Polar coordinates describe points using a distance r from the origin and an angle theta from the positive x-axis. This system is powerful for curves with circular or spiral structure, such as cardioids, roses, and spirals. To study motion, tangents, and slopes on these curves, we need derivatives written in terms of theta rather than only x and y.
Polar derivatives connect the geometry of a rotating radius vector to the familiar slope dy/dx.
Key Facts
- Polar to rectangular conversion: x = r cos(theta), y = r sin(theta).
- For a polar curve r = f(theta), use parametric form: x(theta) = f(theta) cos(theta), y(theta) = f(theta) sin(theta).
- Polar slope formula: dy/dx = (dr/dtheta sin(theta) + r cos(theta)) / (dr/dtheta cos(theta) - r sin(theta)).
- A horizontal tangent occurs when dy/dtheta = 0 and dx/dtheta is not 0.
- A vertical tangent occurs when dx/dtheta = 0 and dy/dtheta is not 0.
- Tangent line at theta = a: y - y(a) = m[x - x(a)], where m = (dy/dx) at theta = a.
Vocabulary
- Polar coordinates
- A coordinate system that locates a point by its distance r from the origin and its angle theta from a reference direction.
- Polar curve
- A curve defined by an equation r = f(theta), where the radius changes as the angle changes.
- Radius vector
- The directed segment from the pole to a point on a polar curve.
- Parametric form
- A way to describe a curve using x and y as separate functions of a parameter, often theta for polar curves.
- Tangent line
- A line that touches a curve at a point and has the same instantaneous direction as the curve there.
Common Mistakes to Avoid
- Using dy/dx = dr/dtheta directly is wrong because r is not y and theta is not x. Convert to x(theta) and y(theta), then use dy/dx = (dy/dtheta)/(dx/dtheta).
- Forgetting the product rule in x = r cos(theta) and y = r sin(theta) gives an incorrect slope. Since r depends on theta, both r and the trig factor must be differentiated.
- Canceling r or dr/dtheta from the polar slope formula without checking terms is wrong because the numerator and denominator are sums. Only common factors of every term can be canceled.
- Calling every zero denominator a vertical tangent is incomplete because the numerator must also be checked. If both dx/dtheta and dy/dtheta are zero, the point may be singular and needs further analysis.
Practice Questions
- 1 For r = 2 + cos(theta), find dy/dx at theta = pi/2.
- 2 For r = 3 sin(theta), find the coordinates of the point and the slope of the tangent line at theta = pi/4.
- 3 Explain why the derivative of a polar curve is found by treating x and y as parametric functions of theta instead of differentiating r = f(theta) as if r were y.