AP Calculus BC - Unit 9: Parametric, Polar, and Vector Functions

Parametric Equations:

A function $y = f(x)$ can be parametrized by setting:

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

How to Eliminate $t$ to Obtain a Cartesian Equation:

Solve for $t$ in terms of $x$, then substitute into the equation for $y$.

Finding Derivatives of Parametric Curves:

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

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

Arc Length of a Parametric Curve:

If a smooth curve $x = f(t), y = g(t)$ is traced from $a$ to $b$, then the arc length is:

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

Polar Coordinates:

The Cartesian and Polar coordinates are related as:

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

Slope of a Polar Curve:

If a curve is given by $r = f(\theta)$, the slope is:

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

Area Enclosed by a Polar Curve:

The area of a region enclosed by $r = f(\theta)$ is:

$$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$$

Vector-Valued Functions:

Vector Operations:

$$\mathbf{u} + \mathbf{v} = \langle x_1 + x_2, y_1 + y_2 \rangle, \quad k\mathbf{v} = k \langle x, y \rangle = \langle kx, ky \rangle$$

Velocity and Acceleration:

The velocity and acceleration vectors are:

$$\mathbf{v}(t) = \langle x'(t), y'(t) \rangle, \quad \mathbf{a}(t) = \langle x''(t), y''(t) \rangle$$

Speed of a Particle:

The speed along the curve is the magnitude of the velocity vector:

$$|\mathbf{v}| = \sqrt{x'(t)^2 + y'(t)^2}$$

Displacement and Distance Traveled:

The displacement vector from $t = a$ to $t = b$ is:

$$\left\langle \int_a^b x'(t) dt, \int_a^b y'(t) dt \right\rangle$$

The total distance traveled is:

$$\int_a^b |\mathbf{v}(t)| dt = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} dt$$