Tangent Plane

The Tangent Plane

Just as a tangent line approximates a 1D curve near a point, a tangent plane approximates a 2D surface near a point $(x_0, y_0)$:

$L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

This is called the linear approximation or linearization of $f$ at $(x_0, y_0)$.

Why It Works

Near $(x_0, y_0)$, $f$ changes approximately linearly:

• Moving $\Delta x$ in the $x$-direction changes $f$ by $\approx f_x \cdot \Delta x$
• Moving $\Delta y$ in the $y$-direction changes $f$ by $\approx f_y \cdot \Delta y$

The tangent plane captures both effects simultaneously.

Error of Approximation

The error $|f(x,y) - L(x,y)|$ is small near $(x_0, y_0)$ — it's $O(\|(x,y)-(x_0,y_0)\|^2)$ for smooth functions.

Examples

For $f(x,y) = x^2 + y^2$ at $(1, 2)$:

• $f_x = 2$, $f_y = 4$, $f(1,2) = 5$
• $L(x,y) = 5 + 2(x-1) + 4(y-2)$
• $L(1.1,\; 2.1) \approx 5 + 0.2 + 0.4 = 5.6$ (exact: $1.21+4.41=5.62$)

Your Task

Implement double tangent_plane(double (*f)(double, double), double x0, double y0, double x, double y, double h) that evaluates the linear approximation $L(x, y)$ at the given $(x, y)$ point, using partial derivatives computed with step size $h$.