Unit 9 Overview (College Board AP® Calculus BC): Exam Questions

For $0\le t\le \pi$, a particle is moving along the curve shown so that its position at time $t$ is $\left(x\right(t),y(t\left)\right)$, where $x\left(t\right)$ is not explicitly given and $y\left(t\right)=2\text{sin}t$. It is known that $\frac{\text{d}x}{\text{d}t}=e^{\text{cos}t}$. At time $t=0$, the particle is at position $(1,0)$.

Find the acceleration vector of the particle at time $t=1$. Show the setup for your calculations.

For $0\le t\le \pi$, find the first time $t$ at which the speed of the particle is $1.5$. Show the work that leads to your answer.

Find the slope of the line tangent to the path of the particle at time $t=1$. Find the $x$-coordinate of the position of the particle at time $t=1$. Show the work that leads to your answers.

Find the total distance traveled by the particle over the time interval $0\le t\le \pi$. Show the setup for your calculations.

Researchers on a boat are investigating plankton cells in a sea. At a depth of $h$ meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by $p\left(h\right)=0.2h^2{e}^{-0.0025h^2}$ for $0\le h\le 30$, and is modeled by $f\left(h\right)$ for $h\ge 30$. The continuous function $f$ is not explicitly given.

Find $p\text{'}\left(25\right)$. Using correct units, interpret the meaning of $p\text{'}\left(25\right)$ in the context of the problem.

Consider a vertical column of water in this sea with horizontal cross sections of constant area $3$ square meters. To the nearest million, how many plankton cells are in this column of water between $h=0$ and $h=30$ meters?

There is a function $u$ such that $0\le f\left(h\right)\le u\left(h\right)$ for all $h\ge 30$ and ${\int }_{30}^{\infty }u\left(h\right)\text{d}h=105$. The column of water in part (b) is $K$ meters deep, where $K>30$. Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to $2000$ million.

The boat is moving on the surface of the sea. At time $t\ge 0$, the position of the boat is the point with coordinates $\left(x\right(t),y(t\left)\right)$, where $x\text{'}\left(t\right)=662\text{sin}\left(5t\right)$ and $y\text{'}\left(t\right)=880\text{cos}\left(6t\right)$. Time $t$ is measured in hours, and $x\left(t\right)$ and $y\left(t\right)$ are measured in meters. Find the total distance traveled by the boat over the time interval $0\le t\le 1$.

A particle moving along a curve in the $xy$-plane has position $\left(x\right(t),y(t\left)\right)$ at time $t$ seconds, where $x\left(t\right)$ and $y\left(t\right)$ are measured in centimeters. It is known that $x\text{'}\left(t\right)=8t-t^2$ and $y\text{'}\left(t\right)=-t+\sqrt{t^{1.2}+20}$. At time $t=2$ seconds, the particle is at the point $(3,6)$.

Find the speed of the particle at time $t=2$ seconds. Show the setup for your calculations.

Find the total distance traveled by the particle over the time interval $0\le t\le 2$. Show the setup for your calculations.

Find the $y$-coordinate of the position of the particle at the time $t=0$. Show the setup for your calculations.

For $2\le t\le 8$, the particle remains in the first quadrant. Find all times $t$ in the interval $2\le t\le 8$ when the particle is moving toward the $x$-axis. Give a reason for your answer.

For time $t\ge 0$, a particle moves in the $xy$-plane with position $\left(x\right(t),y(t\left)\right)$ and velocity vector $\left((t-1)e^{t^2},\text{sin}\left(t^{1.25}\right)\right)$. At time $t=0$, the position of the particle is $(-2,5)$.

Find the speed of the particle at time $t=1.2$. Find the acceleration vector of the particle at time $t=1.2$.

Find the total distance traveled by the particle over the time interval $0\le t\le 1.2$.

Find the coordinates of the point at which the particle is farthest to the left for $t\ge 0$. Explain why there is no point at which the particle is farthest to the right for $t\ge 0$.

Curve $C$ is defined by the polar equation $r\left(\theta \right)=2{\text{sin}}^2\theta$ for $0\le \theta \le \pi$. Curve $C$ and the semicircle $r=\frac{1}{2}$ for $0\le \theta \le \pi$ are shown in the $xy$-plane.

(Note: Your calculator should be in radian mode.)

