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.
Understanding Calculus: Parametric Derivatives
The derivative rule comes from the chain rule. Think of y as changing because t changes, while x changes because that same t changes. The rate of change of y with respect to t measures vertical motion.
The rate of change of x with respect to t measures horizontal motion. Dividing the first rate by the second converts a change per unit of time into a change per unit of horizontal distance.
This explains why the order matters. Reversing the division gives the change in x per unit of y, which describes a different rate.
The signs of the two time derivatives tell a useful story about the path. If both are positive, the point moves up and right, so the curve has positive slope. If one is positive and the other is negative, the point moves in a direction that gives negative slope.
A negative slope does not mean the point is moving downward in every case. It may move left while rising. Since both vertical and horizontal movement are reversed, their ratio can still be positive.
Students often lose this information by looking only at the final equation of the curve. The parameter keeps track of direction and timing.
Special parameter values need careful checking. When the vertical rate is zero but the horizontal rate is not, the point has stopped moving up or down for an instant. This produces a horizontal tangent.
When the horizontal rate is zero but the vertical rate is not, the point has a vertical tangent. A more difficult case occurs when both rates are zero. The usual slope rule then produces an indeterminate form, so it cannot settle the answer.
The curve may have a cusp, a sharp point, a smooth tangent, or another unusual feature. In that situation, examine values just before and after the parameter value. Simplifying the rate expression or using limits can reveal what the curve is doing.
For concavity, the first slope is itself a function of t. To find how that slope changes from left to right on the graph, first differentiate the slope with respect to t. Then divide by the horizontal rate.
This extra division is important because concavity is about change with x, not change with time. A positive second derivative means slopes increase as the curve is followed from left to right.
A negative second derivative means slopes decrease. If the point temporarily moves left, interpret the result using x positions rather than the order of time alone.
Parametric derivatives appear naturally in motion problems. The derivatives of x and y with respect to time are the horizontal and vertical components of velocity. At a given instant, the tangent line points in the direction of motion unless the object is momentarily at rest.
They are useful for describing a thrown ball, a point on a wheel, or an object moving around a track. A reliable method is to find the point first, calculate both time derivatives, check whether the horizontal rate is zero, then form the slope only when it is allowed.
Finally, place the point and slope into the tangent line equation. Keeping the parameter value visible through every step prevents many algebra mistakes.
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.