AP Calculus AB case studies
AP Calculus AB · Put your knowledge to work in context.
AP Calculus AB is a College Board Advanced Placement course and exam equivalent to a first-semester college course in differential and integral calculus. It is designed for high school students who have already built solid foundations in algebra, geometry, trigonometry, analytic geometry, and elementary functions, including linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. The exam measures both procedural fluency and conceptual understanding across eight units: limits and continuity; differentiation definitions and fundamental properties; composite, implicit, and inverse differentiation; contextual applications of derivatives; analytical applications of derivatives; integration and accumulation of change; differential equations; and applications of integration. Questions use analytical, graphical, tabular, and verbal representations, and free-response questions include real-world contexts. The AP Calculus AB case studies are 5 cases and about 2,264 words.
1The Leaking Tank and the Difference Quotient
A cylindrical tank is draining through a valve at its base. A student measures the volume V(t) (in liters) at several times t (in minutes) and records: V(0)=500, V(2)=392, V(4)=300, V(6)=224, V(8)=164.
498 words, about 2 min2Choosing a Linearization: Tangent Line vs. Secant
A physics lab needs to estimate sqrt(4.1) without a calculator. Two students propose methods.
433 words, about 2 min3Related Rates Under a Constraint: The Expanding Balloon
A spherical balloon is being inflated so that its volume increases at a constant 100 cm^3/s. A technician measures the radius when it reaches 5 cm and wants to know how fast the radius is growing at that instant.
406 words, about 2 min4Justifying a Solution Exists: The Intermediate Value and Rolle Arguments
A student is analyzing g(x)=x^3-3x+1 on the interval [-2,2]. She computes g(-2)=-1, g(-1)=3, g(1)=-1, g(2)=3.
444 words, about 2 min5Accumulation and the Meaning of the Definite Integral
Water flows into a reservoir at a rate r(t) measured in liters per minute, where r(t) is positive but decreasing over a 10-minute period. A student computes the left Riemann sum L5 with 5 equal subintervals and the right Riemann sum R5 with the same subintervals, and finds L5=820 L and R5=760 L.
483 words, about 2 min