If the parametric curve $x = f ( t ) , \quad y = g ( t )$ satisfies $g ^ { \prime } ( 4 ) = 0$, then it has a horizontal tangent when $t = 4$.

Question

Quizplus · Accepted Answer

A horizontal tangent occurs when the derivative of the y-coordinate with respect to the x-coordinate is 0. By the chain rule, we have $\frac{dy}{dt} = g'(t)\cdot \frac{dt}{dx}$. At $t = 4$, we know that $g'(4) = 0$, which means that $\frac{dy}{dt}$ is also 0 at $t=4$. Therefore, the tangent line at $t = 4$ is horizontal.

If the Parametric Curve x=f(t),y=g(t)x = f ( t ) , \quad y = g ( t )x=f(t),y=g(t)

If the Parametric Curve $x = f ( t ) , \quad y = g ( t )$