AP Calculus AB Unit 2: Definition of the Derivative (Rates of Change, Limit Definition, Estimation, Continuity vs. Differentiability)

Rate of change

Describes how one quantity changes in response to another.

Average rate of change

The change in a function over an interval, calculated by (\frac{f(b)-f(a)}{b-a}).

Instantaneous rate of change

The rate of change at a specific point, represented as the slope of the tangent line.

Derivative

A limit of average rates of change that represents the instantaneous rate at which a function is changing.

Secant line

A line that connects two points on a graph, representing the average rate of change.

Tangent line

A line that touches the graph at a single point, representing the instantaneous rate of change.

Limit definition of the derivative

(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}) which defines the derivative at point (a).

Continuous function

A function that has no breaks, holes, or jumps at a point or over an interval.

Differentiable function

A function that has a derivative at a specific input value.

Cusp

A pointed end on the graph where the slopes approach (\infty) and (-\infty), leading to a non-existent derivative.

Corner

A sharp turn on a graph where the left-hand and right-hand derivatives are unequal.

Vertical tangent

A tangent line that is vertical, resulting in an infinite slope, so the derivative is undefined.

Forward difference

An estimate of a derivative using values from the point and a small forward step.

Backward difference

An estimate of a derivative using values from the point and a small backward step.

Symmetric difference

An estimate of a derivative using values around a point to balance both sides.

One-sided derivative

The limit of the difference quotient approached from one side, either left or right.

Discontinuous function

A function that has one or more discontinuities, causing the derivative to not exist.

Slope of a tangent line

The derivative of a function at a point, showing the instantaneous rate of change.

Average speed

Total distance divided by total time, representing an average rate of change during a trip.

Marginal cost

The derivative of the cost function, representing the cost incurred by producing one more unit.

Local linearity

The property that near a point, a function behaves like its tangent line.

Interpreting limits

Understanding how values behave as they approach a specific point in the context of rates.

Rounding errors

Mistakes that occur from approximating too early in a calculation involving limits.

Function notation for derivative

Different ways to denote the derivative, such as (f'(x)) or (\frac{dy}{dx}).

Instantaneous velocity

The derivative of the position function with respect to time, indicating the speed at a specific moment.

Fundamental Theorem of Differentiability

States that if a function is differentiable at a point, it must be continuous there.