Contextual Applications of Differentiation — Free AP Calculus AB Review Games.
This unit covers related rates, linearization and L'Hopital's rule — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers related rates, linearization and L'Hopital's rule — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Related Rates
Related rates problems require differentiating an equation with respect to time, where multiple variables are changing simultaneously. Students must identify which rates are given and which are unknown, then apply implicit differentiation with respect to t. The chain rule is always involved since every variable is a function of time.
Key Points
- Draw and label a diagram; write an equation relating the variables before differentiating
- Differentiate both sides with respect to t using implicit differentiation — every variable gets a dx/dt or dy/dt term
- Substitute known values AFTER differentiating, not before
- Common setups: Pythagorean theorem (ladder problems), volume/area formulas (expanding circles, filling cones), similar triangles
A ladder 10 ft long rests against a vertical wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
Set up x² + y² = 100, then differentiate: 2x(dx/dt) + 2y(dy/dt) = 0. At x = 6, y = √(100−36) = 8. Substituting: 2(6)(2) + 2(8)(dy/dt) = 0, so dy/dt = −24/16 = −3/2 ft/s. The negative sign confirms the top is sliding down.
2 Linearization
Linearization uses the tangent line at a known point to approximate the value of a function near that point. The formula is L(x) = f(a) + f′(a)(x − a), where a is the point of tangency. Students must recognize when an approximation is an overestimate or underestimate based on concavity.
Key Points
- L(x) = f(a) + f′(a)(x − a) is the equation of the tangent line used as an approximation
- If f is concave up (f″ > 0), the tangent line lies below the curve → linearization is an underestimate
- If f is concave down (f″ < 0), the tangent line lies above the curve → linearization is an overestimate
- Choose a to be a nearby value where f(a) and f′(a) are easy to compute exactly
Use linearization to approximate √(9.1).
Let f(x) = √x and choose a = 9 since it is nearby and gives an exact value. Then f(9) = 3 and f′(x) = 1/(2√x), so f′(9) = 1/6. Applying the formula: L(9.1) = 3 + (1/6)(9.1 − 9) = 3 + (1/6)(0.1) ≈ 3.0167. Since f″(x) = −1/(4x^(3/2)) < 0 (concave down), this is an overestimate.
3 L'Hopital's Rule
L'Hôpital's Rule resolves indeterminate limits of the forms 0/0 or ∞/∞ by replacing the limit of f(x)/g(x) with the limit of f′(x)/g′(x). Students must verify the indeterminate form exists before applying the rule, and may need to apply it more than once. Other indeterminate forms (0·∞, ∞−∞, 1^∞, 0^0, ∞^0) require algebraic rewriting into 0/0 or ∞/∞ first.
Key Points
- Only apply when direct substitution yields 0/0 or ∞/∞ — always confirm the indeterminate form first
- Differentiate numerator and denominator separately (not the quotient rule)
- Can be applied repeatedly if the result remains indeterminate
- For 0·∞ forms, rewrite one factor as a reciprocal to create a 0/0 or ∞/∞ fraction before applying the rule
Evaluate lim(x→0) (sin 3x) / (5x).
Direct substitution gives 0/0, confirming the indeterminate form. Apply L'Hôpital's Rule: differentiate the numerator to get 3cos(3x) and the denominator to get 5. The limit becomes lim(x→0) 3cos(3x)/5. Substituting x = 0 gives 3cos(0)/5 = 3/5.
Questions, answered.
What is Contextual Applications of Differentiation?
Contextual Applications of Differentiation is Unit 4 of AP Calculus AB, covering related rates, linearization and L'Hopital's rule.
How to study for AP Calculus AB Unit 4?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.