Parametric, Polar, and Vector Functions Cheat Sheet

Key Facts

• For a parametric curve $x=x(t)$ and $y=y(t)$, the slope is $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ when $\frac{dx}{dt}\neq 0$.
• The second derivative of a parametric curve is $\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$ when $\frac{dx}{dt}\neq 0$.
• The arc length of a parametric curve from $t=a$ to $t=b$ is $L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$.
• Polar coordinates convert to rectangular coordinates using $x=r\cos\theta$ and $y=r\sin\theta$.
• The area enclosed by a polar curve from $\theta=\alpha$ to $\theta=\beta$ is $A=\frac{1}{2}\int_{\alpha}^{\beta}r^2\,d\theta$.
• The slope of a polar curve is $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta+r\cos\theta}{\frac{dr}{d\theta}\cos\theta-r\sin\theta}$ when the denominator is not $0$.
• For a vector function $\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle$, velocity is $\mathbf{v}(t)=\mathbf{r}'(t)$ and acceleration is $\mathbf{a}(t)=\mathbf{r}''(t)$.
• Speed is the magnitude of velocity, so $\text{speed}=|\mathbf{v}(t)|=\sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2}$.

Vocabulary

Parametric equation

A pair or set of equations, such as $x=x(t)$ and $y=y(t)$, that describe a curve using a parameter $t$.

Parameter

A variable, often $t$ or $\theta$, that controls the position of a point on a curve.

Polar curve

A curve described by $r=f(\theta)$, where $r$ is distance from the origin and $\theta$ is the angle from the positive $x$-axis.

Vector function

A function such as $\mathbf{r}(t)=\langle x(t),y(t)\rangle$ or $\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle$ that gives position using component functions.

Velocity vector

The derivative of position, $\mathbf{v}(t)=\mathbf{r}'(t)$, which gives both speed and direction of motion.

Arc length

The total distance along a curve, found by integrating speed over the relevant parameter interval.