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

AP Calculus AB/BC Unit 4 Visual Review

A topic-by-topic visual walkthrough of Unit 4: Contextual Applications of Differentiation — motion, related rates, linearization, and L'Hospital's Rule.

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TOPIC 4.1 Meaning of the Derivative in Context f′ is a RATE with units The derivative's units are (units of f) PER (units of x). If V(t) is volume in liters and t in minutes, V′(t) is liters per minute. Always state units when interpreting. Interpret a value in words C′(100) = 4 "When 100 units are made, cost is rising at about $4 per additional unit." Positive f′ → f increasing; negative f′ → f decreasing. Answer in the language of the problem A full interpretation names WHAT is changing, HOW FAST, in which DIRECTION, WHEN, and with correct UNITS. f′ is a rate of change with units — interpret it in the context's words. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.2 Straight-Line Motion position s(t) → velocity v = s′ → acceleration a = v′ = s″ Reading motion SPEED = |v|. Object moves right when v > 0, left when v < 0. It's momentarily AT REST when v = 0. It CHANGES DIRECTION where v changes sign. Speeding up vs. slowing down SPEEDING UP when v and a have the SAME sign. SLOWING DOWN when v and a have OPPOSITE signs. Displacement = ∫v dt (Unit 8). v = s′, a = v′; speeding up when v and a share a sign. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.3 Rates of Change in Applied Contexts Derivatives beyond motion Any quantity changing over time has a rate given by its derivative. • P′(t): population growth rate • dV/dt: how fast a tank fills • dT/dt: how fast something cools Inflow − outflow For a tank: net rate = rate IN − rate OUT. dV/dt = R_in(t) − R_out(t) Amount is rising when in > out; a MAXIMUM occurs when in = out. A classic FRQ setup. Watch the sign of the rate A positive rate means the quantity is increasing; where the rate changes sign, the quantity has a max or min. Any rate is a derivative; a tank's dV/dt = rate in − rate out. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.4 Introduction to Related Rates Linked quantities, linked rates When two quantities are related by an equation, their RATES are related too. Differentiate the equation with respect to TIME t (implicitly). Every variable gets a d/dt factor. Example setup Circle area grows as radius grows: A = πr² dA/dt = 2πr · (dr/dt) Given dr/dt, find dA/dt (or vice versa). The r in the answer comes from the moment. Differentiate with respect to time The chain rule links dA/dt to dr/dt. Related rates = implicit differentiation with t as the variable. Differentiate the relation with respect to t — related quantities have related rates. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.5 Solving Related Rates Problems A reliable procedure 1. Draw & label; note given/wanted rates. 2. Write an equation relating variables. 3. Differentiate BOTH sides w.r.t. t. 4. Plug in values (AT THE MOMENT). 5. Solve for the unknown rate + units. Substitute numbers only AFTER differentiating. Ladder example 10-ft ladder, base slides out at 2 ft/s. x² + y² = 100 2x(dx/dt) + 2y(dy/dt) = 0 At x=6, y=8: solve for dy/dt (top speed of the falling top). dy/dt = −(x/y)(dx/dt) = −1.5 ft/s Relate, differentiate w.r.t. t, THEN substitute the moment's values. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.6 Local Linearity & Linearization L(x) = f(a) + f′(a)(x − a) ≈ f(x) near a The tangent line approximates f Near x = a, the curve looks like its tangent line — use L(x) to estimate f(x). Estimate √4.1 with f(x)=√x at a=4: L(x) = 2 + (1/4)(x − 4) √4.1 ≈ 2 + 0.025 = 2.025 Over- or under-estimate? The tangent line lies BELOW a concave-up curve → linearization UNDERestimates. It lies ABOVE a concave-down curve → OVERestimates. Check the sign of f″ to decide. L(x)=f(a)+f′(a)(x−a); concave up → under-estimate, concave down → over. The Review Hub · AP Calculus AB/BC Unit 4 TOPIC 4.7 L'Hospital's Rule if lim f/g gives 0/0 or ∞/∞: lim f/g = lim f′/g′ Worked example lim(x→0) sin x / x → 0/0 = lim(x→0) cos x / 1 = 1 Differentiate top and bottom SEPARATELY (not the quotient rule), then retake the limit. Repeat if it's still indeterminate. Only for indeterminate forms You may apply it ONLY when the limit is 0/0 or ∞/∞ Other indeterminate forms (0·∞, ∞−∞, 1^∞) must be rewritten as a fraction first. Never use it on a determinate form like 2/0. For 0/0 or ∞/∞, L'Hospital: lim f/g = lim f′/g′ (differentiate top & bottom separately). The Review Hub · AP Calculus AB/BC Unit 4
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How to use the visual review

Spend 30 seconds per slide before clicking next. Look at the diagram, then ask yourself: "Could I set up and solve this kind of problem from memory?"

Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.

This is great for review the night before the exam — fast, visual, and covers every idea you need to recognize in Unit 4.