AP Calculus BC case studies
AP Calculus BC · Put your knowledge to work in context.
AP Calculus BC is a College Board Advanced Placement course and exam described as equivalent to a first-semester college calculus course plus the subsequent single-variable calculus course. It is aimed at high school students who have already completed coursework in algebra, geometry, trigonometry, analytic geometry, and elementary functions, and who intend further study in mathematics, engineering, computer science, or economics. The AP Calculus BC case studies are 5 cases and about 1,960 words.
1The Two-Sided Limit at a Piecewise Junction
A student is analyzing the piecewise function f defined by f(x) = (x^2 - 9)/(x - 3) for x < 3, f(x) = k for x = 3, and f(x) = (x - 3)/(sqrt(x+1) - 2) for x 3. A classmate claims that because the left branch simplifies to x + 3 and the right branch is a rationalized-looking fraction, the two one-sided limits must both equal 6, so setting k = 6 makes f continuous at x = 3.
405 words, about 2 min2Related Rates Under a Leaking Constraint
A conical tank (vertex down) has height 12 m and top radius 4 m. Water leaks out at a constant rate of 3 m³/min.
394 words, about 2 min3Justifying a Series Test Choice
A student must determine whether three series converge, and she has a habit of reaching for the Ratio Test every time. The series are: (A) Σ (n!)/(2^n) from n = 1 to ∞ (B) Σ 1/(n² + 3n) from n = 1 to ∞ (C) Σ (−1)^n /√(n+1) from n = 1 to ∞ Her study partner insists the Ratio Test settles all three because 'it's the most powerful one.' The student suspects each series calls for a different tool and wants to match the right test to each, including checking the hypotheses each test requires before applying it.
347 words, about 2 min4Implicit Differentiation and a Vertical Tangent
A curve is defined implicitly by x² + xy + y² = 7. A student computes dy/dx and wants to find all points on the curve where the tangent line is horizontal and where it is vertical.
391 words, about 2 min5Accumulation, Average Value, and a Modeling Assumption
A factory's pollutant discharge rate is modeled by R(t) = 50 + 30 sin(t/4) gallons per hour, for 0 ≤ t ≤ 24 hours. The plant manager reports 'total discharge = rate × time = 50 × 24 = 1200 gallons' and claims the sinusoidal term 'averages out to zero anyway.' An environmental analyst objects that the average of sin(t/4) over a partial period is not zero, and that the correct total is an accumulation integral.
423 words, about 2 min