AP Calculus AB and BC glossary

Tangent line

The tangent line to a curve at a point is the straight line through that point whose slope equals the derivative there. It is the best linear approximation of the curve near that point.

y - f(a) = f'(a)(x - a)

Building the equation needs exactly two ingredients: the point $(a, f(a))$ and the slope $f'(a)$ . Point-slope form assembles them directly, which is why it is the form worth memorizing.

Near the point of tangency the tangent line and the curve are nearly indistinguishable. That fact is the whole content of linearization and tangent line approximation.

The mistake

Using $f'(x)$ as the slope instead of $f'(a)$ . The slope of a specific tangent line is a number, obtained by evaluating the derivative at the point of tangency.

Appears in: Unit 2: Defining the Derivative, Unit 4: Contextual Applications