AP Calculus BC Unit 4: Contextual Applications of Differentiation (Interpretation, Motion, Related Rates, Linearization, and L’Hospital’s Rule)

Derivative (in context)

A quantity that represents the instantaneous rate of change of an output with respect to an input at a specific input value.

Limit definition of the derivative

The definition f'(a)=lim_{h→0}[f(a+h)−f(a)]/h, turning an average rate of change into an instantaneous rate.

Average rate of change

The change in output over change in input on an interval: (f(b)−f(a))/(b−a).

Instantaneous rate of change

The rate of change at a single input value; equal to the derivative at that point.

Tangent line

The line that touches a differentiable curve at a point and has the same instantaneous slope as the curve there.

Slope of a tangent line

The value of f'(a); interpreted as “rise over run” (change in output per change in input) at x=a.

Differentiable (at a point)

A function is differentiable at a point if its derivative exists there (so it has a well-defined tangent slope).

Units of a derivative

Always “output units per input unit,” which helps interpret what the derivative value means.

Function value vs. derivative value

f(a) is an output amount (with output units), while f'(a) is a rate (with output-per-input units).

Sign of the derivative

If f'(a)>0 the function is increasing at a; if f'(a)<0 it’s decreasing; if f'(a)=0 it’s momentarily not changing.

Magnitude of the derivative

The size |f'(a)| indicates how rapidly the output is changing (large = rapid change, small = slow change).

Horizontal tangent

A tangent line with slope 0 (f'(a)=0); means the quantity is instantaneously not changing at that input.

Symmetric difference quotient

A table-based estimate of a derivative using points on both sides: f'(a)≈[f(a+h)−f(a−h)]/(2h).

One-sided difference quotient

A derivative estimate using data from only one side: f'(a)≈[f(a+h)−f(a)]/h (or a backward version).

Position function (s(t) or x(t))

A function giving location along a line as a function of time.

Velocity (v(t))

The derivative of position: v(t)=s'(t); measures how fast position changes (includes direction).

Acceleration (a(t))

The derivative of velocity: a(t)=v'(t)=s''(t); measures how fast velocity changes.

Displacement

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

Distance traveled

Total amount moved regardless of direction; computed as ∫_a^b |v(t)| dt when applicable.

Speed

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

At rest

A moment when velocity is zero: v(t)=0.

Moving in the positive direction

Occurs when v(t)>0 (often described as moving right).

Moving in the negative direction

Occurs when v(t)<0 (often described as moving left).

Direction change (turning around)

Happens when velocity changes sign (often at a time when v(t)=0, but v(t)=0 alone is not enough).

Speeding up

A situation where speed |v(t)| is increasing; in 1D this occurs when velocity and acceleration have the same sign.

Slowing down

A situation where speed |v(t)| is decreasing; in 1D this occurs when velocity and acceleration have opposite signs.

Sign analysis (interval testing)

Checking the signs of expressions (like v(t) and a(t)) on intervals to decide direction, rest, speeding up, etc.

Concavity (of position)

The shape of a position graph controlled by s''(t); relates to whether velocity is increasing or decreasing.

Local maximum / local minimum (caution)

A point where a function is locally highest/lowest; f'(a)=0 alone does not guarantee a max or min without more information.

Rate interpretation template

A good f'(a) interpretation states: at input a, the output is increasing/decreasing at (number) output-units per input-unit.

Marginal cost

If C(q) is cost in dollars for q items, C'(q) estimates the additional cost per item near q (dollars per item).

Linear approximation of small changes

For small Δx, the change in output is approximated by Δf≈f'(a)Δx near x=a.

Chain rule (rates form)

For indirect dependence: dy/dt = (dy/dx)(dx/dt), linking a rate “through” another variable.

Related rates

Problems where two or more changing quantities are linked by an equation; you’re given one rate and asked for another.

Differentiate with respect to time

The key related-rates move: treat all changing variables as functions of time and differentiate the relation using d/dt.

Implicit differentiation (in related rates)

Differentiating an equation like x^2+y^2=25 with respect to time, producing terms like 2x(dx/dt)+2y(dy/dt).

“Plug in numbers after differentiating” rule

In related rates, substitute the given instant’s values only after taking d/dt, so variables aren’t lost.

ERDS strategy

A workflow for related rates: Equation, Differentiate, Replace (substitute), Solve.

Pythagorean related-rates setup (ladder)

A right-triangle model x^2+y^2=L^2 with constant L, relating horizontal and vertical motion of a ladder.

Similar triangles (related rates)

A geometry tool used to relate variables (like r and h in a cone) so the main equation has fewer variables before differentiating.

Cone volume formula

V=(1/3)πr^2h; often combined with similar triangles to express V in terms of one variable in related rates.

Sphere volume formula

V=(4/3)πr^3; differentiating with respect to time gives dV/dt=4πr^2(dr/dt).

Local linearization (linearization)

Approximating a function near x=a by its tangent line: L(x)=f(a)+f'(a)(x−a).

Tangent-line approximation

Using L(x) to estimate f(x) for x close to a, based on the idea that differentiable functions look linear locally.

Differentials

A notation-based linear approximation method using dy=f'(x)dx to estimate small output changes from small input changes.

Differential error/measurement propagation

Estimating how measurement uncertainty dr affects an output like volume via dV≈(dV/dr)dr.

Concavity and linearization error

Concave up functions lie above their tangent lines (linearization underestimates); concave down functions lie below (overestimates).

L’Hospital’s Rule

A limit method: if a quotient gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (when conditions apply).

Indeterminate form

A limit form that doesn’t determine a value directly, such as 0/0 or ∞/∞ (common triggers for L’Hospital’s Rule).

Repeated application (L’Hospital)

If the limit is still indeterminate after differentiating once, you may differentiate again and re-check until the form becomes determinate.