Find the rate of change of $r$ with respect to $\theta$ at the point on curve $C$ where $\theta =1.3$. Show the setup for your calculations.

Find the area of the region that lies inside curve $C$ but outside the graph of the polar equation $r=\frac{1}{2}$. Show the setup for your calculations.

It can be shown that $\frac{\text{d}x}{\text{d}\theta }=4\text{sin}\theta {\text{cos}}^2\theta -2{\text{sin}}^3\theta$ for curve $C$. For $0\le \theta \le \frac{\pi }{2}$, find the value of $\theta$ that corresponds to the point on curve $C$ that is farthest from the $y$-axis. Justify your answer.

A particle travels along curve $C$ so that $\frac{\text{d}\theta }{\text{d}t}=15$ for all times $t$. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where $\theta =1.3$. Show the setup for your calculations.

The polar curves $r=f\left(\theta \right)=1+\text{sin}\theta \text{cos}\left(2\theta \right)$ and $r=g\left(\theta \right)=2\text{cos}\theta$ for $0\le \theta \le \frac{\pi }{2}$ are shown below. Let $R$ be the region in the first quadrant bounded by the curve $r=f\left(\theta \right)$ and the $x$-axis. Let $S$ be the region in the first quadrant bounded by the curve $r=f\left(\theta \right)$, the curve $r=g\left(\theta \right)$, and the $x$-axis.

Find the area of $R$.

The ray $\theta =k$, where $0<k<\frac{\pi }{2}$, divides $S$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

For each $\theta$ with $0\le \theta \le \frac{\pi }{2}$, let $w\left(\theta \right)$ be the distance between the points with polar coordinates $\left(f\right(\theta ),\theta )$ and $\left(g\right(\theta ),\theta )$. Write an expression for $w\left(\theta \right)$. Find $w_A$, the average value of $w\left(\theta \right)$ over the interval $0\le \theta \le \frac{\pi }{2}$.

Using the information from part (c), find the value of $\theta$ for which $w\left(\theta \right)=w_A$. Is the function $w\left(\theta \right)$ increasing or decreasing at that value of $\theta$? Give a reason for your answer.

A particle moving along a curve in the $xy$-plane is at position $\left(x\right(t),y(t\left)\right)$ at time $t>0$. The particle moves in such a way that $\frac{\text{d}x}{\text{d}t}=\sqrt{1+t^2}$ and $\frac{\text{d}y}{\text{d}t}=\text{ln}(2+t^2)$. At time $t=4$, the particle is at the point $(1,5)$.

Find the slope of the line tangent to the path of the particle at time $t=4$.

Find the speed of the particle at time $t=4$, and find the acceleration vector of the particle at time $t=4$.

Find the $y$-coordinate of the particle's position at time $t=6$.

Find the total distance the particle travels along the curve from time $t=4$ to time $t=6$.

Let $S$ be the region bounded by the graph of the polar curve $r\left(\theta \right)=3\sqrt{\theta }\text{sin}\left({\theta }^2\right)$ for $0\le \theta \le \sqrt{\pi }$, as shown in the figure above.

Find the area of $S$.

What is the average distance from the origin to a point on the polar curve $r\left(\theta \right)=3\sqrt{\theta }\text{sin}\left({\theta }^2\right)$ for $0\le \theta \le \sqrt{\pi }$?

There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.

For $k>0$, let $A\left(k\right)$ be the area of the portion of region $S$ that is also inside the circle $r=k\text{cos}\theta$. Find $\underset{k\to \infty }{\lim }A\left(k\right)$.

The graphs of the polar curves $r=4$ and $r=3+2\text{cos}\theta$ are shown in the figure below. The curves intersect at $\theta =\frac{\pi }{3}$ and $\theta =\frac{5\pi }{3}$.

Let $R$ be the shaded region that is inside the graph of $r=4$ and also outside the graph of $r=3+2\text{cos}\theta$. Write an expression involving an integral for the area of $R$.

Find the slope of the line tangent to the graph of $r=3+2\text{cos}\theta$ at $\theta =\frac{\pi }{2}$.

A particle moves along the portion of the curve $r=3+2\text{cos}\theta$ for $0<\theta <\frac{\pi }{2}$. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of $3$ units per second. Find the rate at which the angle $\theta$ changes with respect to time at the instant when the position of the particle corresponds to $\theta =\frac{\pi }{3}$. Indicate units of measure.