Unit 5 · Analytical Applications of Differentiation
● Core concept · ○ Supporting concept
5.1 Using the Mean Value Theorem
Mean Value Theorem (MVT) ● (core concept) — If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) with f'(c) = [f(b) − f(a)] / (b − a): some point has instantaneous rate equal to the average rate.
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Extreme Value Theorem (EVT) ● (core concept) — If f is continuous on the closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b].
Local (relative) extremum ● (core concept) — A function value f(c) that is greater than (maximum) or less than (minimum) all nearby function values.
Absolute (global) extremum ● (core concept) — The largest (maximum) or smallest (minimum) value of a function over its entire domain or a given interval.
Critical point ● (core concept) — A point c in the domain of f where f'(c) = 0 or f'(c) does not exist; all local extrema occur at critical points.
5.3 Determining Intervals on Which a Function is Increasing or Decreasing
Increasing and decreasing intervals ● (core concept) — A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0 (for f differentiable there).
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
First Derivative Test ● (core concept) — If f' changes from positive to negative at a critical point c, f has a local maximum at c; if it changes from negative to positive, f has a local minimum at c.
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
Candidates Test ● (core concept) — To find absolute extrema of a continuous function on [a, b]: evaluate f at all critical points in (a, b) and at the endpoints a and b; the largest value is the absolute maximum and the smallest is the absolute minimum.
5.6 Determining Concavity of Functions over Their Domains
Concavity ● (core concept) — A function is concave up on an interval where f''(x) > 0 (shaped like a cup) and concave down where f''(x) < 0 (shaped like a cap).
Point of inflection ● (core concept) — A point where the concavity of a function changes (and the tangent line exists there).
5.7 Using the Second Derivative Test to Determine Extrema
Second Derivative Test ● (core concept) — At a critical point c with f'(c) = 0: if f''(c) > 0, f has a local minimum at c; if f''(c) < 0, f has a local maximum at c. If f''(c) = 0, the test is inconclusive.
5.10 Introduction to Optimization Problems
Optimization ● (core concept) — Finding the maximum or minimum value of a quantity by writing it as a function of one variable, then using derivatives to locate extrema.