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], $f_{avg} = \frac{1}{b-a} \int_a^b f(x)\,dx.$

Integrable (on an interval)

A function is integrable on [a,b] if its definite integral $\int_a^b f(x) \,dx$ 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 \nint_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 \nint_a^b f(x) \,dx.

Riemann sum

A sum of the form $\sum f(x_i^*)\Delta x$ that approximates $\int_a^b f(x) \,dx$; 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 $\Delta x = \frac{b-a}{n}.$

Mean Value Theorem for Integrals

If f is continuous on [a,b], then there exists c in [a,b] such that $f(c) = \frac{1}{b-a} \int_a^b f(x) \,dx.$

Signed area

Area counted with sign: regions above the x-axis contribute positively to $\int f$, 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 $A(t)=\int_0^t f(x) \,dx$, then $A′(t)=f(t)$.

Net Change Theorem

If $F'(x)=f(x)$ on [a,b], then $\int_a^b f(x) \,dx = F(b)-F(a),$ 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 $s(b)-s(a)=\int_a^b v(t) \,dt.$

Total distance traveled

Total path length over [a,b], computed by \nint_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., $\int_0^t a(u) \, du$) used to avoid confusion with the outside variable.

Accumulation function

A function defined by an integral such as $A(t)=\int_0^t v(x) \,dx$, 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).