For the parametric curve x = t^2, y = t^3, dy/dx expressed in terms of t is which expression?

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Multiple Choice

For the parametric curve x = t^2, y = t^3, dy/dx expressed in terms of t is which expression?

Explanation:
For a parametric curve x(t) and y(t), the slope dy/dx is found by dy/dt divided by dx/dt (when dx/dt ≠ 0). Here, dx/dt = 2t and dy/dt = 3t^2. So dy/dx = (3t^2)/(2t) = (3/2) t for t ≠ 0. This shows the slope varies linearly with t. At t = 0 the direct quotient is not defined because dx/dt = 0, but the limit of dy/dx as t → 0 is 0, matching the expression (3/2) t evaluated near 0. Therefore the expression in terms of t is (3/2) t.

For a parametric curve x(t) and y(t), the slope dy/dx is found by dy/dt divided by dx/dt (when dx/dt ≠ 0). Here, dx/dt = 2t and dy/dt = 3t^2. So dy/dx = (3t^2)/(2t) = (3/2) t for t ≠ 0. This shows the slope varies linearly with t. At t = 0 the direct quotient is not defined because dx/dt = 0, but the limit of dy/dx as t → 0 is 0, matching the expression (3/2) t evaluated near 0. Therefore the expression in terms of t is (3/2) t.

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