Unit 5 (AP Calculus AB): Mean Value Theorem, Extreme Values, and Extrema on Closed Intervals

Mean Value Theorem (MVT)

If a function is continuous on a closed interval and differentiable on the corresponding open interval, then there exists at least one interior point where the tangent slope equals the secant slope.

Secant slope

The slope of the secant line through the points (a, f(a)) and (b, f(b)), calculated as (f(b) - f(a)) / (b - a).

Instantaneous rate of change

The slope of the tangent line at a specific point, represented as f'(c).

Continuity on [a,b]

The function does not have any holes or jumps in the interval [a,b].

Differentiability on (a,b)

The function has a derivative at every point in the open interval (a,b); there are no corners, cusps, or vertical tangents.

Rolle’s Theorem

A special case of MVT where f(a) = f(b); guarantees a c in (a,b) such that f'(c) = 0.

Absolute maximum

The highest function value of f on a particular domain.

Absolute minimum

The lowest function value of f on a particular domain.

Relative maximum

Where f(c) is larger than nearby values within some neighborhood around c.

Relative minimum

Where f(c) is smaller than nearby values within some neighborhood around c.

Critical point

A point where f'(c)=0 or f'(c) does not exist.

First Derivative Test

A method to classify relative extrema by examining the sign of f' on either side of a critical point.

Candidates Test (Closed Interval Method)

A method to find absolute extrema on [a,b] by evaluating f at endpoints and critical points.

EVT (Extreme Value Theorem)

If f is continuous on a closed interval [a,b], then f attains both an absolute max and an absolute min on that interval.

Fermat’s Theorem

If f has a relative extremum at an interior point c and f'(c) exists, then f'(c)=0.

Piecewise function

A function defined by multiple sub-functions, often leading to critical points and complicated behavior.

Discontinuity

A point where a function is not defined or does not meet the criteria of continuity.

Endpoints

The values of a function at a closed interval that could potentially yield absolute extrema.

Function behavior near critical points

Describes how the function's value increases or decreases around a critical point based on the first derivative's sign.

Inflection point

A point where the function changes concavity, but not necessarily an extremum.

Horizontal tangent

A tangent line at a point where the slope (f'(c)) is zero.

Asymptote

A line that a graph approaches but never touches.

Limit

The value that a function approaches as the input approaches a specified value.

Graph interpretation

Analyzing the shape and behavior of a function's graph to determine relative or absolute extrema.

Negative slope

Indicates that the function is decreasing in that interval.

Positive slope

Indicates that the function is increasing in that interval.