AP Calculus BC Unit 8 Notes: Average Value and One-Dimensional Motion with Integrals

Average value of a function (on [a,b])

A single representative “average height” of f(x) over an interval, computed by taking total signed accumulation (area) and dividing by the interval length.

Average value formula

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

Signed area (net area)

The value of a definite integral where area above the x-axis counts positive and area below the x-axis counts negative.

Interval length (b−a)

The denominator in the average value formula; it represents the width of the interval over which the function is averaged.

Average rate of change

The slope of the secant line on [a,b], given by (f(b)−f(a))/(b−a); uses only endpoint values, not an integral.

Secant line

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

Mean Value Theorem for Integrals (Average Value Theorem)

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.

Units check (for average value)

A verification that average value has the same units as f(x): ∫ f(x) dx has “(units of f)·(units of x),” and dividing by (b−a) returns units of f.

Average of endpoints misconception

The common error of using (f(a)+f(b))/2 instead of the correct average value (1/(b−a))∫_a^b f(x) dx.

Average value of |f(x)|

An average that may be needed in applications where the quantity cannot be negative (e.g., speed), computed as (1/(b−a))∫_a^b |f(x)| dx.

Position function

A function s(t) (or x(t)) giving an object’s location along a line as a function of time t.

Velocity function

v(t) = s′(t); the instantaneous rate of change of position with respect to time.

Acceleration function

a(t) = v′(t) = s″(t); the instantaneous rate of change of velocity with respect to time.

Displacement

Net change in position on [a,b], given by s(b)−s(a) = ∫_a^b v(t) dt; can be negative.

Distance traveled

Total ground covered on [a,b], given by ∫_a^b |v(t)| dt; always nonnegative and requires handling direction changes.

Direction change (in motion)

A time when velocity changes sign (often found by solving v(t)=0); used to split integrals for distance traveled.

Change in velocity from acceleration

Integrating acceleration gives velocity change: v(b)−v(a) = ∫_a^b a(t) dt.

Velocity from acceleration with initial condition

Given v(t0)=v0, velocity can be written v(t)=v0 + ∫_{t0}^t a(u) du (integral gives change, initial condition anchors the value).

Position from velocity with initial condition

Given s(t0)=s0, position can be written s(t)=s0 + ∫_{t0}^t v(u) du.

Fundamental Theorem of Calculus (motion interpretation)

Connects derivatives and integrals: if v(t)=s′(t), then ∫ v accumulates to position change; for F(t)=∫_{t0}^t v(u) du, F′(t)=v(t).

Accumulation function

A function defined by an integral with variable upper limit, e.g., F(t)=∫_{t0}^t v(u) du, representing total accumulated change up to time t.

Average velocity

The average value of v(t) on [a,b]: vavg = (1/(b−a))∫a^b v(t) dt, which also equals (s(b)−s(a))/(b−a).

Average speed

(1/(b−a))∫_a^b |v(t)| dt; equals average velocity only if v(t) does not change sign (no direction change).

Speeding up vs slowing down

An object speeds up when v(t) and a(t) have the same sign, and slows down when they have opposite signs.

Riemann sum / Trapezoidal sum (integral approximation)

Numerical methods to estimate a definite integral from a table of values, used to approximate quantities like displacement or distance when no formula is given.