightarrow c = 2 - a; \\
{} f'(-1) = 0 = 3a + 2b + c = 3a + 2b + 2 - a; \\
{} \qquad \Rightarrow -2 = 2a + 2b; \\
{} \qquad \Rightarrow b = -1 - a; \\
{} f'(-1) = 12 = 3a - 2b + c = 3a + 2 + 2a + 2 - a = 4a + 4; \\
{} \qquad \Rightarrow 8 = 4a; \\
{} \qquad \Rightarrow a = 2; \\
{} \qquad \Rightarrow b = -1 - 2 = -3; \\
{} \qquad \Rightarrow c = 2 - 2 = 0; \\
{} \qquad \Rightarrow d = 1 + 2 + 3 = 6; \\
{} \qquad \Rightarrow f(x) = 2x^3 - 3x^2 + 6;
>>

=helper MyBook::Helper::Gnuplot

# File automatically generated by Plot.pm

# Global settings
set samples 10000
unset border
set xtics axis
set ytics axis
set xzeroaxis lt -1
set yzeroaxis lt -1

# Coordinate system settings
set title ""
set xlabel "\nx\n"
set ylabel "y\n"
set xrange [ -2.000000 : 2.000000 ]
set yrange [ -1.000000 : 11.000000 ]
set grid
set xtics 1.000000
set ytics 1.000000

# Function definitions
func0(x) = 2.*x**3. - 3.*x**2. + 6.
func1(x) = 12.*x + 13.
func2(x) = 5

# Plotting
plot func0(x) t "f" w l lt 1, func1(x) t "g" w l lt 2, func2(x) t "t" w l lt 3

=hend