## The Derivative

**What is a Limit?**

**When does a limit exist?**

**Evaluating Limits**

**Limits and Infinity**

**Continuity**Intermediate Value Theorem

**Tangent Lines**

**The Difference Quotient**

**Definition of the Derivative**

**The Derivative at a Point**

**Calculus Grapher**

**Chapter 2 Study Guide (MS Word)**

**AP Calculus (BC) Topics**

II.

**Derivatives**

A.

**Concept of the derivative**.

- Derivative presented graphically, numerically, and analytically.
- Derivative interpreted as an instantaneous rate of change.
- Derivative defined as the limit of the difference quotient.
- Relationship between differentiability and continuity.

**Derivative at a point**.

- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
- Tangent line to a curve at a point and local linear approximation.
- Instantaneous rate of change as the limit of average rate of change.
- Approximate rate of change from graphs and tables of values.

**Derivative as a function**.

- Corresponding characteristics of graphs of f and f'.
- Relationship between the increasing and decreasing behavior of f and the sign of f'.
- The Mean Value Theorem and its geometric consequences.
- Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

**Second derivatives**.

- Corresponding characteristics of the graphs of f, f', and f''.
- Relationships between the concavity of f and the sign of f''.
- Points of inflection as places where concavity changes.