- Topics At a Glance
- Derivative as a Limit of Slopes
- Slope of a Line Between Two Points
- Slope of a Line Between Two Points on a Function
- Slope at One Point?
- Estimating Derivatives Given the Formula
- Estimating Derivatives from Tables
- Finding Derivatives Using Formulas
- Derivatives as an Instantaneous Rate of Change
- Average Rate of Change
- Instantaneous Rate of Change
- Units, Words, and Notation
**Tangent Lines****How Tangent Lines Look**- When Tangent Lines Don't Exist
- Tangent Lines and Derivatives
- Tangent Line Approximation
- Finding Tangent Lines
- Using Tangent Lines to Approximate Function Values
- Differentiability and Continuity
- The Derivative Function
- Graphs of
*f*(*x*) and*f*' (*x*) - Theorems
- Rolle's Theorem
- The Mean Value Theorem
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
- Appendix: Speed v. Velocity

The *tangent line to f at a* is the line approached by the secant lines between

Tangent lines and secant lines are (usually) different things.

**A secant line must hit two points on a graph.** A secant line between the points on the graph where *x* = *a* and *x* = *b* will hit the graph at *x* = *a* and *x* = *b*.

**A tangent line occurs at one single point on a graph and has the same slope as the graph at that point.**

Visually, the tangent line to *f* at *a* bounces off the graph at *x* = *a*:

A tangent line may also cross through the graph somewhere else. The important thing that makes a tangent line a tangent line is that it grazes the graph at that one special point. This line is tangent to *f* at *x* = *a* because it bounces off there:

However, this line is also a secant line between *x* = *a* and *x* = *b*:

A tangent line usually doesn't "cross over" the graph from one side to the other. However, it may cross over the graph at *x* = *a* in cases like this:

Here, the tangent line isn't so much "bouncing off" as it is "laying along" the graph of *f*.

This phenomenon has to do with something called **concavity.**

When looking to see if a line is tangent to *f* at *a*, we're looking to see if the line "bounces off" or "lays along" the graph of *f* at *a*.

This function has a tangent line with infinite slope at *x* = *a*:

If we have a function *f* that's already a line, the tangent line to *f* at any point *a* will be *f* again:

.

Example 1

Draw the tangent line to |

Exercise 1

Determine whether each line is tangent to *f* at *x* = *a*, a secant line, or both a tangent line and a secant line.

Exercise 2

For the function *f* and point *a*, draw the tangent line to *f* at *a*:

Exercise 3

For the function *f* and point *a*, draw the tangent line to *f* at *a*:

Exercise 4

For the function *f* and point *a*, draw the tangent line to *f* at *a*:

Exercise 5

Determine if each line is tangent to *f* at *a*.