Chapter 10, Problem 1RWDT

To determine

To check: The existence of vertical tangent at $t=2$ .

Answer to Problem 1RWDT

Ye, the curve has a vertical tangent at $t=2$ .

Explanation of Solution

Given information: It is given that $\frac{dx}{dt}=0$ at $t=2$ .

Formula used:

The slope of the tangent to the curve $\left(x\left(t\right),y\left(t\right)\right)$ at the point $t$ is given as $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ .

Calculation:

The slope of the tangent to the curve at $t=2$ is $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ .

Substitute $0$ for $\frac{dx}{dt}$ in the above equation of slope.

  $\begin{array}{c}\frac{dy}{dx}=\frac{dy/dt}{0}\\ =\infty \end{array}$

At $t=2$ , then the slope of the curve is infinity. This implies that the curve has a vertical tangent at $t=2$ .

Thus, it is necessary that the parametric curve has vertical tangent at $t=2$ .