For a parametric curve x(t), y(t), the slope dy/dx at a given t is given by which expression?

For a parametric curve x(t), y(t), the slope dy/dx at a given t is given by which expression?

• A. dy/dx = dx/dt / dy/dt
• B. dy/dx = dy/dt / dx/dt
• C. dy/dx = (dx/dt)(dy/dt)
• D. dy/dx = dy/dt

Answer: (B)

The slope of a parametric curve at a point is found by using the chain rule: dy/dt = (dy/dx) · (dx/dt). Solve for dy/dx to get dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This makes sense because as t advances a small amount dt, y changes by dy ≈ (dy/dt) dt and x changes by dx ≈ (dx/dt) dt, so their ratio gives the slope dy/dx. If dx/dt = 0, the tangent is vertical and the slope is undefined; if both derivatives vanish, you’d inspect the curve more closely (sometimes by elimination of t). For example, with x(t)=t^2 and y(t)=t^3, dy/dx = (3t^2)/(2t) = (3/2)t for t ≠ 0, illustrating the ratio of rates along the parameter.

The slope of a parametric curve at a point is found by using the chain rule: dy/dt = (dy/dx) · (dx/dt). Solve for dy/dx to get dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This makes sense because as t advances a small amount dt, y changes by dy ≈ (dy/dt) dt and x changes by dx ≈ (dx/dt) dt, so their ratio gives the slope dy/dx. If dx/dt = 0, the tangent is vertical and the slope is undefined; if both derivatives vanish, you’d inspect the curve more closely (sometimes by elimination of t). For example, with x(t)=t^2 and y(t)=t^3, dy/dx = (3t^2)/(2t) = (3/2)t for t ≠ 0, illustrating the ratio of rates along the parameter.