Parametric, polar and vector functions Knowledge-Based Study

Study notes

1. Parametric Equations

   A planar curve can often be represented parametrically by expressing both coordinates as functions of an independent parameter, typically denoted as $t$:

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

   Derivatives and Tangent Lines

   To analyze the slope of the tangent line to a parametric curve, apply the chain rule. If $x$ and $y$ are differentiable functions of $t$ and $\frac{dx}{dt} \neq 0$, the first derivative is:

   $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

   The second derivative measures concavity and is computed by differentiating $\frac{dy}{dx}$ with respect to $t$, then dividing by $\frac{dx}{dt}$:

   $$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$$

   Arc Length

   The arc length $L$ of a smooth curve traversed exactly once from $t = a$ to $t = b$ is given by the definite integral:

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

   Vector-Valued Functions

   Motion in a two-dimensional plane is naturally described using a position vector $\vec{r}(t)$:

   $$\vec{r}(t) = \langle x(t), y(t) angle = x(t)\hat{\mathbf{i}} + y(t)\hat{\mathbf{j}}$$

   Velocity, Speed, and Acceleration

   Differentiating the position vector with respect to time yields the velocity vector, speed scalar, and acceleration vector:

   • Velocity vector: $\vec{v}(t) = \vec{r}'(t) = \langle x'(t), y'(t) angle$
   • Speed: $\|\vec{v}(t)\| = \sqrt{(x'(t))^2 + (y'(t))^2}$
   • Acceleration vector: $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t) = \langle x''(t), y''(t) angle$

   Total Distance Traveled

   The total distance traveled by an object along its path from $t = t_1$ to $t = t_2$ equals the integral of its speed over the given interval:

   $$\text{Total Distance} = \int_{t_1}^{t_2} \|\vec{v}(t)\|\, dt = \int_{t_1}^{t_2} \sqrt{(x'(t))^2 + (y'(t))^2}\, dt$$

   Notice that this formulation is identical to the arc length of the parametric path traced by the particle.

   Polar Coordinate Foundations

   In the polar coordinate system, points are represented as $(r, \theta)$, where $r$ is the directed distance from the origin (pole) and $\theta$ is the angle measured counterclockwise from the positive $x$-axis (polar axis). The relationships connecting rectangular and polar coordinates are:

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

   Slopes of Polar Curves

   Given a polar equation $r = f(\theta)$, the curve can be treated as a set of parametric equations with parameter $\theta$:

   $$x(\theta) = f(\theta)\cos\theta, \quad y(\theta) = f(\theta)\sin\theta$$

   Applying the product rule yields:

   $$\frac{dx}{d\theta} = \frac{dr}{d\theta}\cos\theta - r\sin\theta$$

   $$\frac{dy}{d\theta} = \frac{dr}{d\theta}\sin\theta + r\cos\theta$$

   The slope of the tangent line in rectangular coordinates is:

   $$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}$$

   A critical distinction exists between $\frac{dr}{d\theta}$ and $\frac{dy}{dx}$: $\frac{dr}{d\theta}$ represents the rate of change of the distance from the origin with respect to $\theta$, whereas $\frac{dy}{dx}$ is the geometric slope of the tangent line in the $xy$-plane.

   Area Enclosed by Polar Curves

   The area of an infinitesimal polar sector of radius $r$ subtended by an angle $d\theta$ is $\frac{1}{2}r^2\, d\theta$. Consequently, the area bounded by a continuous curve $r = f(\theta)$ and radial rays $\theta = \alpha$ and $\theta = \beta$ (where $\beta - \alpha \leq 2\pi$) is given by:

   $$A = \frac{1}{2}\int_{\alpha}^{\beta} [f(\theta)]^2\, d\theta = \frac{1}{2}\int_{\alpha}^{\beta} r^2\, d\theta$$

   Area Between Two Polar Curves

   When computing the area of a region bounded between an outer curve $r_2 = f_2(\theta)$ and an inner curve $r_1 = f_1(\theta)$ where $r_2 \geq r_1 \geq 0$ over $[\alpha, \beta]$, subtract the inner sector areas from the outer sector areas:

   $$A = \frac{1}{2}\int_{\alpha}^{\beta} \left( [r_2(\theta)]^2 - [r_1(\theta)]^2 \right)\, d\theta$$

   Polar Arc Length

   By substituting $x = r\cos\theta$ and $y = r\sin\theta$ into the parametric arc length formula and simplifying with trigonometric identities, the arc length of a polar curve from $\theta = \alpha$ to $\theta = \beta$ becomes:

   $$L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta$$