3 Parametric Equations Online Quiz Questions

True or false: In a parametric curve, the parameter must always represent time.

For the parametric equations $x=t+1$ and $y=t^2$, which rectangular equation is obtained by eliminating $t$?

For $x=t^2$ and $y=t^3$, find the value of $\frac{dy}{dx}$ at $t=2$.

True or false: A horizontal tangent to a parametrically defined curve occurs when $dy/dt=0$ and $dx/dt\ne0$.

For $x=3\cos t$ and $y=3\sin t$, which equation represents the tangent line at $t=\frac{\pi}{4}$?

Select all correct formulas for $\frac{d^2y}{dx^2}$ for a parametrically defined curve when $x'(t)\ne0$.

For the projectile motion $x(t)=20t$ and $y(t)=30t-4.9t^2$, find the tangent slope $\frac{dy}{dx}$ at $t=2$.

For $x=t^2$ and $y=t^3$, what is the concavity of the curve at $t=2$?

Select all statements that correctly describe motion with position $\mathbf r(t)=\langle x(t),y(t)\rangle$.

For the parametric curve $x=t^2$, $y=t$, on $0\le t\le2$, find the area under the curve. Show the setup, evaluate the integral, and state why the signed area equals the geometric area here.

Which parametrization represents the line through $(2,-1)$ with direction vector $\langle3,4\rangle$?

True or false: If $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$ at the same parameter value, the usual parametric derivative formula automatically determines the tangent.

A curve is parametrized by $x=t+1$ and $y=t^2$. Which rectangular equation and tracing direction are correct as $t$ increases?

For the parametrized curve $x=t^2+1$, $y=t^3$, what is the slope $\frac{dy}{dx}$ at $t=1$?

For $x=2\cos t$ and $y=2\sin t$, what type of tangent occurs at $t=0$?

The circle $x=4\cos t$, $y=4\sin t$, $0\le t\le2\pi$, is traversed once. What is its exact arc length?

Find the area under the parametrized curve $x=t^3$, $y=t^2$ for $0\le t\le1$.

For the projectile model $x(t)=20t$, $y(t)=30t-4.9t^2$, what is the vertical component of velocity at $t=2$ seconds?