Learn Extracted exam questions AP Calculus BC 2026 Free Response
2026 Free Response
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Male birds of a certain species arrive at a nesting area over a thirty-day period. The rate at which the male birds arrive at the nesting area at time $t$ days is modeled by a differentiable function $M$, where $M(t)$ is measured in number of birds per day. Selected values of $M(t)$ are shown in the table.
| $t$ (days) | 0 | 5 | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|---|
| $M(t)$ (birds per day) | 2 | 7 | 16 | 6 | 5 | 2 | 0 |
(Note: Your calculator should be in radian mode.)
Approximate $M'(7.5)$ using the average rate of change of $M$ over the interval $5 \le t \le 10$. Show the work that leads to your answer, and indicate units of measure.
Use a midpoint Riemann sum with the three subintervals $[0,10]$, $[10,20]$, and $[20,30]$ to approximate $\displaystyle\int_0^{30} M(t)\,dt$. Show the work that leads to your answer.
Interpret the meaning of $\displaystyle\int_0^{30} M(t)\,dt$ in the context of the problem.
The rate at which female birds of the same species arrive at the same nesting area, in birds per day, is modeled by the function $F$ defined as follows.
On the interval $15
Let $S$ be the shaded region bounded by the graph of the polar curve
[Graph in the $xy$-plane showing a polar curve traced for $0 \le \theta \le \pi$. The curve produces two lobes above the $x$-axis: a smaller bump straddling the origin $O$ to the left of the $y$-axis and dipping slightly below near the $y$-axis, and a larger shaded lobe labeled $S$ to the right of the $y$-axis, bulging out to about $x=3$ and reaching up to about $y=3$ before returning to the $x$-axis around $x\approx 4$. The $y$-axis is marked with a tick at 1.]
It can be shown that $r'(\theta) = 4\cos(2\theta) - 2\sin(2\theta)$.
(Note: Your calculator should be in radian mode.)
Find the area of region $S$. Show the setup for your calculations.
There is a point on the curve at which the slope of the line tangent to the curve is $-\dfrac{3}{7}$. At this point, $\dfrac{dy}{d\theta} = \dfrac{3\sqrt{2}}{2}$. Find $\dfrac{dx}{d\theta}$ at this point. Show the work that leads to your answer.
Find the value of $\theta$ in the interval $0 < \theta < \dfrac{\pi}{2}$ at which $r$ has a critical point.
Use a derivative test to determine whether the critical point is the location of a relative minimum, a relative maximum, or neither for $r$.
Find the average distance from the origin to a point on the polar curve
A pie is taken from a hot oven and put on a table. The internal temperature of the pie at time $t$ minutes can be modeled by the function $H$ that satisfies the differential equation
Explain why the following could not be a slope field for the differential equation $\dfrac{dH}{dt} = -\dfrac{1}{15}(H-20)$.
[Slope field graph: horizontal axis $t$ from 0 to about 28 (gridlines at 5, 10, 15, 20, 25), vertical axis $H$ from about 25 to 70 (gridlines at 30, 40, 50, 60, 70). Short line segments are drawn at each grid point. Near the top of the grid (around $H=70$) the segments are steep, close to vertical, positive slope; moving down the grid the segments rotate, becoming shallow near $H=30$, with negative slope by the bottom rows. At a given height $H$, the segments look the same regardless of $t$ (slopes vary only with $H$, not with $t$, matching the autonomous equation) — but the segments are shown with a mix of orientations inconsistent with the required sign pattern of $\dfrac{dH}{dt}=-\frac{1}{15}(H-20)$.]
Find the slope of the line tangent to the graph of $H$ at time $t=0$. Show the work that leads to your answer.
It can be shown that $\dfrac{d^2H}{dt^2} = \dfrac{1}{225}(H-20)$. The line tangent to the graph of $H$ at time $t=0$ is used to approximate $H(5)$, the internal temperature of the pie at time $t=5$. Is this approximation an overestimate or an underestimate for the actual value of $H(5)$? Give a reason for your answer.
Use separation of variables to find an expression for $H(t)$, the particular solution to the given differential equation with initial condition $H(0)=75$.
Let $f$ be a twice-differentiable function on the closed interval $[-4,4]$ with $f(2)=3$. The graph of $f'$, the derivative of $f$, is shown.
[Graph of $f'$ in the $xy$-plane, $x$-axis from $-4$ to $4$ (gridlines at each integer), $y$-axis from $-1$ to $5$. The curve passes through labelled points: $(-4,-0.5)$, a local minimum at $(-3,-1)$, rising through the origin area, up to a local maximum at $(1,3)$, back down through $(2,1.5)$, continuing down to a local minimum at approximately $(3,0)$ on the $x$-axis, then rising steeply up to $(4,5)$. Caption: "Graph of $f'$".]
For $x>0$, the function $g$ is defined by $g(x) = f(x) - \ln x$. Find $g'(2)$. Show the work that leads to your answer.
Find all values of $x$ on the open interval $0
For $-4 \le x \le 4$, on what open intervals, if any, is the graph of $f$ both increasing and concave down? Give a reason for your answer.
For $-4 \le x \le 4$, find the value of $x$ at which $f$ has an absolute minimum and the value of $x$ at which $f$ has an absolute maximum. Give reasons for your answers.
Let $f$ be the function defined by $f(x) = \sqrt[3]{x-1}$, and let $g$ be the function defined by $g(x) = e^{(-2x+4)}$. The graphs of $f$ and $g$ intersect at the point $(2,1)$. Let $R$ be the region bounded by the graph of $f$, the $x$-axis, and the vertical line $x=2$, as shown in the figure.
[Graph in the $xy$-plane: $y$-axis marked at 1, $x$-axis marked at 1, 2, 3. The curve $y=f(x)$ starts near $x=1$ (where it crosses the $x$-axis) and rises to the right, passing through $(2,1)$; the curve $y=g(x)$ decreases from upper left, also passing through $(2,1)$, and continues decreasing toward the right past $x=3$. The shaded region $R$ is bounded below by the $x$-axis, on the left by the curve $y=f(x)$, and on the right by the vertical line $x=2$, lying between $x=1$ and $x=2$.]
Evaluate $\displaystyle\int_1^2 f(x)\,dx$. Show the work that leads to your answer.
Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid generated when region $R$ is rotated about the $x$-axis.
Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of region $R$.
Evaluate $\displaystyle\int_2^{\infty} g(x)\,dx$. Show the work that leads to your answer.
The Maclaurin series for a function $g$ is given by
For $x>0$, the Maclaurin series for $g$ is an alternating geometric series. Find $g(3)$. Show the work that leads to your answer.
The function $f$ is defined by $f(x) = g'(x)$. Write the first four nonzero terms of the Maclaurin series for $f$.
The second-degree Taylor polynomial for $f$ about $x=0$ is used to approximate $f\left(\dfrac{5}{2}\right)$ as $-\dfrac{3}{10}$, where $f$ is the function defined in part B. Justify that $\left|f\left(\dfrac{5}{2}\right) - \left(-\dfrac{3}{10}\right)\right| \le \dfrac{1}{5}$.
The function $h$ is defined by $h(x) = 25g(x) - 2e^x$. Write the first three nonzero terms of the Maclaurin series for $e^x$.
Write the first three nonzero terms of the Maclaurin series for $h$.