Chapter 25 — Quiz

Ten questions on parametric curves, their calculus, and applications. Try each before opening the answer. Section references point back to index.md.

1. For a parametric curve $x = x(t)$, $y = y(t)$ with $\dot x \neq 0$, the slope $dy/dx$ equals:

  • A) $\dot y$
  • B) $\dot x$
  • C) $\dot y / \dot x$
  • D) $\ddot y / \ddot x$
Answer**C) $\dot y/\dot x$.** The chain rule gives $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$; solving for $dy/dx$ yields $\dot y/\dot x$. *Section 25.3.*

2. The arc length of a parametric curve traced once on $[a, b]$ is:

  • A) $\int_a^b \dot x \, dt$
  • B) $\int_a^b \dot y \, dt$
  • C) $\int_a^b \sqrt{\dot x^2 + \dot y^2} \, dt$
  • D) $\int_a^b (\dot x^2 + \dot y^2) \, dt$
Answer**C).** The arc-length element is $ds = \sqrt{(dx)^2 + (dy)^2} = \sqrt{\dot x^2 + \dot y^2}\,dt$. *Section 25.4.*

3. The second derivative $d^2y/dx^2$ of a parametric curve is:

  • A) $\ddot y / \ddot x$
  • B) $\dfrac{\frac{d}{dt}(dy/dx)}{\dot x}$
  • C) $\dot y / \dot x$ differentiated once more in $t$
  • D) $\ddot y \cdot \ddot x$
Answer**B).** Treat $dy/dx$ as a new function of $t$ and divide its $t$-derivative by $\dot x$ — **not** by $\ddot x$. Writing $\ddot y/\ddot x$ is the single most common error in this chapter. *Section 25.3.*

4. One arch of the cycloid $x = r(t - \sin t)$, $y = r(1 - \cos t)$ has arc length:

  • A) $\pi r$
  • B) $4r$
  • C) $8r$
  • D) $2\pi r$
Answer**C) $8r$** — a clean integer multiple of the radius, with no $\pi$, found via the half-angle identity $2 - 2\cos t = 4\sin^2(t/2)$. *Section 25.7.*

5. Ignoring air resistance, a projectile's range $R = \dfrac{v_0^2 \sin(2\theta)}{g}$ is maximized at launch angle:

  • A) $30^\circ$
  • B) $45^\circ$
  • C) $60^\circ$
  • D) $90^\circ$
Answer**B) $45^\circ$**, where $\sin(2\theta) = 1$, giving $R_{\max} = v_0^2/g$. *Section 25.6.*

6. A parametric curve can describe loops and self-crossings that no graph $y = f(x)$ can. True or false?

  • A) True
  • B) False
Answer**A) True.** Freeing both coordinates to depend on $t$ describes any path; a single $(x,y)$ point may be visited more than once, with a different tangent each time. *Sections 25.1–25.3.*

7. Eliminating the parameter from $x = t$, $y = t^2$ gives:

  • A) $y = x$
  • B) $y = x^2$
  • C) $y = x^3$
  • D) it cannot be eliminated
Answer**B) $y = x^2$**, a parabola. Here $t = x$, so $y = x^2$. *Section 25.2.*

8. Revolving the curve $x = x(t)$, $y = y(t)$ ($y \ge 0$) about the $x$-axis generates a surface of area:

  • A) $\int_a^b 2\pi\,y\,\sqrt{\dot x^2 + \dot y^2}\,dt$
  • B) $\int_a^b 2\pi\,x\,\sqrt{\dot x^2 + \dot y^2}\,dt$
  • C) $\int_a^b \pi y^2 \, dt$
  • D) $\int_a^b 2\pi\,y\,dt$
Answer**A).** Each point traces a circle of circumference $2\pi y$; multiply by the band width $ds = \sqrt{\dot x^2+\dot y^2}\,dt$. (Revolving about the $y$-axis uses radius $x$ instead.) *Section 25.5.*

9. Which condition locates a vertical tangent on a parametric curve?

  • A) $\dot y = 0$ and $\dot x \neq 0$
  • B) $\dot x = 0$ and $\dot y \neq 0$
  • C) $\dot x = \dot y = 0$
  • D) $\ddot x = 0$
Answer**B).** No horizontal motion ($\dot x = 0$) with vertical motion ($\dot y \neq 0$) gives a vertical tangent. (When *both* vanish, suspect a cusp.) *Section 25.3.*

10. The perimeter of the ellipse $x = a\cos t$, $y = b\sin t$ (with $a \neq b$):

  • A) equals $\pi(a + b)$ exactly
  • B) equals $2\pi\sqrt{ab}$ exactly
  • C) has no elementary closed form and is evaluated numerically
  • D) equals $4(a + b)$
Answer**C).** The integral $\int_0^{2\pi}\sqrt{a^2\sin^2 t + b^2\cos^2 t}\,dt$ is a (non-elementary) *elliptic integral*; we evaluate it numerically or with Ramanujan's approximation. *Section 25.4.*

Scoring

  • 9–10 correct: Excellent — you command the parametric calculus of slope, length, and area.
  • 7–8 correct: Good. Review the section flagged on any miss.
  • Below 7: Re-read §25.3 (slope and second derivative) and §25.4 (arc length), the two engines behind everything else, then retry.