The curvature κ(t) of r(t) = (t, t^2, 0) is given by which expression?

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Multiple Choice

The curvature κ(t) of r(t) = (t, t^2, 0) is given by which expression?

Curvature measures how sharply a curve bends and for a space curve r(t) with r' ≠ 0 is κ(t) = ||r' × r''|| / ||r'||^3. For r(t) = (t, t^2, 0), we have r'(t) = (1, 2t, 0) and r''(t) = (0, 2, 0). The cross product is r' × r'' = (0, 0, 2), so its magnitude is 2. The speed is ||r'|| = sqrt(1 + 4t^2). Thus κ(t) = 2 / (1 + 4t^2)^{3/2}.

Curvature measures how sharply a curve bends and for a space curve r(t) with r' ≠ 0 is κ(t) = ||r' × r''|| / ||r'||^3. For r(t) = (t, t^2, 0), we have r'(t) = (1, 2t, 0) and r''(t) = (0, 2, 0). The cross product is r' × r'' = (0, 0, 2), so its magnitude is 2. The speed is ||r'|| = sqrt(1 + 4t^2). Thus κ(t) = 2 / (1 + 4t^2)^{3/2}.

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