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Math101Calculus with Parametric Curves
A rigorous, example-driven guide to calculus with parametric curves, including hypotheses, method choice, verification, and practice.
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
- Determine the parameter interval and tracing direction.
- Compute $x'$ and $y'$ and identify stationary or singular parameter values.
- Form $dy/dx=y'/x'$ only where $x'\ne0$.
- For concavity, differentiate that ratio with respect to $t$ and divide by $x'$.
- Use speed for arc length and split intervals if smoothness or tracing changes.
