Unit 5: Analytical Applications of Differentiation

Derivative (local slope)

A measure of instantaneous rate of change at a point; equals the slope of the tangent line to the graph at that point.

Average rate of change

The change in output over change in input on [a,b]: (f(b)−f(a))/(b−a); slope of the secant line.

Instantaneous rate of change

The rate of change at a single point x=c; given by the derivative f'(c).

Secant line

A line through two points on a curve; its slope on [a,b] is (f(b)−f(a))/(b−a).

Tangent line

A line that touches a curve at a point and matches its local slope; slope equals f'(c).

Mean Value Theorem (MVT)

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).

MVT hypotheses

The required conditions for applying MVT: continuity on the closed interval [a,b] and differentiability on the open interval (a,b).

Continuity on [a,b] (for MVT/EVT)

No jumps, holes, or vertical asymptotes on the interval; required for MVT and for EVT on closed intervals.

Differentiability on (a,b)

The function has a derivative at every interior point; rules out corners, cusps, and vertical tangents inside the interval.

Corner (nondifferentiable point)

A point where left and right slopes exist but are not equal, so the derivative does not exist (e.g., |x| at 0).

Cusp

A sharp point where the slope becomes unbounded or changes abruptly; derivative does not exist there.

Vertical tangent

A tangent line with infinite/undefined slope; indicates nondifferentiability at that point.

Rolle’s Theorem

A special case of MVT: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c in (a,b) satisfies f'(c)=0.

Horizontal tangent

A tangent line with slope 0; occurs where f'(c)=0 (when the derivative exists).

Existence guarantee (in calculus theorems)

A conclusion that at least one point/value must occur (not necessarily unique), assuming hypotheses are met.

Non-uniqueness of c in MVT

MVT guarantees at least one c where the tangent slope matches the secant slope; there may be multiple such c values.

Common MVT mistake (interval issues)

Applying MVT when f is not continuous on [a,b] or not differentiable on (a,b), or picking a c not in (a,b).

Local (relative) maximum

A point x=c where f(c) is greater than nearby function values.

Local (relative) minimum

A point x=c where f(c) is less than nearby function values.

Absolute (global) maximum

The largest function value on an entire interval.

Absolute (global) minimum

The smallest function value on an entire interval.

Critical point / critical number

A domain value c where f'(c)=0 or f'(c) does not exist; a candidate location for extrema.

Candidate for extrema

A point (often a critical point or endpoint) that must be tested/evaluated to determine if an extremum occurs.

Extreme Value Theorem (EVT)

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

Closed interval requirement (EVT)

EVT requires [a,b] to be closed; continuity alone does not guarantee absolute extrema on an open interval.

Closed Interval Method (Candidate’s Test)

To find absolute extrema on [a,b]: find critical points in (a,b), evaluate f at those and endpoints, then compare values.

Endpoint check (absolute extrema)

For absolute maxima/minima on [a,b], endpoints must be evaluated along with critical points.

First derivative sign rule (increasing)

If f'(x)>0 on an interval, then f is increasing on that interval.

First derivative sign rule (decreasing)

If f'(x)<0 on an interval, then f is decreasing on that interval.

Sign chart (test-interval method)

A method: use critical numbers to split the number line, test f'(x) in each interval to determine increasing/decreasing behavior.

First Derivative Test

At a critical number c: +→− sign change in f' gives a local max; −→+ gives a local min; no sign change gives no local extremum.

Critical point with no extremum

A point where f'(c)=0 but f' does not change sign; often a horizontal-tangent inflection point (e.g., f(x)=x^3 at 0).

One-to-one via derivative

If f'(x)>0 everywhere on an interval, f is strictly increasing there and cannot repeat output values (so it’s one-to-one).

Constant function criterion (via MVT)

If f'(x)=0 for all x on an interval, then f is constant on that interval.

Second derivative

f''(x) measures how the slope f'(x) changes; used for concavity and inflection point analysis.

Concave up

Where f''(x)>0; slopes are increasing as x increases.

Concave down

Where f''(x)<0; slopes are decreasing as x increases.

Inflection point

A point where concavity changes (from up to down or down to up); must be confirmed by a sign change in f''.

Possible inflection point

A value where f''(x)=0 or f''(x) is undefined; it becomes an inflection point only if concavity changes.

Second Derivative Test

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

Inconclusive (second derivative test)

When f'(c)=0 and f''(c)=0; the second derivative test cannot determine max/min, so another method is needed.

Concavity from f' graph

If f' is increasing on an interval, then f is concave up there; if f' is decreasing, then f is concave down.

Interpreting f'' via f'

Because f'' is the slope of f', the sign of f'' tells whether f' is rising (positive) or falling (negative).

Curve sketching pipeline

A structured approach: domain, intercepts (if needed), f' for extrema/increase-decrease, f'' for concavity/inflection, then endpoint/asymptotic behavior.

Quotient rule

Derivative of f(x)=u/v: f'=(v·u'−u·v')/v^2; used for rational functions like x/(x^2+1).

Derivative sign controlled by numerator (rational case)

If the denominator of f' is always positive, the sign of f' (and increase/decrease) depends only on the numerator.

Optimization problem

A calculus application where you maximize or minimize an objective quantity subject to constraints and a feasible domain.

Objective function

The function Q(x) representing the quantity to maximize or minimize in an optimization problem.

Constraint equation

An equation relating variables (often from geometry or resources) used to rewrite the objective in one variable.

Feasible domain

The set of variable values that satisfy constraints and practical restrictions (e.g., lengths > 0); endpoints may need checking.