Finding the Tangent Line at a Point

Calculus 1 · Axiom Academy

EXAMPLE Finding the Tangent Line at a Point Use the derivative to write the equation of the tangent line to a curve at a given point Find the equation of the line tangent to the curve f(x) = x^2 - 3x + 1 at the point where x = 2 . The tangent touches the curve at and matches its steepness there. Nice work — you built a tangent line from scratch. Here's what carried the solution: The slope is the derivative: f'(a) gives the slope of the tangent line at x = a . You also need a point: evaluate f(a) to get the y -coordinate where the line touches. Point-slope assembles it: y - f(a) = f'(a)(x - a) turns one point and one slope into the line. Here: f(2) = -1 and f'(2) = 1 give y = x - 3 . A tangent line is the best linear approximation of a curve near a point — the foundation of linearization and Newton's method.

This is the written version of the interactive lesson above. See the full Calculus 1 course.