AP Calculus AB Unit 8 Notes: Average Value, Net Change, and Motion via Integration

Average value of a function

The continuous average of a function’s outputs over an interval, found by total accumulation divided by interval length.

Average value formula

For integrable f on [a,b], favg = (1/(b−a))∫a^b f(x) dx.

Integrable (on an interval)

A function is integrable on [a,b] if its definite integral ∫_a^b f(x) dx exists (so total accumulation is well-defined).

Total accumulation (via an integral)

The net amount built up over an interval, represented by a definite integral ∫_a^b f(x) dx.

Interval length

The length of [a,b], equal to b−a; it is the denominator in the average value formula.

Equal-area rectangle interpretation

The average value is the constant height of a rectangle with base (b−a) whose area equals ∫_a^b f(x) dx.

Riemann sum

A sum of the form Σ f(xi*)Δx that approximates ∫a^b f(x) dx; it becomes exact as the partition gets finer.

Sample point (x_i*)

A chosen point in the i-th subinterval where the function is evaluated in a Riemann sum.

Subinterval

One of the smaller intervals formed when [a,b] is partitioned into n pieces for a Riemann sum.

Δx (delta x)

The width of each equal subinterval in a partition of [a,b], given by Δx = (b−a)/n.

Mean Value Theorem for Integrals

If f is continuous on [a,b], then there exists c in [a,b] such that f(c) = (1/(b−a))∫_a^b f(x) dx.

Signed area

Area counted with sign: regions above the x-axis contribute positively to ∫ f, and regions below contribute negatively.

Average rate of change

The slope of the secant line on [a,b], computed as (f(b)−f(a))/(b−a); not the same as average value.

Endpoint average misconception

The incorrect idea that average value equals (f(a)+f(b))/2; this only matches average value in special cases.

Fundamental Theorem of Calculus (FTC)

Connects derivatives and integrals; in accumulation form, if A(t)=∫_0^t f(x)dx, then A′(t)=f(t).

Net Change Theorem

If F′(x)=f(x) on [a,b], then ∫_a^b f(x) dx = F(b)−F(a), meaning “integral of a rate = total change.”

Position function (s(t))

A function giving location along a line at time t.

Velocity (v(t))

The rate of change of position: v(t)=s′(t); can be positive or negative depending on direction.

Acceleration (a(t))

The rate of change of velocity: a(t)=v′(t)=s′′(t).

Displacement

Signed change in position over [a,b], given by s(b)−s(a)=∫_a^b v(t) dt.

Total distance traveled

Total path length over [a,b], computed by ∫_a^b |v(t)| dt (requires splitting where v changes sign).

Initial condition

A given value like v(0)=2 or s(0)=5 used to determine the constant introduced when integrating.

Dummy variable (of integration)

A placeholder variable inside an integral (e.g., ∫_0^t a(u) du) used to avoid confusion with the outside variable.

Accumulation function

A function defined by an integral such as A(t)=∫_0^t v(x) dx, representing accumulated change up to time t.

Speeding up condition

A particle’s speed |v(t)| increases when velocity and acceleration have the same sign (v(t) and a(t) both positive or both negative).