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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