Mathematics

Mathematics

Problem

For the parametric curve x=f(t),  y=g(t)x=f(t),\;y=g(t), which of the following gives the correct condition for a horizontal tangent at a parameter value t=t0t=t_0? a) dxdt=0  and  dydt≠0\dfrac{dx}{dt}=0 \;\text{and}\; \dfrac{dy}{dt}\neq 0 b) dydt=0  and  dxdt≠0\dfrac{dy}{dt}=0 \;\text{and}\; \dfrac{dx}{dt}\neq 0 c) dydt=0\dfrac{dy}{dt}=0 \text{(regardless of dxdt\dfrac{dx}{dt})} d) dxdt=0  and  dydt=0\dfrac{dx}{dt}=0 \;\text{and}\; \dfrac{dy}{dt}=0
Intuitive explanation

A horizontal tangent means the curve is flat—neither rising nor falling—at that instant. The slope of a parametric curve is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, so you need the numerator to be zero while the denominator is nonzero. This isolates the condition: g′(t0)=0g'(t_0) = 0 and f′(t0)≠0f'(t_0) \neq 0.

Intuitive explanation

A horizontal tangent means the curve is flat—neither rising nor falling—at that instant. The slope of a parametric curve is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, so you need the numerator to be zero while the denominator is nonzero. This isolates the condition: g′(t0)=0g'(t_0) = 0 and f′(t0)≠0f'(t_0) \neq 0.

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