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Cheat Sheet
AP Calculus AB/BC Unit 4 Cheat SheetA one-page visual summary of Contextual Applications of Differentiation — motion, related rates, linearization, L’Hospital’s Rule, and every exam trap, on a single screen.
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The basics
What it covers: Interpreting derivatives in context, straight-line motion, related rates, linear approximation, and L’Hospital’s Rule.
Exam weight: About 10–15% of the AB exam and 6–9% of the BC exam.
The big question: How do derivatives model and solve real-world rate problems?
Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.
Key topics at a glance
Derivative in Context
A derivative is an instantaneous rate of change with units (output)/(input). Always state meaning and units.
Straight-Line Motion
v(t) = s′(t) , a(t) = v′(t) = s″(t) . Speed = |v(t)| . Speeding up when v and a share a sign.
Motion Sign Analysis
At rest: v = 0. Changes direction where v changes sign. Displacement = s(b) − s(a).
Related Rates Setup
Write an equation relating the quantities, differentiate w.r.t. t , then plug in known values and solve for the unknown rate.
Related Rates Tips
Each variable gets its rate (dx/dt, etc.) by the chain rule. Use geometry: Pythagorean, cone V = (1/3)πr²h, sphere V = (4/3)πr³.
Linear Approximation
L(x) = f(a) + f′(a)(x − a). Use a nearby point a. Overestimate if concave down, underestimate if concave up.
L’Hospital’s Rule
For 0/0 or ∞/∞ : lim f/g = lim f′/g′. Check the form first; rewrite other indeterminate forms as quotients.
Differentials
dy = f′(x) dx — the tangent-line change; the engine behind linear approximation.
The key terms and formulas you must know
Derivative in context — instantaneous rate of change with proper units.Velocity — v(t) = s′(t); signed.Acceleration — a(t) = v′(t) = s″(t).Speed — |v(t)|; increasing when v and a share a sign.Related rates — differentiate a relating equation w.r.t. t.d/dt[x²] — 2x(dx/dt).Linearization — L(x) = f(a) + f′(a)(x − a).Differential — dy = f′(x) dx.L’Hospital’s Rule — lim f/g = lim f′/g′ for 0/0 or ∞/∞.Indeterminate form — 0/0, ∞/∞ (and forms rewritten into these).
Key themes to remember
Units matter. Interpreting a derivative always means stating what it measures and its units.Motion is derivatives stacked. Position → velocity → acceleration.Related rates = implicit differentiation in time. Differentiate the relationship w.r.t. t.The tangent line approximates. Linearization trades exactness for a quick estimate near a point.Check the form before L’Hospital. The rule only applies to true indeterminate forms.
Common exam traps
Speed vs. velocity. Speed is |v|; a negative velocity can still mean increasing speed.Speeding up needs matching signs of v and a — not just a > 0.Don’t forget dx/dt. Every variable in related rates carries its own time-rate factor.Plug in numbers only after differentiating, not before, in related rates.L’Hospital only for 0/0 or ∞/∞. Applying it to a determinate limit gives wrong answers.Differentiate numerator and denominator separately — do NOT use the quotient rule for L’Hospital.Concavity sets over/under. A linear approximation is not exact; note the direction of error.