3 Parametric Equations Free Online FlashCards

What is a parametric curve?

A parametric curve describes coordinates using a parameter: $x=x(t)$ and $y=y(t)$, usually over an interval $a\le t\le b$.

Eliminate $t$: $x=t+1,\ y=t^2$.

From $x=t+1$, solve $t=x-1$. Substitution into $y=t^2$ gives $y=(x-1)^2$.

How do you parametrize a line?

A line through $(x_0,y_0)$ with direction vector $\langle a,b\rangle$ is $x=x_0+at$, $y=y_0+bt$.

How do you parametrize a circle?

A circle centered at $(h,k)$ with radius $r$ is $x=h+r\cos t$, $y=k+r\sin t$, for $0\le t\le2\pi$.

What direction does the standard circle parametrization trace?

For the unit circle $x=\cos t$, $y=\sin t$, increasing $t$ from $0$ to $2\pi$ traces the circle counterclockwise.

What is the slope of a parametric curve?

When $dx/dt\ne0$, the slope is $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{y'(t)}{x'(t)}$.

How do you find a parametric tangent line?

At $t=t_0$, use $(x(t_0),y(t_0))$ and the slope $m=\left.\frac{dy}{dx}\right|_{t=t_0}$: $y-y(t_0)=m\bigl(x-x(t_0)\bigr)$.

When are parametric tangents horizontal or vertical?

A horizontal tangent occurs when $dy/dt=0$ and $dx/dt\ne0$. A vertical tangent occurs when $dx/dt=0$ and $dy/dt\ne0$.

What is the second derivative for a parametric curve?

Provided $x'(t)\ne0$, $\frac{d^2y}{dx^2}=\frac{d(dy/dx)/dt}{dx/dt}=\frac{x'(t)y''(t)-y'(t)x''(t)}{[x'(t)]^3}$.

What is the arc-length formula for a parametric curve?

For $x=x(t)$, $y=y(t)$ on $[a,b]$, arc length is $L=\int_a^b\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt$.

What is the signed-area formula for a parametric curve?

The signed area is $A=\int_a^b y(t)x'(t)\,dt$. It can be negative when the curve is below the $x$-axis or when $x(t)$ decreases.

What is the velocity vector of parametric motion?

For position $\mathbf r(t)=\langle x(t),y(t)\rangle$, velocity is $\mathbf v(t)=\mathbf r'(t)=\langle x'(t),y'(t)\rangle$.