AP Calculus BC Unit 5: Analytical Applications of Differentiation (Comprehensive Study Notes)

Derivative

A function f'(x) that measures the instantaneous rate of change (tangent slope) of f(x) with respect to x.

Instantaneous rate of change

How fast f(x) is changing at a specific x-value; equals the derivative f'(x) at that point.

Tangent line

The line that touches a curve at a point and has slope f'(c) at x=c.

Increasing on an interval

For any x1<x2 in the interval, f(x1)<f(x2); the graph goes uphill left to right.

Decreasing on an interval

For any x1f(x2); the graph goes downhill left to right.

Derivative test for increasing

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

Derivative test for decreasing

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

Critical number

A value c in the domain of f where either f'(c)=0 or f'(c) does not exist.

Critical point

The point (c,f(c)) on the graph corresponding to a critical number c.

Relative (local) extrema

A local maximum or minimum: the highest or lowest value of f in a neighborhood around a point.

Local maximum

A point x=c where f changes from increasing to decreasing (often where f' changes from + to −).

Local minimum

A point x=c where f changes from decreasing to increasing (often where f' changes from − to +).

First Derivative Test

Classifies a critical number c by checking the sign of f'(x) on each side: +→− gives local max; −→+ gives local min; no sign change gives neither.

Sign change (of f')

A switch in f'(x) from positive to negative or negative to positive across a point; used to confirm local extrema.

Sign chart

A table/number line showing where f'(x) (or f''(x)) is positive, negative, or zero to determine behavior on intervals.

Test point

A chosen x-value in an interval used to determine the sign of f'(x) (or f''(x)) on that interval.

Even exponent factor (in f')

A factor like (x-a)^2 in f'(x) that becomes 0 at x=a but does not change sign across a, often meaning no local extremum there.

Endpoint

A boundary value a or b of a closed interval [a,b]; must be checked when finding absolute maxima/minima.

Absolute extremum

The greatest (absolute maximum) or least (absolute minimum) function value on an entire interval.

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].

Candidate’s Test

To find absolute extrema on [a,b]: evaluate f at interior critical numbers and at endpoints a and b, then compare all values.

Closed interval

An interval [a,b] that includes both endpoints; required (with continuity) for EVT guarantees.

Continuous on [a,b]

f has no breaks, holes, or jumps anywhere on the interval [a,b]; a key condition for EVT and MVT.

Bounded interval

An interval with finite length (a and b are finite), such as [a,b].

Secant line

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

Average rate of change

The slope of the secant line: (f(b)-f(a))/(b-a).

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) with f'(c)=0.

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).

Differentiable on (a,b)

f has a derivative at every point in the open interval (a,b); excludes corners, cusps, vertical tangents, and discontinuities.

Horizontal tangent

A tangent line with slope 0; occurs where f'(c)=0 (but does not automatically mean a max/min).

Concavity

How the graph bends; determined by whether slopes are increasing or decreasing, using the second derivative.

Second derivative

f''(x)= (f'(x))'; measures how the slope f'(x) changes and determines concavity.

Concave up

f''(x)>0 on an interval; slopes increase left to right and the graph bends like a cup.

Concave down

f''(x)<0 on an interval; slopes decrease left to right and the graph bends like a cap.

Inflection point

A point where concavity changes (concave up to concave down or vice versa); requires a sign change in f''.

Second Derivative Test

If f'(c)=0 and f''(c) exists: f''(c)>0 ⇒ local min; f''(c)<0 ⇒ local max; f''(c)=0 ⇒ inconclusive.

Inconclusive result

When a test (often the second derivative test with f''(c)=0) cannot determine whether a critical point is a max, min, or neither.

f' graph above x-axis

Where f'(x)>0; implies f is increasing on that interval.

f' graph below x-axis

Where f'(x)<0; implies f is decreasing on that interval.

f' touches x-axis (no sign change)

When f'(c)=0 but f' does not cross the axis; indicates no local extremum at c (often due to an even-multiplicity zero).

Optimization

Finding the maximum or minimum of a quantity under constraints, typically by building an objective function and checking critical points and endpoints.

Objective function

The function representing what you want to maximize or minimize (e.g., area, cost, profit).

Constraint

An equation/condition linking variables (e.g., 2x+2y=200) used to rewrite the objective in one variable.

Feasible domain

The set of allowed input values based on the problem’s constraints (often a closed interval to test endpoints).

Position function s(t)

A function giving location along a line at time t.

Velocity v(t)

v(t)=s'(t); the rate of change of position. Positive means moving right, negative means moving left.

Acceleration a(t)

a(t)=v'(t)=s''(t); the rate of change of velocity.

Speed |v(t)|

The magnitude of velocity; always nonnegative and equals |v(t)|.

Displacement

Net change in position over [a,b]: s(b)-s(a).

Distance traveled

Total distance over [a,b], computed as ∫_a^b |v(t)| dt (often by splitting at times where v(t)=0).