Learn Extracted exam questions AP Calculus AB 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.
How many female birds of this species arrive at the nesting area from $t=15$ to $t=45$? Show the setup for your calculations, and round your answer to the nearest integer.
On the interval $15 < t < 30$, the difference in the rates at which male and female birds of this species arrive at the nesting area can be modeled by the differentiable function $D(t)=M(t)-F(t)$, where $F$ is the function defined in part C. Is there a time $t$ in the interval $15 < t < 20$ when $D(t)=0$? Justify your answer.
The function $f$ is defined by $f(x)=1.43^{x}+0.57$, and the function $g$ is defined by $g(x)=\dfrac{14x+12}{x+12}$. The graphs of $f$ and $g$ intersect at the points $(1,2)$ and $(a,b)$, as shown in Figure 1.
[Figure 1: A graph with x-axis from 0 to 3+ and y-axis from 0 to 5+. The curve $y=g(x)$ and the curve $y=f(x)$ both start near $(0,1)$–$(0,1.5)$, rise, and intersect at the labeled point $(1,2)$ and again at the labeled point $(a,b)$ (near $x=3$, $y \approx 3.8$), with $f$ above $g$ just after the second intersection. The region $R$ is shaded/bounded between the curve $y=g(x)$, the $x$-axis, the $y$-axis, and the vertical line $x=1$ (region labeled $R$ near the origin below $g$). The shaded region between the two curves from $x=0$ to $x=a$ is shown shaded gray between $(1,2)$ and $(a,b)$.]
For $x \ge 0$, the equation $y=g(x)$ can be rewritten as $x=h(y)=\dfrac{12y-12}{14-y}$, where $h$ is a function of $y$. The graph of $x=h(y)$ and the horizontal line $y=3.5$ are shown in Figure 2.
[Figure 2: A graph with x-axis from 0 to 3+ and y-axis from 0 to 5+. The curve $x=h(y)$ starts near $(0,1)$ and increases, crossing the horizontal line $y=3.5$ near $x=3$. The region $T$ is the region bounded by the curve $x=h(y)$, the $y$-axis, and the line $y=3.5$, labeled $T$ to the left of the curve.]
END OF PART A
Let $R$ be the region bounded by the graph of $g$, the $x$-axis, the $y$-axis, and the vertical line $x=1$, as shown in Figure 1. Find the area of region $R$. Show the setup for your calculations.
Region $R$, described in part A, is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a rectangle whose height is $\dfrac{1}{3}$ times the length of its base in region $R$. Write, but do not evaluate, an integral expression that gives the volume of the solid.
The shaded region in Figure 1 is bounded by the graphs of $f$ and $g$ on the interval from $x=0$ to $x=a$. Find the area of the shaded region. Show the setup for your calculations.
Let $T$ be the region bounded by the graph of $x=h(y)$, the $y$-axis, and the horizontal line $y=3.5$, as shown in Figure 2. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when region $T$ is revolved about the $y$-axis.
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 $\dfrac{dH}{dt}=-\dfrac{1}{15}(H-20)$, where $H(t)$ is measured in degrees Celsius and $H(0)=75$. For $t>0$, it is known that $20 < H(t) < 75$.
Explain why the following could not be a slope field for the differential equation $\dfrac{dH}{dt}=-\dfrac{1}{15}(H-20)$.
[Slope field diagram: $H$-axis (vertical) from 30 to 70+ in increments of 10, $t$-axis (horizontal) from 0 to 25+ in increments of 5. At every grid point across the full range of $t$ shown, the line segments have the same positive slope for a given horizontal row and the slopes appear to depend only on $t$ (segments in each row are parallel and tilt more steeply positive at greater height $H$, but they do not appear to flatten toward zero as $H \to 20$ nor do the segments vary with $t$ within a row as required) — segments are drawn as short parallel dashes tilting up-and-to-the-right throughout, uniformly across each horizontal band of $H$ values, without change as $t$ increases.]
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 $f'$: x-axis from $-4$ to $4$, y-axis from $-1$ to $5$. The curve passes through labeled points $(-4,-0.5)$, a local minimum at $(-3,-1)$, rises through the $x$-axis between $x=-2$ and $x=-1$, continues rising to a local maximum at $(1,3)$, then decreases through $(2,1.5)$, crosses the $x$-axis at $x=3$ (touching down to a minimum of 0 at $x=3$), then rises sharply to $(4,5)$.]
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 $0 < x < 3$ at which the graph of $f$ has a point of inflection. Give a reason for your answer.
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.
A remote-controlled toy car moves back and forth along a straight path so that its velocity at time $t$ is given by the function $v$, where $v(t)$ is measured in feet per second and $t$ is measured in seconds.
The graph of $v(t)$ is shown.
[Graph of $v(t)$: t-axis from 0 to 12 in increments of 2 (labels at 2,4,6,8,10,12); v-axis unlabeled (vertical). The curve starts at the origin $(0,0)$, rises to a local maximum somewhere between $t=2$ and $t=4$, returns to and touches the $t$-axis at $t=4$, stays at $v=0$ (flat on the $t$-axis) from $t=4$ to $t=6$, then dips below the axis into a trough with minimum between $t=8$ and $t=10$, and rises back up to touch the $t$-axis at $t=12$.]
Find the acceleration of the car at time $t=1$ second. Show the work that leads to your answer.
Is the car speeding up or slowing down at time $t=1$ second? Give a reason for your answer.
Find the distance, in feet, that the car traveled over the time interval $0 \le t \le 4$ seconds. Show the work that leads to your answer.
Find the average velocity of the car over the time interval $6 \le t \le 12$ seconds. Show the work that leads to your answer.
The function $f$ is twice differentiable. The table gives values of $f$ and its derivative $f'$ at selected values of $x$.
| $x$ | 0 | 2 | 3 | 6 |
|---|---|---|---|---|
| $f(x)$ | $-1$ | 3 | 8 | 5 |
| $f'(x)$ | $-5$ | 4 | 9 | $-2$ |
Find $\displaystyle\lim_{x\to 2} \dfrac{f(x)}{x}$, or state that the limit does not exist.
Let $g(x)=f(f(x))$. Find $g'(2)$. Show the work that leads to your answer.
Let $h$ be a differentiable function such that $h(0)=10$ and $h'(x)=f'(3x)$. Find $h(2)$. Show the work that leads to your answer.
Let $k$ be the function defined by $k(x)=\displaystyle\int_0^x t^2 f(t)\,dt$. Find $k'(x)$.
Find $k''(3)$. Show the work that leads to your answer.