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Parametric Curves

Position by time — path and slope · BC Calculus · greeting

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Owl · A warehouse robot reports its coordinates every second: x(t) and y(t). The path on the floor is the curve traced as t runs. You might eliminate t to get y in terms of x — but you don't always have to. Slope along the path is dy/dx = (dy/dt)/(dx/dt) when dx/dt ≠ 0. Speed is how fast the tip moves: √((dx/dt)² + (dy/dt)²). A robot reports x(t) and y(t) — what's its path and slope? We'll plot a few points by t, compute dy/dx carefully, and respect zeros of dx/dt.

Larry’s gold answer — Show text to read it out loud.

Larry · gold answer · For x = t², y = 2t with t from −1 to 2, I'd make a t-table, plot (x,y) pairs, and connect them in time order — without needing y in terms of x first.

Owl: A warehouse robot reports its coordinates every second: x(t) and y(t). The path on the floor is the curve traced as t runs. You might eliminate t to get y in terms of x — but you don't always have to. Slope along the path is dy/dx = (dy/dt)/(dx/dt) when dx/dt ≠ 0. Speed is how fast the tip moves: √((dx/dt)² + (dy/dt)²). A robot reports x(t) and y(t) — what's its path and slope? We'll plot a few points by t, compute dy/dx carefully, and respect zeros of dx/dt.

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