ap-calculus-bc

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2025
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2024 free-response

6 maths questions

Q1 Connected Rates of Change Table-Based Estimation with Rate of Change Interpretation View
The temperature of coffee in a cup at time $t$ minutes is modeled by a decreasing differentiable function $C$, where $C(t)$ is measured in degrees Celsius. For $0 \leq t \leq 12$, selected values of $C(t)$ are given in the table shown.
\begin{tabular}{ c } $t$
(minutes)
& 0 & 3 & 7 & 12 \hline
$C(t)$
(degrees Celsius)
& 100 & 85 & 69 & 55 \hline \end{tabular}
(a) Approximate $C'(5)$ using the average rate of change of $C$ over the interval $3 \leq t \leq 7$. Show the work that leads to your answer and include units of measure.
(b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of $\int_{0}^{12} C(t)\, dt$. Interpret the meaning of $\frac{1}{12} \int_{0}^{12} C(t)\, dt$ in the context of the problem.
(c) For $12 \leq t \leq 20$, the rate of change of the temperature of the coffee is modeled by $C'(t) = \frac{-24.55 e^{0.01t}}{t}$, where $C'(t)$ is measured in degrees Celsius per minute. Find the temperature of the coffee at time $t = 20$. Show the setup for your calculations.
(d) For the model defined in part (c), it can be shown that $C''(t) = \frac{0.2455 e^{0.01t}(100 - t)}{t^2}$. For $12 < t < 20$, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.
Q2 Variable acceleration (vectors) View
A particle moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ seconds, where $x(t)$ and $y(t)$ are measured in centimeters. It is known that $x'(t) = 8t - t^2$ and $y'(t) = -t + \sqrt{t^{1.2} + 20}$. At time $t = 2$ seconds, the particle is at the point $(3, 6)$.
(a) Find the speed of the particle at time $t = 2$ seconds. Show the setup for your calculations.
(b) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 2$. Show the setup for your calculations.
(c) Find the $y$-coordinate of the position of the particle at the time $t = 0$. Show the setup for your calculations.
(d) For $2 \leq t \leq 8$, the particle remains in the first quadrant. Find all times $t$ in the interval $2 \leq t \leq 8$ when the particle is moving toward the $x$-axis. Give a reason for your answer.
Q3 Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View
The depth of seawater at a location can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$, where $H(t)$ is measured in feet and $t$ is measured in hours after noon $(t = 0)$. It is known that $H(0) = 4$.
(a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, $y = H(t)$, through the point $(0, 4)$.
(b) For $0 < t < 5$, it can be shown that $H(t) > 1$. Find the value of $t$, for $0 < t < 5$, at which $H$ has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your answer.
(c) Use separation of variables to find $y = H(t)$, the particular solution to the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$ with initial condition $H(0) = 4$.
Q4 Indefinite & Definite Integrals Accumulation Function Analysis View
The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x = -6$, and the $x$- and $y$-axes. Region $R$ has area 12.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.
Q5 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) View
The function $f$ is twice differentiable for all $x$ with $f(0) = 0$. Values of $f'$, the derivative of $f$, are given in the table for selected values of $x$.
$x$0$\pi$$2\pi$
$f'(x)$560

(a) For $x \geq 0$, the function $h$ is defined by $h(x) = \int_{0}^{x} \sqrt{1 + \left(f'(t)\right)^2}\, dt$. Find the value of $h'(\pi)$. Show the work that leads to your answer.
(b) What information does $\int_{0}^{\pi} \sqrt{1 + \left(f'(x)\right)^2}\, dx$ provide about the graph of $f$?
(c) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f(2\pi)$. Show the computations that lead to your answer.
(d) Find $\int (t + 5)\cos\left(\frac{t}{4}\right)\, dt$. Show the work that leads to your answer.
Q6 Sequences and series, recurrence and convergence Series convergence and power series analysis View
The Maclaurin series for a function $f$ is given by $\sum_{n=1}^{\infty} \frac{(n+1)x^n}{n^2 6^n}$ and converges to $f(x)$ for all $x$ in the interval of convergence. It can be shown that the Maclaurin series for $f$ has a radius of convergence of 6.
(a) Determine whether the Maclaurin series for $f$ converges or diverges at $x = 6$. Give a reason for your answer.
(b) It can be shown that $f(-3) = \sum_{n=1}^{\infty} \frac{(n+1)(-3)^n}{n^2 6^n} = \sum_{n=1}^{\infty} \frac{n+1}{n^2}\left(-\frac{1}{2}\right)^n$ and that the first three terms of this series sum to $S_3 = -\frac{125}{144}$. Show that $\left|f(-3) - S_3\right| < \frac{1}{50}$.
(c) Find the general term of the Maclaurin series for $f'$, the derivative of $f$. Find the radius of convergence of the Maclaurin series for $f'$.
(d) Let $g(x) = \sum_{n=1}^{\infty} \frac{(n+1)x^{2n}}{n^2 3^n}$. Use the ratio test to determine the radius of convergence of the Maclaurin series for $g$.