Problem 88 - Essential Calculus

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 \(t=0\).

Step-by-Step Solution

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 1: 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 \(\frac{dy/dt}{dx/dt} = \frac{2t}{3t^{2}} =\frac{2}{3t}\).

Step 2: 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.

Step 3: 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.