Problem Sets: Parametric Equations and Polar Coordinates (参数方程与极坐标)

Last Update: 2026-03-31

• Given the curve $y=y(x)$ defined by \(\begin{cases} x = t + \cos t, \\ e^{y} + ty + \sin t = 1, \end{cases}\)find the tangent line at $t=0$.

• Given the curve $\begin{cases} x = \ln(1 + t^2), \ y = t - \arctan t, \end{cases}$, find $\dfrac{d^2y}{dx^2}$.

• Given the curve $y = y(x)$ defined by \(\begin{cases} x - e^x \sin t + 1 = 0, \\ y = \displaystyle \int_{0}^{t} \sqrt{1 + u^2} \, du. \end{cases}\)
  1. Find the tangent line at $t = 0$.
  2. Find $\dfrac{d^2y}{dx^2}$.

• Find $\dfrac{dy}{dx}$ and $\dfrac{d^{2}y}{dx^{2}}$ along the general curve $x = x(t), y = y(t)$.

• Find the arc length of the curve $x = \dfrac{1}{4}\ln(1 + t^{2})$, $y = \arctan t$, from $t = 0$ to $t = 2$.

• Let $D$ be the region bounded by $y = \sqrt{1 - x^{2}}$ ($0 \le x \le 1$) and the astroid $\begin{cases} x = \cos^{3} t \ y = \sin^{3} t \end{cases}$ ($0 \le t \le \frac{\pi}{2}$). Compute the volume and surface area of the solid formed by rotating $D$ about the $x$-axis.

• Consider the cycloid \(x = r(\theta - \sin\theta), \quad y = r(1 - \cos\theta)\)
  1. Find the tangent to the cycloid at the point where $\theta = \pi/3$.
  2. 1. Find the area under one arch of the cycloid.
  3. Find the length of one arch of the cycloid.

• Find the area enclosed by the curve $x^{2} + y^{2} -ax = a \sqrt{x^{2} + y^{2}}$

• Find the area of the planar region enclosed by the curve $y^{2} = x^{2} - x^{4}$.

• Let the curve $C$ be defined by the equation: \(x^{3} + y^{3} - \frac{3}{2}xy = 0.\)
  1. It is known that the curve $C$ has an oblique asymptote. Find the equation of this oblique asymptote.
  2. Find the area of the planar region bounded by the curve $C$.

• Find the area of one petal of the three-leaved rose $r = \sin 3\theta$.

• Find the area of one petal of the eight-leaved rose $r = \sin 4\theta$.

• Find the area of the region that lies inside the circle $r = 3\sin\theta$ and outside the cardioid $r = 1 + \sin\theta$.

• Find the area inside the circle $r = \sin\theta$ and outside the cardioid $r = 1 - \cos\theta$.

• Find the arc length of $r = \sin^{3}(\theta/3)$ from $\theta = 0$ to $\theta = 3\pi/2$.

• Investigate $r = 1 + \sin\theta$ for horizontal and vertical tangents.

• A curve called the folium of Descartes is defined by the parametric equations \(x = \frac{3t}{1 + t^{3}}, \quad y = \frac{3t^{2}}{1 + t^{3}}\)
  1. Show that if $(a, b)$ lies on the curve, then so does $(b, a)$; that is, the curve is symmetric with respect to the line $y = x$. Where does the curve intersect this line?
  2. Find the points on the curve where the tangent lines are horizontal or vertical.
  3. Show that the line $y = -x - 1$ is a slant asymptote.
  4. Sketch the curve.
  5. Show that a Cartesian equation of this curve is \(x^{3} + y^{3} = 3xy.\)
  6. Show that the polar equation can be written in the form \(r = \dfrac{3 \sec \theta \tan \theta}{1 + \tan^{3} \theta}\)
  7. Find the area enclosed by the loop of this curve.
  8. Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.)