Tangent line to a curve
A tangent line to a curve is the straight line that touches the curve at a single point and has the same slope as the curve at that point. In calculus, that slope is the value of the derivative at the point.
A tangent line to a curve is the straight line that just grazes the curve at one point, matching the curve's direction there. It's the line the curve "looks like" if you zoom in far enough at that point — the best straight-line approximation to the function nearby.
Calculus makes this precise: the slope of the tangent line at x = a is the derivative f′(a). Formally, it's the limit of the slopes of secant lines through (a, f(a)) and a nearby point as that second point slides toward the first. Once you have the slope, the tangent line's equation comes from point-slope form: y − f(a) = f′(a)(x − a).
For example, take f(x) = x² at the point (3, 9). The derivative is f′(x) = 2x, so the slope at x = 3 is f′(3) = 6, and the tangent line is y − 9 = 6(x − 3), or y = 6x − 9. A horizontal tangent (slope 0) marks a critical point — a candidate maximum or minimum — while a vertical tangent means the derivative is undefined there.
Tangent lines are the geometric heart of the derivative, and the AP Calculus AB exam tests them constantly: finding a tangent line's equation at a point, locating horizontal tangents, and using the tangent line for local linear approximation of function values.
Key takeaways
- The tangent line touches a curve at one point and matches the curve's slope there.
- Its slope equals the derivative at that point: slope = f′(a) at x = a.
- Write its equation with point-slope form: y − f(a) = f′(a)(x − a).
- Horizontal tangents (f′ = 0) mark critical points; tangent lines also provide local linear approximations.