Question 1

Given the parametric equations $$x = t^3$ - t$$, $$y = t^2$ - 2$$, which of the following expressions correctly gives $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2t}{3t^2$ - 1}$$

Question 2

For the parametric curve $$x = t^3$ - t$$, $$y = t^2$ - 2$$, at which value of $$t$$ does the tangent line become vertical?

Accepted Answer

$$t = 0 or$$ $$t = -1$$

Question 3

For the same parametric curve, at which value(s) of $$t is $$the tangent line horizontal?

Accepted Answer

$$t = 0$$

Question 4

Suppose you are asked to find the area enclosed by the parametric curve $$x = t^3$ - t$$, $$y = t^2$ - 2$$ and the $$x-$$axis for $$t in$$ $$[-2,2]. $$Which integral correctly represents this area?

Accepted Answer

$$\int_{-2}^{2} (t^2$ - 2) \cdot \frac{d}{dt}(t^3$ - t) \, dt$$

Question 5

Which of the following best describes the process for finding the equation of the tangent line to a parametric curve at a given $$t$$?

Accepted Answer

Find $$\frac{dy}{dt}$$ and $$\frac{dx}{dt}$$, compute $$\frac{dy}{dx}$$, evaluate at $$t$$, and use the point-slope form.

Question 6

If a parametric curve has $$\frac{dx}{dt} = 0$$ and $$\frac{dy}{dt} 
eq 0 at$$ $$t = t_0$, what does this imply about the tangent line at that point?

Accepted Answer

The tangent line is vertical.

Question 7

Given $$x = t^3$ - t$$, $$y = t^2$ - 2$$, what is the equation of the tangent line at $$t = 1$$?

Accepted Answer

$$y - (-1) = 1 \cdot (x - 0)$$

Question 8

A roller coaster's path is modeled by $$x = t^3$ - t$$, $$y = t^2$ - 2. At $$what value of $$t is $$the roller coaster moving horizontally?

Accepted Answer

$$t = 0$$

Question 9

Which of the following is the correct formula for the area enclosed by a parametric curve $$x = f(t)$$, $$y = g(t)$$ for $$t in$$ $$[a, b]$$?

Accepted Answer

$$\int_{a}^{b} y \frac{dx}{dt} dt$$

Question 10

Suppose $$x = t^3$ - t$$, $$y = t^2$ - 2. $$What is the value of $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$\frac{4}{11}$$

Question 11

If $$x = t^3$ - t$$ and $$y = t^2$ - 2$$, what is the $$x-$$coordinate when the tangent is horizontal?

Accepted Answer

$$x = 0$$

Question 12

Which of the following statements is true about the tangent lines to the curve $$x = t^3$ - t$$, $$y = t^2$ - 2$$?

Accepted Answer

The tangent is vertical when $$\frac{dx}{dt} = 0.$$

Question 13

A particle moves along the curve $$x = t^3$ - t$$, $$y = t^2$ - 2. At$$ $$t = -1$$, what is the slope of the tangent line?

Accepted Answer

$$-1$$

Question 14

Which of the following best describes the geometric meaning of $$\frac{dy}{dx}$$ for a parametric curve?

Accepted Answer

It is the slope of the tangent line to the curve at a given point.

Question 15

If $$x = t^3$ - t$$ and $$y = t^2$ - 2$$, what is the $$y-$$coordinate when the tangent is vertical at $$t = 1$$?

Accepted Answer

$$-1$$

Homework: Parametric Equations, Tangent Lines, and Area Enclosed by Curves

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