Q: For the parametric curve $x = f(t)$, $y = g(t)$, at what values of $t$ is the tangent line horizontal?

When $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$

Q: If $x = f(t)$ and $y = g(t)$, at what values of $t$ is the tangent line vertical?

When $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$

Q: If a curve is described by $x = f(t)$, $y = g(t)$, and $\frac{dx}{dt} = 0$ at $t = t_0$, what does this imply about the tangent line at $t_0$?

The tangent line is vertical (provided $\frac{dy}{dt} \neq 0$)

Question 1

Suppose a curve is defined parametrically by $x = f(t)$ and $y = g(t)$, where $f$ and $g$ are differentiable functions. What is the formula for the slope of the tangent line at a point on the curve?

Accepted Answer

$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Question 2

Given the parametric equations $x = r(	heta - \sin	heta)$ and $y = r(1 - \cos	heta)$ for a cycloid, which integral gives the area under one arch of the cycloid?

Accepted Answer

$\int_{0}^{2\pi} y \, dx$

Question 3

For the parametric curve $x = f(t)$, $y = g(t)$, at what values of $t$ is the tangent line horizontal?

Accepted Answer

When $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} 
eq 0$

Question 4

Which formula gives the length $L$ of a curve described by $x = f(t)$, $y = g(t)$, $a \leq t \leq b$?

Accepted Answer

$L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Question 5

A curve is given by $x = \cos t$, $y = \sin t$, $0 \leq t \leq 2\pi$. What is the total length of the curve?

Accepted Answer

$2\pi$

Question 6

For the parametric equations $x = r \cos t$, $y = r \sin t$, $0 \leq t \leq \pi$, what is the surface area generated by rotating the curve about the $x$-axis?

Accepted Answer

$4\pi r^2$

Question 7

If $x = f(t)$ and $y = g(t)$, at what values of $t$ is the tangent line vertical?

Accepted Answer

When $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} 
eq 0$

Question 8

Given $x = r(	heta - \sin	heta)$ and $y = r(1 - \cos	heta)$, what is $\frac{dx}{d	heta}$?

Accepted Answer

$r(1 - \cos	heta)$

Question 9

Which of the following is the correct formula for the surface area $S$ generated by rotating a parametric curve $x = f(t)$, $y = g(t)$ about the $x$-axis?

Accepted Answer

$S = \int_{a}^{b} 2\pi y \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Question 10

A parametric curve is given by $x = t^2$, $y = t^3$, $0 \leq t \leq 2$. What is the value of $\frac{dy}{dx}$ at $t = 1$?

Accepted Answer

$\frac{3}{2}$

Question 11

Suppose a particle moves along a path given by $x = t^2$, $y = t^3$. At $t = 0$, what is the direction of the tangent line?

Accepted Answer

Vertical

Question 12

Which theorem allows you to compute the length of a curve described by parametric equations $x = f(t)$, $y = g(t)$, where $f$ and $g$ are continuous and differentiable?

Accepted Answer

Theorem 5

Question 13

For the parametric equations $x = r \cos t$, $y = r \sin t$, $0 \leq t \leq \pi$, what real-world object does the surface generated by rotating this curve about the $x$-axis represent?

Accepted Answer

A sphere

Question 14

Given $x = t^2$, $y = t^3$, $0 \leq t \leq 2$, what is the length of the curve from $t = 0$ to $t = 2$?

Accepted Answer

$\int_{0}^{2} \sqrt{4t^2 + 9t^4} \, dt$

Question 15

If a curve is described by $x = f(t)$, $y = g(t)$, and $\frac{dx}{dt} = 0$ at $t = t_0$, what does this imply about the tangent line at $t_0$?

Accepted Answer

The tangent line is vertical (provided $\frac{dy}{dt} 
eq 0$)

Question 16

For the parametric curve $x = r(	heta - \sin	heta)$, $y = r(1 - \cos	heta)$, what is the value of $\frac{dy}{d	heta}$?

Accepted Answer

$r \sin	heta$

Question 17

Which of the following integrals gives the surface area generated by rotating the parametric curve $x = f(t)$, $y = g(t)$ about the $y$-axis?

Accepted Answer

$S = \int_{a}^{b} 2\pi x \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Question 18

A medical device traces a path given by $x = 2t$, $y = t^2$ for $0 \leq t \leq 3$. What is the total distance the device travels?

Accepted Answer

$\int_{0}^{3} \sqrt{4 + 4t^2} \, dt$

Section 1.2 - Calculus with Parametric Curves

Study Guide - Practice Questions

Study Guide - Flashcards