9Unit 9: Parametric, Polar & Vector-Valued Functions (BC)

When a curve is traced by $x(t)$ and $y(t)$, the slope is still rise over run — but both rise and run are measured against the parameter $t$.

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To get $\dfrac{d^2y}{dx^2}$ you differentiate the SLOPE $\dfrac{dy}{dx}$ again with respect to $x$ — but since everything is in terms of $t$, differentiate with respect to $t$ and divide by $\dfrac{dx}{dt}$ once more.

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The length of a parametric path adds up tiny hypotenuses: each has horizontal piece $x'(t)\,dt$ and vertical piece $y'(t)\,dt$, so its length is $\sqrt{x'^2+y'^2}\,dt$.

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A vector-valued function $\langle x(t),y(t)\rangle$ traces a moving point.

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Integrating a velocity vector recovers position, one component at a time.

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Planar motion bundles two 1-D motions.

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A polar curve $r=f(\theta)$ is really a parametric curve: $x=r\cos\theta,\ y=r\sin\theta$.

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Sweeping a ray from the origin, each thin slice is nearly a circular sector of radius $r(\theta)$ and tiny angle $d\theta$.

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For the region between an outer and inner polar curve, each sector is a washer: subtract the inner sector area from the outer over the angles where the region lives.

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