2025 AP Calculus AB Free-Response Questions
Advanced PlacementAP Calculus ABFree ResponseMathematics
6 questions54 pts
6 gradable54 rubric criteria0 dependencies
q1Question 1
q1
Question 1
1. An invasive species of plant appears in a fruit grove at time t = 0 and begins to spread. The function C defined by C(t) = 7.6 arctan(0.2t) models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that C'(t) = 38/(25 + t²). (Note: Your calculator should be in radian mode.) A. Find the average number of acres affected by the invasive species from time t = 0 to time t = 4 weeks. Show the setup for your calculations. B. Find the time t when the instantaneous rate of change of C equals the average rate of change of C over the time interval 0 ≤ t ≤ 4. Show the setup for your calculations. C. Assume that the invasive species continues to spread according to the given model for all times t > 0. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression. D. At time t = 4 weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function A, defined by A(t) = C(t) − ∫₄ᵗ 0.1 · ln(x) dx, models the number of acres affected by the species over the time interval 4 ≤ t ≤ 36. At what time t, for 4 ≤ t ≤ 36, does A attain its maximum value? Justify your answer.
Rubric Criteria (9)
P1Presents the average-value setup \(\frac{1}{4}\int_0^4 C(t)\,dt\), or an equivalent definite integral together with evidence of division by 4.
1.0000pts
P2States that the average number of acres affected from \(t=0\) to \(t=4\) is \(2.778\) acres, or an equivalent value accurate to three decimal places by rounding or truncation.
1.0000pts
P3Presents the average rate of change of \(C\) on \([0,4]\) as \(\frac{C(4)-C(0)}{4-0}\), or an equivalent expression or value such as \(\frac{C(4)}4\) or \(1.282008\).
1.0000pts
P4Uses an equation equating the instantaneous rate \(C'(t)=\frac{38}{25+t^2}\) to the average rate of change and obtains \(t=2.154\) weeks, or an equivalent value accurate to three decimal places.
1.0000pts
P5Writes a limit expression describing the end behavior, such as \(\lim_{t\to\infty}C'(t)=\lim_{t\to\infty}\frac{38}{25+t^2}\) or \(\lim_{t\to\infty}C(t)\).
1.0000pts
P6Evaluates the limit of the rate of change as \(0\), without relying solely on arithmetic expressions involving infinity.
1.0000pts
P7Considers a critical point for \(A\) by setting \(A'(t)=0\), equivalently \(C'(t)-0.1\ln t=0\) or \(C'(t)=0.1\ln t\), or discusses a sign change of \(A'\).
1.0000pts
P8Justifies the global maximum on \(4\le t\le36\) by correctly evaluating \(A\) at \(t=4\), \(t=11.441700\), and \(t=36\), or by showing that \(A'>0\) before \(11.442\) and \(A'<0\) after it, or an equivalent global argument.
1.0000pts
P9States that \(A\) attains its maximum at \(t=11.442\) weeks, or an equivalent value such as \(11.441\), accurate to three decimal places.
1.0000pts
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