Unit 5: Analytical Applications of Differentiation

Extrema

Maximum and minimum values of a function (plural of extremum).

Extremum

A maximum or minimum value of a function (singular of extrema).

Absolute (Global) Maximum

The highest value of f(x) on a specified domain; no other value on the domain is larger.

Absolute (Global) Minimum

The lowest value of f(x) on a specified domain; no other value on the domain is smaller.

Local (Relative) Maximum

A function value f(c) that is greater than nearby function values (not necessarily the greatest on the whole domain).

Local (Relative) Minimum

A function value f(c) that is less than nearby function values (not necessarily the least on the whole domain).

Instantaneous Rate of Change

The rate a function changes at a specific point; given by the derivative f'(c).

Derivative (f'(x))

A function that gives the slope of the tangent line (instantaneous rate of change) at each x where it exists.

Critical Number

A number c in the domain of f such that f'(c)=0 or f'(c) does not exist.

Critical Point

A point (c, f(c)) on the graph where c is a critical number (often a candidate for local extrema).

Horizontal Tangent

A tangent line with slope 0; occurs where f'(c)=0 (when f is differentiable there).

Corner (or Cusp)

A sharp point on a graph where the derivative does not exist, even if the function is continuous.

Vertical Tangent Line

A tangent line with infinite/undefined slope; a common reason f'(c) does not exist.

Domain

The set of x-values for which a function is defined; critical numbers must lie in the domain.

Continuous on [a,b]

The function has no breaks, jumps, or holes on the entire interval from a to b (including endpoints).

Differentiable on (a,b)

The derivative exists for every x strictly between a and b (no corners/cusps/vertical tangents inside).

Closed Interval

An interval [a,b] that includes both endpoints a and b.

Open Interval

An interval (a,b) that does not include endpoints a or b.

Bounded Interval

An interval of finite length (both endpoints are finite numbers).

Extreme Value Theorem (EVT)

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

EVT Conditions

To guarantee absolute extrema: f must be continuous on [a,b], and [a,b] must be closed and bounded.

Attains (a maximum/minimum)

The function actually reaches the value at some x in the domain (not just approaches it).

Unbounded (near a point)

Function values grow without limit (toward ±∞), so an absolute max/min may not exist.

Closed Interval Method (Candidate’s Test)

Procedure to find absolute extrema on [a,b] by evaluating f at all interior critical numbers and at endpoints a and b, then comparing values.

Candidate x-values

The x-values you must check for absolute extrema: interior critical numbers plus endpoints (when the interval is closed).

Endpoint

A boundary point a or b of an interval [a,b]; absolute extrema can occur here even if f' is not zero.

Secant Line

A line through two points on the graph, such as (a,f(a)) and (b,f(b)); its slope is the average rate of change.

Average Rate of Change

The slope of the secant line on [a,b]: (f(b)-f(a))/(b-a).

Tangent Line

A line that touches the curve at a point with slope equal to the derivative at that point (when it exists).

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a).

MVT Geometric Interpretation

There is at least one point where the tangent line is parallel to the secant line over [a,b].

Rolle’s Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) such that f'(c)=0.

First Derivative Sign Test (Increasing/Decreasing Test)

If f'(x)>0 on an interval, f is increasing there; if f'(x)<0, f is decreasing there.

Increasing (on an interval)

As x increases, f(x) increases on that interval; typically indicated by f'(x)>0.

Decreasing (on an interval)

As x increases, f(x) decreases on that interval; typically indicated by f'(x)<0.

Sign Chart (for f' or f'')

A number-line analysis showing where a derivative is positive/negative to determine increasing/decreasing or concavity.

Test Point Method

Choosing a sample x in each interval (split by critical numbers) to determine the sign of f'(x) or f''(x).

First Derivative Test

Classifies a critical number c: if f' changes +→−, local max; if f' changes −→+, local min; if no sign change, no local extremum.

False Positive Critical Point

A point where f'(c)=0 but f does not have a local max/min because f' does not change sign (e.g., f(x)=x^3 at x=0).

Second Derivative (f''(x))

The derivative of f'(x); measures how the slope f' is changing.

Concavity

Whether a graph bends upward or downward; determined by the sign of f'' (when it exists).

Concave Up

Graph bends like a cup; slopes increase; typically f''(x)>0 on the interval.

Concave Down

Graph bends like a frown; slopes decrease; typically f''(x)<0 on the interval.

Inflection Point

A point on the graph where concavity changes (concave up to down or down to up).

Inflection Point Candidates

x-values where f''(x)=0 or f''(x) does not exist; must check for an actual sign change in f''.

Second Derivative Test

If f'(c)=0 and f''(c)>0, local min; if f''(c)<0, local max; if f''(c)=0, inconclusive.

Inconclusive (Second Derivative Test)

When f'(c)=0 and f''(c)=0, the second derivative test cannot classify the point; more analysis is needed.

Monotonicity

Overall increasing/decreasing behavior of a function on intervals, often determined from the sign of f'(x).

Derivative Notation (dy/dx)

An alternative notation for f'(x), meaning the derivative of y with respect to x.

Qualitative Sketch

A rough graph built from key features (increasing/decreasing, local extrema, concavity, inflection points) rather than exact plotting.