Derivatives and Area in Polar Form

Parametric Equations, Polar Coordinates, and Vectors

By the end of this lesson, you will be able to work with parametric equations and their calculus, convert between polar and rectangular coordinates, and apply vector concepts to motion.

Parametric Equations

Parametric equations express $x$ and $y$ as functions of a third variable $t$ (the parameter).

$$x = f(t), \quad y = g(t)$$

Example: $x = t^2$, $y = t^3 - t$ traces a curve as $t$ varies.

Sketching: make a table of $t$, $x$, $y$ values; plot and connect in order.

Calculus with Parametric Curves

Slope of tangent line:

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$$

(valid where $f'(t) \neq 0$)

Second derivative:

$$\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right] = \frac{\frac{d}{dt}\left[\frac{dy}{dx}\right]}{dx/dt}$$

Arc length of a parametric curve from $t = a$ to $t = b$:

$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}, dt$$

Speed: $\sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2}$

Example: Find $\dfrac{dy}{dx}$ for $x = t^2$, $y = t^3$ at $t = 2$.

$\dfrac{dx}{dt} = 2t$, $\dfrac{dy}{dt} = 3t^2$

$\dfrac{dy}{dx} = \dfrac{3t^2}{2t} = \dfrac{3t}{2}$

At $t = 2$: $\dfrac{dy}{dx} = 3$.

Polar Coordinates

Polar coordinate system: each point is described by $(r, \theta)$, where $r$ is the distance from the origin (pole) and $\theta$ is the angle from the positive $x$-axis.

Conversion formulas:

$$x = r\cos\theta, \quad y = r\sin\theta$$

$$r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}$$

Common polar curves:

Equation	Curve
$r = a$	Circle of radius $
$r = a\cos\theta$	Circle of diameter $
$r = a\sin\theta$	Circle of diameter $