AP-Style Practice Questions
Topics 8.1 – 8.12AB
Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part — each question below is labeled accordingly.
The average value of $f(x)=x^{2}$ on $[0,3]$ is
A particle moves along the $x$-axis with velocity $v(t)=t-2$ for $0\le t\le 3$. The displacement of the particle on this interval is
A particle moves with velocity $v(t)=t-2$ for $0\le t\le 3$. The total distance traveled is
Water flows into a tank at a rate of $r(t)=4+\sin(t)$ gallons per minute, for $0\le t\le 6$. The tank initially contains $50$ gallons. The amount in the tank at $t=6$ is closest to
The area enclosed by $y=x$ and $y=x^{2}$ is
Which integral gives the area enclosed by $x=y^{2}$ and $x=y+2$?
The total area of the regions enclosed between $y=x^{3}-x$ and the $x$-axis is
The base of a solid is the region in the $xy$-plane bounded by $y=x$, $y=0$, and $x=2$. Cross sections perpendicular to the $x$-axis are squares. The volume of the solid is
The base of a solid is the region bounded by $y=\sqrt{x}$ and the $x$-axis on $[0,4]$. Cross sections perpendicular to the $x$-axis are equilateral triangles. The volume is
The region bounded by $y=\sqrt{x}$, $y=0$, and $x=4$ is revolved about the $x$-axis. The volume is
The region bounded by $y=x^{2}$, $y=0$, and $x=2$ is revolved about the line $y=-1$. Which integral gives the volume?
Let $R$ be the region enclosed by $y=x$ and $y=x^{2}$. The volume of the solid formed when $R$ is revolved about the $x$-axis is
Let $R$ be the region bounded by $y=x^{2}$ and $y=4$. Which integral gives the volume of the solid generated when $R$ is revolved about the line $y=5$?
The rate at which people enter a park is modeled by $E(t)$ people per hour, where $t$ is hours since opening. Selected values:
| $t$ (hr) | $0$ | $2$ | $4$ | $6$ | $8$ |
|---|---|---|---|---|---|
| $E(t)$ | $100$ | $240$ | $380$ | $300$ | $150$ |
Using a left Riemann sum with the four subintervals of equal length, the approximate total number of people who entered during the $8$ hours is
If $f(x)=4x$, the average value of $f$ on $[1,3]$ is
A particle has velocity $v(t)=3t^{2}-6t$ and initial position $x(0)=2$. Then $x(2)=$
The area enclosed by $y=4-x^{2}$ and the $x$-axis is
The base of a solid is the region under $y=\sqrt{x}$ on $[0,4]$. Cross sections perpendicular to the $x$-axis are semicircles with diameter in the base. The volume is
Free-response answers must include setup, units, and interpretation in context where appropriate. A calculator is permitted unless marked otherwise.
A particle moves along the $x$-axis with velocity $v(t)=t^{2}-4t+3$ for $0\le t\le 4$. The particle is at position $x=2$ when $t=0$.
Let $R$ be the region in the first quadrant bounded by the graphs of $y=\sin(\pi x)$ and $y=x-x^{2}$.
Water is being pumped into a tank at a rate of $P(t)=20+5\sin(t/2)$ gallons per minute. At the same time, water is leaking out at a rate of $L(t)=2+0.5t$ gallons per minute, for $0\le t\le 30$. At time $t=0$, the tank contains $400$ gallons of water.
Let $R$ be the region enclosed by the graphs of $y=e^{-x^{2}}$ and $y=\dfrac{1}{2}$.
A car travels along a straight road for $12$ seconds. The car's velocity $v(t)$, in meters per second, is differentiable. Selected values are given.
| $t$ (sec) | $0$ | $3$ | $6$ | $9$ | $12$ |
|---|---|---|---|---|---|
| $v(t)$ (m/s) | $0$ | $12$ | $20$ | $15$ | $6$ |