Unit 2: Differentiation — The Definition of the Derivative

Average Rate of Change

The change in the value of a function over an interval, defined as ( AROC = \frac{f(b) - f(a)}{b - a} ).

Instantaneous Rate of Change

The rate at which a function is changing at a specific moment, represented by the slope of the tangent line.

Derivative

The mathematical expression for the slope of the tangent line at any value ( x ), denoted as ( f'(x) ).

Limit Definition of Derivative

The derivative can be defined using limits: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).

Tangent Line

A line that touches a curve at a single point without crossing it.

Secant Line

A line that intersects a curve at two or more points.

Symmetric Difference Quotient

An estimate for the derivative using points on either side of the target point, defined as ( f'(c) \approx \frac{f(b) - f(a)}{b - a} ).

Continuity

A function is continuous at a point if there are no breaks, holes, or jumps in the graph at that point.

Differentiability

A function is differentiable at a point if its derivative exists at that point.

Fundamental Theorem of Differentiability

If a function is differentiable at ( x = c ), then it must also be continuous at ( x = c ).

Discontinuity

A condition where a function has a hole, jump, or asymptote, making the derivative undefined.

Corner (Sharp Turn)

A point on a graph where the left-hand derivative and the right-hand derivative differ.

Cusp

An extreme sharp turn on a graph where slopes approach ( \infty ) and ( -\infty ) from opposite sides.

Vertical Tangent

A tangent line that is strictly vertical at a point, resulting in an undefined slope.

Example of Average Rate of Change

The slope between two points on a graph, calculated as ( \frac{f(b) - f(a)}{b - a} ).

Limit

A mathematical concept that describes the value that a function approaches as the input approaches a certain point.

Derivative at a Point

The derivative of a function at a specific point ( a ) can be found using the limit: ( f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} ).

Algebraic Cancellation Error

A common mistake where students incorrectly cancel terms in a limit expression, leading to wrong derivatives.

Average vs. Instantaneous

Average rates are computed over intervals, whereas instantaneous rates are calculated at specific points.

Notation of the Derivative

Different notations include ( f'(x) ) (Lagrange), ( \frac{dy}{dx} ) (Leibniz), and ( \frac{d}{dx}[f(x)] ) (Operator).

The 'Point' Trap in Limits

A common mistake in limits where students approach 0 instead of the intended point ( a ).

Sharp Turn

A point on a graph where the derivative exists, but the limits from either side do not match.

Continuity Condition

A condition where a function must be continuous to be differentiable, but not vice versa.

Mistake Correction Chart

A tool to help differentiate between average and instantaneous rates, correct limit approaches, and ensure proper algebraic cancellation.

Kinematic Connection

How calculus relates to motion by analyzing rates of change in position over time.

Three Main Characteristics of Graphs

Include differentiability, continuity, and the presence of critical points.