Parametric curves describe motion by giving x and y as separate functions of a parameter, usually time t. Instead of writing y directly as a function of x, you track a moving point (x(t), y(t)) as it traces a path. This approach is powerful for curves that loop, turn back, or fail the vertical line test.
It connects calculus to motion, geometry, and physics in a natural way.
Calculus on parametric curves works by combining rates of change with respect to t. The slope dy/dx comes from dividing dy/dt by dx/dt, while the second derivative measures how that slope changes along the curve. Area and arc length are found by summing thin geometric pieces as t changes over an interval.
These tools let you compute tangents, concavity, swept area, and distance traveled along complicated paths.
Key Facts
- A parametric curve is written as x = x(t), y = y(t), where t ranges over an interval.
- Slope of a parametric curve: dy/dx = (dy/dt)/(dx/dt), as long as dx/dt is not 0.
- Second derivative: d2y/dx2 = (d/dt(dy/dx))/(dx/dt), as long as dx/dt is not 0.
- Horizontal tangent occurs when dy/dt = 0 and dx/dt is not 0.
- Vertical tangent occurs when dx/dt = 0 and dy/dt is not 0.
- Arc length from t = a to t = b: L = ∫[a,b] sqrt((dx/dt)^2 + (dy/dt)^2) dt.
Vocabulary
- Parameter
- A parameter is an independent variable, often t, that determines the coordinates of a point on a curve.
- Parametric curve
- A parametric curve is a set of points traced by equations x = x(t) and y = y(t) over a chosen interval of t.
- Tangent vector
- A tangent vector is the vector <dx/dt, dy/dt> that points in the instantaneous direction of motion along the curve.
- Arc length
- Arc length is the total distance traveled along a curve over a parameter interval.
- Concavity
- Concavity describes whether the curve bends upward or downward as measured by d2y/dx2.
Common Mistakes to Avoid
- Using dy/dt as the slope, which is wrong because slope compares vertical change to horizontal change. For a parametric curve, use dy/dx = (dy/dt)/(dx/dt).
- Forgetting to check dx/dt before dividing, which is wrong because dy/dx is undefined when dx/dt = 0. A zero dx/dt can indicate a vertical tangent or a more complicated point.
- Computing d2y/dx2 as d2y/dt2 divided by d2x/dt2, which is wrong because the second derivative with respect to x must account for how x changes with t. Use d2y/dx2 = (d/dt(dy/dx))/(dx/dt).
- Using ∫ y dt for area, which is wrong because area with respect to the x-axis must include horizontal change. Use A = ∫ y(t) x'(t) dt when the curve is traversed left to right, and interpret signs carefully.
Practice Questions
- 1 For x = t^2 + 1 and y = t^3 - 3t, find dy/dx at t = 2.
- 2 For x = 3t and y = 2t^2 from t = 0 to t = 2, find the area under the curve using A = ∫ y(t) x'(t) dt.
- 3 A curve has x'(t) = 0 and y'(t) = 5 at t = 1. Explain what kind of tangent occurs there and why the usual slope formula cannot be evaluated.