AP Calculus AB and BC
Secant Line vs Tangent Line
A secant line passes through two points on a curve and its slope is the average rate of change. A tangent line touches at a single point and its slope is the derivative there. Shrinking the gap between the two points turns the secant into the tangent.
Secant line
Use when: You have two points and want an average rate of change over the interval between them.
Tangent line
Use when: You have one point and want the instantaneous rate of change, or a linear approximation there.
Side by side
| Secant | Tangent | |
|---|---|---|
| Points of contact | Two | One, locally |
| Slope | ||
| Needs calculus | No | Yes |
| Related theorem | Mean Value Theorem | Linearization |
The limit connecting them is the definition of the derivative. Writing the secant slope as and letting go to zero is exactly the difference quotient.
The Mean Value Theorem ties them back together: on a suitable interval some tangent line is parallel to the secant line joining the endpoints.
A tangent can cross the curve
Touching at one point is a local description. A tangent line may intersect the curve again elsewhere, and at an inflection point it crosses right at the point of tangency.
Frequently asked questions
Can a secant line and a tangent line be the same?
For a straight line, yes, since every secant and tangent coincide with the line itself.
In the CED: Unit 2: Defining the Derivative