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

The slope of the tangent to the curve at the point where $t = 4$ is given by $dy/dx$ evaluated at $t = 4$, which is $dy/dt / dx/dt$ by the chain rule. If $g'(4) = 0$, then $dy/dt = g'(4) = 0$, which means the slope of the tangent is 0 and hence the tangent is horizontal. Therefore, the statement is true.

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 $x = f ( t ) , \quad y = g ( t )$