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Unit 4 · Contextual Applications Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Calculus AB/BC Unit 4 Essentials

The must-know terms and core concepts for Unit 4: Contextual Applications of Differentiation. Every vocabulary word, formula, and idea you need to master.

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Key Concept 1
A derivative is a rate of change — read its meaning and units in context
In applied problems a derivative measures an instantaneous rate of change of one quantity with respect to another, and it carries units of (output units)/(input units). The clearest example is straight-line motion: position s(t) has derivative velocity v(t) = s′(t), and velocity has derivative acceleration a(t) = s″(t). Speed = |v(t)|, and an object is speeding up exactly when velocity and acceleration share the same sign.
Rate of Change Motion Speed vs. Velocity
Key Concept 2
Related rates connect the rates of changing quantities through time
When several quantities change together, their rates are linked. To solve a related rates problem, write an equation relating the quantities, differentiate both sides with respect to time t (so each variable picks up a rate like dx/dt by the chain rule), substitute the known values, and solve for the unknown rate. The setup usually comes from geometry — Pythagorean relationships, or cone and sphere volume formulas — so identifying the right equation is the key step.
Related Rates d/dt Geometry Setup
Key Concept 3
Linearization approximates values; L’Hospital’s Rule evaluates indeterminate limits
The tangent line gives a quick estimate: linearization L(x) = f(a) + f′(a)(x − a) approximates f near a known point a, over- or under-estimating depending on concavity. And when a limit gives an indeterminate form 0/0 or ∞/∞, L’Hospital’s Rule replaces it with the limit of the derivatives: lim f/g = lim f′/g′ — always check the form first.
Linearization Concavity L’Hospital
Derivative in context
An instantaneous rate of change with units (output)/(input).
Context
Units of a derivative
Units of the output quantity divided by units of the input quantity.
Context
Position function
s(t), giving location along a line at time t.
Motion
Velocity
v(t) = s′(t); signed rate of change of position.
Motion
Acceleration
a(t) = v′(t) = s″(t); rate of change of velocity.
Motion
Speed
|v(t)|; increasing when v and a share a sign.
Motion
At rest / direction change
v(t) = 0; direction changes where v changes sign.
Motion
Displacement
s(b) − s(a); net change in position.
Motion
Related rates
Rates of connected quantities linked by a differentiated equation.
Related Rates
Differentiate w.r.t. t
Applying d/dt so each variable gains a rate factor (chain rule).
Related Rates
Linearization
L(x) = f(a) + f′(a)(x − a); tangent-line approximation.
Approximation
Differential
dy = f′(x) dx; approximate change in y.
Approximation
Over/underestimate
Set by concavity: over if concave down, under if concave up.
Approximation
Indeterminate form
A limit form like 0/0 or ∞/∞ with no immediate value.
L’Hospital
L’Hospital’s Rule
lim f/g = lim f′/g′ for indeterminate 0/0 or ∞/∞.
L’Hospital