Question 1

Given the parametric equations $x = t^2 - 1$, $y = t^3 - t$, which of the following expressions gives the slope of the tangent line at any point $t$?

Accepted Answer

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

Question 2

For the parametric curve $x = t^2 - 1$, $y = t^3 - t$, at which value of $t$ is the tangent line horizontal?

Accepted Answer

$t = 0$ or $t = 1$ or $t = -1$ or $t = \sqrt{\frac{1}{3}}$ or $t = -\sqrt{\frac{1}{3}}$

Question 3

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

Accepted Answer

$t = 0$

Question 4

What is the equation of the tangent line to the curve $x = t^2 - 1$, $y = t^3 - t$ at $t = -2$?

Accepted Answer

$y + 6 = -7(x + 3)$

Question 5

Which integral represents the area enclosed by the parametric curve $x = t^2 - 1$, $y = t^3 - t$ and the $x$-axis for $t$ in $[-1, 1]$?

Accepted Answer

$\int_{-1}^{1} (t^3 - t) \cdot 2t \, dt$

Question 6

Suppose a particle moves along the curve $x = t^2 - 1$, $y = t^3 - t$. What is the speed of the particle at $t = 1$?

Accepted Answer

$\sqrt{4 + 4} = 2\sqrt{2}$

Question 7

If $x = t^2 - 1$ and $y = t^3 - t$, what is $\frac{d^2y}{dx^2}$ at $t = 1$?

Accepted Answer

$\frac{3}{2}$

Question 8

Which of the following best describes the geometric meaning of $\frac{dy}{dx}$ for a parametric curve $x = f(t)$, $y = g(t)$?

Accepted Answer

It is the slope of the tangent line to the curve at the point corresponding to $t$.

Question 9

For the parametric curve $x = t^2 - 1$, $y = t^3 - t$, which of the following is true about the symmetry of the curve?

Accepted Answer

The curve is symmetric about the origin.

Question 10

A chemical reaction's rate is modeled by the parametric equations $x = t^2 - 1$, $y = t^3 - t$, where $x$ is time and $y$ is concentration. At what time is the concentration changing most rapidly?

Accepted Answer

$t = 0$

Question 11

Given $x = t^2 - 1$, $y = t^3 - t$, what is the $y$-coordinate when the tangent line is vertical?

Accepted Answer

$y = 0$

Question 12

If $x = t^2 - 1$, $y = t^3 - t$, what is the area enclosed by the curve and the $x$-axis for $t$ in $[-1, 1]$?

Accepted Answer

$0$

Question 13

Which of the following is the correct formula for the arc length of the parametric curve $x = t^2 - 1$, $y = t^3 - t$ from $t = a$ to $t = b$?

Accepted Answer

$\int_a^b \sqrt{(2t)^2 + (3t^2 - 1)^2} \, dt$

Question 14

Suppose $x = t^2 - 1$, $y = t^3 - t$ models the path of a robot. At $t = 0$, what is the direction of motion (as a vector)?

Accepted Answer

$(0, -1)$

Study Notes: Parametric Equations, Tangent Lines, and Area Enclosed by Curves

Study Guide - Practice Questions

Study Guide - Flashcards