Calculus with Parametric Curves

The central idea

For a differentiable parametric curve $x=x(t)$, $y=y(t)$ with $x'(t)\ne0$, $dy/dx=y'(t)/x'(t)$. Where this first derivative is differentiable and $x'\ne0$, $d^2y/dx^2=[d/dt(dy/dx)]/x'(t)$. Arc length is $\int\sqrt{(x')^2+(y')^2}dt$.

Definitions, hypotheses, and notation

A regular parameter value has $(x',y')\ne(0,0)$. If both components vanish, the ratio $y'/x'$ is indeterminate and local analysis may reveal a cusp, tangent, or self-intersection; canceling a common factor can sometimes expose the limiting slope, but it must be justified.

Signed area under a parametrized arc can be written $\int y\,dx=\int y(t)x'(t)dt$. The sign of $x'$ records left-to-right or right-to-left orientation. Thus parameter direction matters for signed integrals even though geometric arc length uses nonnegative speed.

Conceptual meaning

The parameter supplies motion along a plane curve. The tangent vector is $\langle x',y'\rangle$; dividing its vertical component by its horizontal component gives graph slope. The second derivative divides by $x'$ again because it differentiates with respect to $x$, not $t$.

A dependable method and decision rule

1. Determine the parameter interval and tracing direction.
2. Compute $x'$ and $y'$ and identify stationary or singular parameter values.
3. Form $dy/dx=y'/x'$ only where $x'\ne0$.
4. For concavity, differentiate that ratio with respect to $t$ and divide by $x'$.
5. Use speed for arc length and split intervals if smoothness or tracing changes.

Fully worked example

Graphical or geometric meaning

The tangent vector points in the direction of increasing parameter. Scaling the parameter speed does not alter the tangent-line slope, because both derivative components scale together, but it does alter how quickly the point traverses the curve.

Common mistakes and why they fail

Verification and reasonableness checks

• Eliminate the parameter when possible and compare an explicit derivative.
• Verify the tangent line passes through the parametric point.
• Check denominator values before reporting finite slope.

Eliminate the parameter only when it helps

For $x=x(t)$ and $y=y(t)$, $dy/dx=(dy/dt)/(dx/dt)$ when $dx/dt\ne0$. A zero $dx/dt$ can indicate a vertical tangent only after checking $dy/dt$ and local behavior. For $d^2y/dx^2$, differentiate $dy/dx$ with respect to $t$ and divide by $dx/dt$ again. Eliminating $t$ may reveal the curve but can erase orientation, starting point, and repeated tracing. Mark the interval and direction before computing. Verify tangent direction against the velocity vector $(x'(t),y'(t))$, and ensure the selected parameter actually gives the intended branch. These checks distinguish geometric shape from the motion encoded by its parameterization.

Practice

1. If $x=t,y=t^2$, find $dy/dx$.
2. For $x=\cos t,y=\sin t$, find slope where $\sin t\ne0$.
3. What does $x'=0,y'\ne0$ usually indicate?

Connections and next steps

• Parametric Equations
• Arc Length
• Polar Coordinates in Calculus