Parametric equations describe a curve by giving the coordinates of a moving point as functions of a parameter, usually t. Instead of writing y directly as a function of x, we write x = x(t) and y = y(t). This is useful for motion, rotating objects, projectiles, and curves that fail the vertical line test.
Parametric derivatives let us find the slope of the curve at each moment of the motion.
The key idea is that slope compares vertical change to horizontal change, so dy/dx can be found by comparing dy/dt to dx/dt. If dx/dt is not zero, then dy/dx = (dy/dt)/(dx/dt). The second derivative tells how the slope changes as x changes, not just as t changes.
This makes parametric calculus powerful for analyzing tangents, concavity, cusps, and changing direction along a curve.
Key Facts
- For x = x(t) and y = y(t), dy/dx = (dy/dt)/(dx/dt), if dx/dt != 0.
- A tangent line at parameter value t = a has slope m = y'(a)/x'(a), if x'(a) != 0.
- The tangent line can be written as y - y(a) = m[x - x(a)].
- The second derivative is d2y/dx2 = d/dx(dy/dx) = [d/dt(dy/dx)]/(dx/dt).
- A horizontal tangent occurs when dy/dt = 0 and dx/dt != 0.
- A vertical tangent occurs when dx/dt = 0 and dy/dt != 0.
Vocabulary
- Parametric equation
- An equation that defines x and y separately as functions of a parameter, such as x = x(t) and y = y(t).
- Parameter
- A variable, often t, that controls the position of a point moving along a curve.
- Parametric derivative
- The derivative dy/dx found from parametric equations by dividing dy/dt by dx/dt.
- Tangent line
- A line that touches a curve at a point and has the same instantaneous direction as the curve there.
- Concavity
- The way a curve bends, described by the sign of the second derivative d2y/dx2.
Common Mistakes to Avoid
- Using dy/dt as the slope, which is wrong because slope must compare vertical change to horizontal change. The correct slope is dy/dx = (dy/dt)/(dx/dt).
- Forgetting to check dx/dt = 0, which is wrong because the formula dy/dx = (dy/dt)/(dx/dt) is undefined when dx/dt is zero. This may indicate a vertical tangent or a special point needing closer analysis.
- Computing the second derivative as d2y/dt2 divided by d2x/dt2, which is wrong because d2y/dx2 measures change of slope with respect to x. Use d2y/dx2 = [d/dt(dy/dx)]/(dx/dt).
- Plugging t into the slope but not into the point, which gives an incomplete or incorrect tangent line. A tangent line needs both the slope at t = a and the point (x(a), y(a)).
Practice Questions
- 1 For x = t^2 + 1 and y = t^3 - 3t, find dy/dx at t = 2.
- 2 For x = 3cos t and y = 3sin t, find dy/dx at t = pi/4 and write the tangent line at that point.
- 3 A parametric curve has dx/dt = 0 and dy/dt != 0 at t = a. Explain what this means for the tangent line and why dy/dx is not defined by the usual formula.