Tangent Line

Tangent Line

The tangent line at a point $(x_0, f(x_0))$ is the line that just touches the curve there, with slope equal to $f'(x_0)$.

Equation

$y = f(x_0) + f'(x_0) \cdot (x - x_0)$

This is the point-slope form of a line, using the derivative as the slope.

Linearization

The tangent line is also called the linear approximation (or linearization) of $f$ near $x_0$:

$f(x) \approx f(x_0) + f'(x_0) \cdot (x - x_0) \quad \text{for } x \text{ near } x_0$

This is the foundation of calculus-based approximation. It says: near any point, a smooth curve looks like a straight line.

Example

For $f(x) = x^2$ at $x_0 = 3$:

• $f(3) = 9$
• $f'(3) = 6$
• Tangent line: $y = 9 + 6(x - 3) = 6x - 9$
• At $x = 4$: $y = 15$ (exact: $f(4) = 16$, error = 1 — small for nearby $x$)

Normal Line

The normal line is perpendicular to the tangent:

$y = f(x_0) - \frac{1}{f'(x_0)} \cdot (x - x_0)$

Your Task

Implement double tangent_y(double (*f)(double), double x0, double x, double h) that returns the y-value of the tangent line at $x_0$, evaluated at $x$. Use the central difference formula to compute $f'(x_0)$.