Parametric Equations (College Board AP® Calculus BC): Exam Questions

For $0\le t\le 10$ a particle moves along a curve in the $xy$-plane so that its position at time $t$ is $\left(x\left(t\right),y\left(t\right)\right)$, where $\frac{dx}{dt}=\ln \left(4+\sqrt{t}\right)$. At time $t=0$ the position of the particle is $\left(3,6\right)$.

Find the $x$-coordinate of the position of the particle at time $t=2$. Show the work that leads to your answer.

A particle is traveling along a curve in the $xy$-plane so that its position at time $t$ is $\left(x\left(t\right),y\left(t\right)\right)$, where $\frac{dx}{dt}=t{e}^t+1$ and $y\left(t\right)=5t^4-2t^2$.

Find the time at which the speed of the particle is 4. Show the work that leads to your answer.

At time $t\ge 0$, a particle is moving along a curve in the $xy$-plane with its position given by $\left(x\left(t\right),y\left(t\right)\right)$ where $x\left(t\right)=6t+e^t$. The equation for $y\left(t\right)$ is not explicitly given, but it is known that $y\text{'}\left(t\right)=30+5\ln \left(2t+1\right)$.

There is a point on the curve at which the line tangent to the curve has a slope of 1. Find the time at which the particle is at this point.

A particle is moving along a curve in the $xy$-plane with position $\left(x\left(t\right),y\left(t\right)\right)$ at time $t\ge 0$, where $\frac{dy}{dt}=e^{\cos t}$. At time $t=0$ the position of the particle is $\left(5,7\right)$.

Find the $y$-coordinate of the position of the particle at time $t=\frac{\pi }{2}$. Show the work that leads to your answer.

A particle travels along a curve so that its position at time $t$ is $\left(x\left(t\right),y\left(t\right)\right)$. At time $t=2$, the particle has a position of $\left(-3,-1\right)$. It is known that $\frac{dx}{dt}=\sin \left(\frac{2t}{1+t}\right)$ and $\frac{dy}{dt}=\ln \left(3t+e^t\right)$.

Write an equation for the line tangent to the curve at the point where the particle is at time $t=2$.

For $0\le t\le 4$, a particle is moving along a curve in the $xy$-plane with its position given by $\left(x\left(t\right),y\left(t\right)\right)$. The equation for $y\left(t\right)$ is not explicitly given, but it is known that $y\text{'}\left(t\right)=-t^2+2\ln \left(1+6t\right)$ and that the particle remains in the first quadrant during this time.

Find all the times $t$ in the interval $0\le t\le 4$ when the particle is moving towards the $x$-axis. Give a reason for your answer.

A particle moves along a curve in the $xy$-plane with position $\left(x\left(t\right),y\left(t\right)\right)$ where time $t\ge 0$ is measured in seconds, and $x\left(t\right)$ and $y\left(t\right)$ are measured in meters. It is known that $x\text{'}\left(t\right)=\sqrt{t}+\sin \left(\frac{t}{1+t}\right)$. At time $t=5$ seconds, the particle is at the point $\left(-2,-3\right)$.

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

At time $t\ge 0$, the position of a particle moving along a curve in the $xy$-plane is given by the parametric functions $\left(x\left(t\right),y\left(t\right)\right)$, where $\frac{dx}{dt}=\sqrt{t^2+3\cos t}$. The graph of $y\left(t\right)$ is shown below, formed using two straight line segments.

Find the total distance traveled by the particle over the time interval $0\le t\le 10$.

A curve is defined by the parametric equations $x\left(t\right)=2e^t-t^5$ and $y\left(t\right)=4+3e^t$.

Find an expression for $\frac{d^{2}y}{d{x}^2}$ in terms of $t$. Write your answer in the form $\frac{a{t}^3{e}^t\left(b-t\right)}{{\left(g\left(t\right)\right)}^3}$ where $a$ and $b$ are positive integers and $g\left(t\right)$ is a function to be found.

A curve is defined parametrically by $x=e^t\cos t$ and $y=e^t\sin t$.

Find the length of the curve from $t=0$ to $t=\pi$. Give your answer in the form $p\left(e^q-1\right)$ where $p$ and $q$ are real numbers to be found.

At time $t$, the position of a particle moving in the $xy$-plane is given by the parametric functions $\left(x\right(t),y(t\left)\right)$, where $\frac{\text{d}x}{\text{d}t}=t^2+\text{sin}\left(3t^2\right)$. The graph of $y$, consisting of three line segments, is shown in the figure below. At $t=0$, the particle is at position $(5,1)$.

Find the position of the particle at $t=3$.

Find the slope of the line tangent to the path of the particle at $t=3$.

Find the speed of the particle at $t=3$.

Find the total distance traveled by the particle from $t=0$ to $t=2$.

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.

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.

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 $t=6$.