Calculus Review: Local Extrema, Intervals, and Average Rate of Change (Video Notes)

Flashcards covering local and absolute extrema, intervals of increase/decrease, endpoints vs interior points, the Extreme Value Theorem, and the average rate of change and secant line concepts from the lecture.

Define a local maximum of f at x = c

There exists an open interval around c such that f(c) ≥ f(x) for all x in that interval (the inequality may be non-strict).

Define a local minimum of f at x = c

There exists an open interval around c such that f(c) ≤ f(x) for all x in that interval (the inequality may be non-strict).

What is an absolute maximum of f on its domain?

The greatest value of f(x) over the entire domain of f.

What is an absolute minimum of f on its domain?

The smallest value of f(x) over the entire domain of f.

State the Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f has both an absolute maximum and an absolute minimum on [a,b].

What is an interval of increase?

An interval on which f(x1) < f(x2) for any x1 < x2 in the interval.

What is an interval of decrease?

An interval on which f(x1) > f(x2) for any x1 < x2 in the interval.

What is the average rate of change of f on [a,b]?

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

What does a positive average rate of change indicate?

The function increases on the interval from a to b.

What does a negative average rate of change indicate?

The function decreases on the interval from a to b.

What does a zero average rate of change indicate?

The function is constant on the interval from a to b.

What is a secant line?

A straight line through two points on the graph, typically (a, f(a)) and (b, f(b)); its slope equals the average rate of change on [a,b].

What is a closed interval [a,b]?

An interval including its endpoints a and b; extrema on [a,b] may occur at interior points or at a and b.

What is an open interval?

An interval that does not include its endpoints; used when defining local extrema around a point c.

How do absolute maxima/minima differ from local maxima/minima?

Absolute maxima/minima are the highest/lowest values over the entire domain; local maxima/minima are the highest/lowest within a neighborhood of a point.

What graph shapes illustrate local extrema?

A W-shaped graph for a local maximum and a local minimum; a U-shaped graph for a minimum.