Question 1

Which of the following best describes a smooth parametric curve?

Accepted Answer

A curve that has a tangent line at every point and the tangent turns smoothly as the point moves along the curve.

Question 2

Given the parametric equations $$\left\{ \begin{array}{l} x = t^2$ \ y = t^3$ \end{array} ight.$$, what happens at $$t = 0$$?

Accepted Answer

There is a cusp at $$(0,0)$$ because both $$\frac{dx}{dt}$$ and $$\frac{dy}{dt}$$ vanish.

Question 3

For the parametric curve $$\left\{ \begin{array}{l} x = t^3$ - 3t \ y = t^3$ - 12t \end{array} ight.$$, at which values of $$t$$ does the curve have a horizontal tangent?

Accepted Answer

$$t = \pm 2$$

Question 4

For the same curve $$\left\{ \begin{array}{l} x = t^3$ - 3t \ y = t^3$ - 12t \end{array} ight.$$, at which values of $$t$$ does the curve have a vertical tangent?

Accepted Answer

$$t = \pm 1$$

Question 5

Given $$x = t^4$ - t^2$ and y$$ = $$t^3 - 2t$, what is the slope $$\frac{dy}{dx}$$ at t = -1$?

Accepted Answer

$$\frac{1}{2}$$

Question 6

Which of the following is the correct formula for the second derivative $$\frac{d^2$y}{dx^2$}$$ for a parametric curve $$x = f(t)$$, $$y = g(t)$$?

Accepted Answer

$$\frac{g''(t)f'(t) - f''(t)g'(t)}{(f'(t))^3$}$$

Question 7

For the parametric curve $$x = t^3$ - 3t$$, $$y = t^2$, on the interval t$$ \in [-2, 2]$$, at which points does the curve have a vertical tangent?

Accepted Answer

$$(2, 1)$$ and $$(-2, 1)$$

Question 8

If $$x = f(t)$$ and $$y = g(t)$$ are parametric equations with $$f'(t_0$) = 0$$ and $$g'(t_0$) 
eq 0$$, what can be said about the tangent at $$t_0$$?

Accepted Answer

The curve has a vertical tangent at $$t_0.$$

Question 9

Given the parametric equations $$x = t - \cos t$$, $$y = 1 - \sin t$$, what is the slope of the tangent line at $$t = \frac{\pi}{4}$$?

Accepted Answer

$$-\frac{1}{1 + \frac{1}{\sqrt{2}}}$$

Question 10

For the curve $$x = t^3$ - 3t$$, $$y = t^2$, what is the concavity of the curve for t$$ \in (-1, 1)$$?

Accepted Answer

Concave up

Question 11

Which of the following is the parametric equation for the tangent line to $$x = f(t)$$, $$y = g(t) at$$ $$t = t_0$?

Accepted Answer

$$\left\{ \begin{array}{l} x = f(t_0$) + f'(t_0$)(t - t_0$) \ y = g(t_0$) + g'(t_0$)(t - t_0$) \end{array} ight.$$

Question 12

Suppose a parametric curve $$x = f(t)$$, $$y = g(t)$$ has $$f'(t_0$) = 0$$ and $$g'(t_0$) = 0. $$What is most likely to occur at $$t_0$$?

Accepted Answer

A cusp

Question 13

For the curve $$x = t^3$ - 3t$$, $$y = t^2$, as t$$ 	o +\infty$$, what is the approximate relationship between x$ and y$?

Accepted Answer

$$y \approx (x^2$)^{1/3}$

Question 14

Which of the following statements about the concavity of a parametric curve is correct?

Accepted Answer

If $$\frac{d^2$y}{dx^2$} \u003e 0$$, the curve is concave up.

Question 15

A parametric curve $$x = f(t)$$, $$y = g(t) is $$said to be smooth on an interval $$I if$$:

Accepted Answer

Both $$f'(t)$$ and $$g'(t)$$ are continuous and not simultaneously zero on $$I.$$

Smooth Parametric Curves and Their Slopes

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