Chapter 28 — Quiz

Ten questions covering the core of vector-valued functions: component-wise calculus, the velocity/speed/acceleration distinction, arc length, the unit tangent and normal, and curvature. Answer each, then check the explanation and the section it comes from. Aim to explain each answer, not just recognize it.

1. The derivative of a vector-valued function $\mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle$ is computed by:

  • A) Taking the magnitude $|\mathbf{r}(t)|$ first
  • B) Differentiating each component separately
  • C) Forming the gradient $\nabla \mathbf{r}$
  • D) Crossing $\mathbf{r}$ with $\mathbf{r}'$
Answer**B.** Limits, derivatives, and integrals of a vector function all act **component-wise** — the limit of a vector is the vector of limits, so $\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t)\rangle$. This is exactly why all of single-variable calculus carries over untouched. *Section 28.3.*

2. A vector-valued function $\mathbf{r}(t)$ and a vector field are:

  • A) The same object with different names
  • B) Different: $\mathbf{r}(t)$ takes one number and returns one vector (a curve); a vector field attaches a vector to every point of space
  • C) Different: $\mathbf{r}(t)$ is always two-dimensional, a field always three
  • D) Different only in physics, not in mathematics
Answer**B.** This is the single most common confusion in vector calculus, flagged on the chapter's first page. $\mathbf{r}(t)$ has *one* input $t$ and traces *one* curve; a **vector field** (Chapter 34) attaches a vector to *every point*, filling a region with arrows. Don't let the shared word "vector" blur them. *Section 28.1.*

3. For the helix $\mathbf{r}(t) = \langle \cos t, \sin t, t\rangle$, the speed $|\mathbf{r}'(t)|$ equals:

  • A) $0$ B) $1$ C) $\sqrt{2}$ D) $t$
Answer**C.** $\mathbf{r}'(t) = \langle -\sin t, \cos t, 1\rangle$, so $|\mathbf{r}'| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}$ — a *constant*. The particle climbs the helix at uniform speed. *Section 28.4.*

4. A car rounds a bend at a steady 30 mph. Which statement is correct?

  • A) Velocity is constant, so acceleration is zero
  • B) Speed is constant but velocity is changing, so the car is accelerating
  • C) Both speed and velocity are constant
  • D) Acceleration points along the direction of motion
Answer**B.** Speed (a scalar) is constant, but velocity (a vector) is changing direction as the car turns — so there is acceleration even though the speedometer never moves. This velocity-versus-speed distinction is the conceptual core of the section, and the acceleration here is purely the *turning* (normal) part. *Section 28.4.*

5. The arc length of a curve $\mathbf{r}(t)$ on $[a,b]$ is:

  • A) $\int_a^b \mathbf{r}'(t)\,dt$ B) $\int_a^b |\mathbf{r}'(t)|\,dt$ C) $|\mathbf{r}(b) - \mathbf{r}(a)|$ D) $\int_a^b |\mathbf{r}(t)|\,dt$
Answer**B.** Arc length integrates the **speed** $|\mathbf{r}'(t)|$ (a scalar), not the velocity vector $\mathbf{r}'(t)$ — "length is accumulated speed," the Net Change Theorem again. Option C gives only the straight-line distance between endpoints, which undercounts a curved path. *Section 28.5.*

6. The unit tangent vector is defined as:

  • A) $\mathbf{r}'(t)$ B) $\mathbf{r}''(t)$ C) $\dfrac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$ D) $\mathbf{r}'(t)\times\mathbf{r}''(t)$
Answer**C.** $\mathbf{T}(t) = \mathbf{r}'(t)/|\mathbf{r}'(t)|$ is the velocity with its length scrubbed out — direction of motion only, length $1$. Reparametrizing by arc length makes the velocity *equal* $\mathbf{T}$, since unit-speed motion has $|d\mathbf{r}/ds| = 1$. *Section 28.6.*

7. A circle of radius $R$ has curvature:

  • A) $R$ B) $1/R$ C) $1/R^2$ D) $0$
Answer**B.** $\kappa = 1/R$ — a small circle (small $R$) bends sharply (large $\kappa$); a vast circle barely bends. A straight line is the limit $R\to\infty$, giving $\kappa = 0$. The radius of curvature $\rho = 1/\kappa$ is the radius of the osculating circle. *Section 28.8.*

8. The most practical formula for the curvature of a space curve is:

  • A) $\kappa = |\mathbf{r}''(t)|$ B) $\kappa = \dfrac{|\mathbf{r}'(t)\times\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$ C) $\kappa = \dfrac{|\mathbf{r}'(t)|}{|\mathbf{r}''(t)|}$ D) $\kappa = |\mathbf{r}'(t)|\,|\mathbf{r}''(t)|$
Answer**B.** The cross-product formula $\kappa = |\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^3$ avoids the painful quotient-rule differentiation of $\mathbf{T}$ and is almost always the one to reach for. For a plane curve $y=f(x)$ it collapses to $\kappa = |f''|/(1+f'^2)^{3/2}$. *Section 28.8.*

9. When acceleration is split as $\mathbf{a} = a_T\mathbf{T} + a_N\mathbf{N}$, the normal component is:

  • A) $a_N = \kappa\,|\mathbf{v}|$ B) $a_N = \kappa\,|\mathbf{v}|^2$ C) $a_N = |\mathbf{v}|^2/\kappa$ D) $a_N = dv/dt$
Answer**B.** $a_N = \kappa|\mathbf{v}|^2$ is the *turning* part; the tangential part $a_T = dv/dt = \frac{\mathbf{v}\cdot\mathbf{a}}{|\mathbf{v}|}$ is the *speeding-up* part. Because $a_N$ scales with the **square** of speed, doubling your speed through a bend quadruples the sideways force. There is no third (binormal) component — acceleration always lies in the $\mathbf{T}$–$\mathbf{N}$ plane. *Section 28.9.*

10. A charged particle moving through a uniform magnetic field traces a:

  • A) Straight line B) Parabola C) Helix D) Hyperbola
Answer**C.** The Lorentz force $q\,\mathbf{v}\times\mathbf{B}$ is always perpendicular to the velocity, so it changes direction but never speed. The velocity component along $\mathbf{B}$ drifts unchanged while the perpendicular component is forced into a circle — drift plus circle is a **helix**, the chapter's running example, produced here by physics (and the mechanism behind the aurora). *Section 28.11.*

Scoring Guide

  • 9–10 correct — Excellent. You command the geometry of space curves: the velocity/speed distinction, arc length as accumulated speed, and curvature as the rate of turning. Move on to Chapter 29.
  • 7–8 correct — Solid. Reread the sections flagged on your misses, especially the $a_T$/$a_N$ split (§28.9) and the curvature formulas (§28.8).
  • 5–6 correct — Shaky. Rework the helix from scratch: velocity, speed, $\mathbf{T}$, $\kappa$, and the acceleration decomposition. The helix touches every idea in the chapter.
  • Below 5 — Review. Restudy §28.3–28.4 (component-wise calculus and the velocity/speed/acceleration trio) before continuing; everything later builds on them.