Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The curve given by \(x=t^{3}, y=t^{2}\) has a horizontal tangent at the origin because \(d y / d t=0\) when \(t=0\).

Short answer

The statement is false. The curve \(x=t^{3}, y=t^{2}\) does not have a horizontal tangent at the origin because the derivative \(\frac{dy}{dx}\) is undefined when \(t=0\).

Step-by-step solution

Compute the direction derivative dy/dx

The derivative dy/dt is given as the derivative of \(y=t^{2}\) with respect to t, which is \(2t\). Now, the derivative dx/dt is the derivative of \(x=t^{3}\) with respect to t, which is \(3t^{2}\). The direction derivative dy/dx is the ratio \(\frac{dy/dt}{dx/dt} = \frac{2t}{3t^{2}} =\frac{2}{3t}\).

Evaluate dy/dx at the origin

We need to evaluate dy/dx when \(t=0\). Substituting \(t=0\) in dy/dx we get \(\frac{2}{3*0}\), which is undefined. This suggests that the slope of the tangent at the origin is not defined, which means it does not have a horizontal tangent at the origin.

Explanation for the false statement

The statement is false because the derivative \(\frac{dy}{dx}\) is not equal to \(0\) at \(t=0\), which must be the case for it to have a horizontal tangent at the origin. Instead, the derivative is undefined which contradicts the provided statement.