Unit 9: Parametric, Polar, and Vector-Valued Functions (BC-only)
This BC-only unit generalizes single-variable calculus to curves traced in more flexible ways. Instead of always writing \(y\) as an explicit function of \(x\), we let both coordinates depend on a parameter or describe curves through angle and radius.
Parametric equations
A parametric curve is given by
\[x = f(t), \qquad y = g(t).\]The same geometric curve can be traced in different ways depending on how \(t\) changes.
Derivatives for parametric curves
If \(dx/dt \ne 0\), then
\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt}.\]Horizontal tangent:
\[\frac{dy}{dt} = 0, \qquad \frac{dx}{dt} \ne 0\]Vertical tangent:
\[\frac{dx}{dt} = 0, \qquad \frac{dy}{dt} \ne 0.\][Image Placeholder: parametric curve with tangent vectors and repeated tracing]
Second derivative for parametric curves
\[\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/ \frac{dx}{dt}.\]Speed and arc length
For a particle moving with position
\[\langle x(t), y(t) \rangle,\]speed is
\[\sqrt{[x'(t)]^2 + [y'(t)]^2}.\]Arc length from \(t=a\) to \(t=b\):
\[L = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2}\,dt.\]Polar coordinates
Connections to rectangular coordinates:
\[x = r\cos\theta, \qquad y = r\sin\theta\] \[r^2 = x^2 + y^2.\]Different polar pairs can describe the same point because adding \(2\pi\) to \(\theta\) changes nothing and negative \(r\) reflects through the origin.
Slope in polar form
If \(r=f(\theta)\), then
\[\frac{dy}{dx} = \frac{r'(\theta)\sin\theta + r(\theta)\cos\theta} {r'(\theta)\cos\theta - r(\theta)\sin\theta}.\]Area in polar coordinates
Area swept from \(\theta=a\) to \(\theta=b\):
\[A = \frac12 \int_a^b [r(\theta)]^2\,d\theta.\][Image Placeholder: sector approximation leading to polar area formula]
Arc length in polar form
If \(r=f(\theta)\), then arc length is
\[L = \int_a^b \sqrt{[r(\theta)]^2 + [r'(\theta)]^2}\,d\theta.\]Vector-valued functions
A vector-valued function in the plane is
\[\mathbf{r}(t) = \langle x(t), y(t) \rangle\]and in space:
\[\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle.\]Then
\[\mathbf{r}'(t)\]gives velocity and
\[\mathbf{r}''(t)\]gives acceleration.
Common mistakes
- Forgetting that \(dy/dx\) for parametric or polar curves is a ratio of derivatives.
- Declaring a vertical tangent whenever the denominator is zero without checking the numerator.
- Losing track of the interval of parameter values or angles actually tracing the region.
- Forgetting that polar curves can retrace themselves.