Unit 13: Parametric Equations
A curve traced by a moving point, x and y each written as their own function of time.
6 topics · 25 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Defining a Curve Parametrically
- How it worksHow do x(t) and y(t) together trace out a path as t sweeps through its range of values?
- Worked exampleFind the point traced by x = t2, y = 2t - 1 at t = 3.
- Common mistakeWhy doesn't a parametric curve have to pass the vertical line test the way y = f(x) does?
- Going furtherWhat happens to a parametric curve's path when it's traced with t running backward instead of forward?
Your turn1 of 2
A curve is defined by x = t2, y = 4t - 4. Find the y-coordinate of the point traced at t = 5.Eliminating the Parameter to Find a Rectangular Equation
- How it worksHow does solving one equation for t and substituting it into the other eliminate the parameter entirely?
- Worked exampleFind the rectangular equation for x = t + 1, y = t2 - 2 by eliminating t.
- Worked exampleFind the rectangular equation for x = 3cos(t), y = 3sin(t) using a Pythagorean identity.
- Common mistakeWhy doesn't eliminating the parameter always preserve the same restricted piece of the original curve?
- Going furtherWhat happens to a rectangular equation's domain when the eliminated parameter t was restricted to positive values?
Your turn1 of 2
Find the rectangular equation (solve for y in terms of x) for x = t + 4, y = t2 + 4 by eliminating t.The Slope of a Parametric Curve
- How it worksHow does dividing dy/dt by dx/dt recover the slope of a parametric curve at a point?
- Worked exampleFind dy/dx for the parametric curve x = t2, y = t3 at t = 2.
- Common mistakeWhy doesn't a vertical tangent line on a parametric curve mean the curve has stopped or broken?
- Going furtherWhat happens to the slope formula for a parametric curve at a point where dx/dt equals zero?
Your turn1 of 2
Find dy/dx for the parametric curve x = 3t2, y = 3t3, as a function of t.Concavity of a Parametric Curve
- How it worksHow does differentiating dy/dx with respect to t, then dividing by dx/dt again, produce d2y/dx2?
- Worked exampleFind d2y/dx2 for the parametric curve x = t2, y = t3.
- Common mistakeWhy isn't d2y/dx2 for a parametric curve just the second derivative of y(t) divided by that of x(t)?
- Going furtherWhat happens to the concavity formula when dx/dt equals zero at the point in question?
Your turn1 of 2
Find d2y/dx2, as a function of t, for the parametric curve x = 3t2, y = 3t3.Arc Length of a Parametric Curve
- How it worksHow does the Pythagorean theorem on a tiny piece of a parametric path turn into an integral of √((dx/dt)2 + (dy/dt)2)?
- Worked exampleFind the arc length of the parametric curve x = cos(t), y = sin(t) for t from 0 to π.
- Common mistakeWhy doesn't the parametric arc length formula work if you only know dy/dt and not dx/dt?
- Ties togetherHow does the parametric arc length formula connect back to the arc length formula for y = f(x) when x = t?
Your turn1 of 2
Find the arc-length integrand ds/dt = √((dx/dt)2 + (dy/dt)2) for the parametric curve x = 3t2, y = 3t3, as a function of t.Surface Area from a Revolved Parametric Curve
- How it worksHow does multiplying the parametric arc length element by the circumference swept out build the surface area integral?
- Worked exampleFind the surface area generated by revolving x = cos(t), y = sin(t) for t from 0 to π/2 around the x-axis.
- Worked exampleFind the surface area generated by revolving the parametric curve x = t, y = t3 for t from 0 to 1 around the x-axis.
- Common mistakeWhy doesn't the surface area formula for a parametric curve just reuse the y = f(x) version with x and y swapped?
Your turn1 of 2
For the parametric curve x = 3t2, y = 3t3 revolved around the x-axis, write the surface-area integrand 2·π·y*√((dx/dt)2+(dy/dt)2), as a function of t.