Chapter 28 — Exercises

These problems build the vector-valued-function toolkit one layer at a time: first reading and graphing curves, then component-wise calculus, then the physics of motion, and finally the geometry of tangents, normals, and curvature. Work them in order — the later tiers lean on the earlier ones.

Difficulty tiers. ⭐ warm-up (one idea, direct application) · ⭐⭐ standard (combine two ideas) · ⭐⭐⭐ challenging (multi-step or conceptual) · ⭐⭐⭐⭐ deep (proof, modeling, or open-ended).

Verify your hand work against the symbolic and numerical recipes of Section 28.10 whenever a problem gets messy. Selected answers appear in the back-of-book answers-to-selected.md.

Part A — Reading and Graphing Space Curves (§28.1–28.2)

A1. ⭐ Write the vector-valued function for the straight line through $(2,-1,3)$ in the direction $\langle 1,4,-2\rangle$. Where is the point when $t=2$?

A2. ⭐ Identify the curve $\mathbf{r}(t) = \langle 3\cos t,\; 3\sin t,\; 5\rangle$. What is its shape, where does it sit, and what is the role of the constant $z=5$?

A3. ⭐ For the helix $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$, by how much does the curve rise during one full turn ($t$ from $0$ to $2\pi$)?

A4. ⭐⭐ Sketch (by hand, reasoning component-wise) the curve $\mathbf{r}(t)=\langle t,\; \cos t,\; \sin t\rangle$. Which axis does it spiral around, and why does this differ from the standard helix of §28.2?

A5. ⭐⭐ The twisted cubic is $\mathbf{r}(t)=\langle t, t^2, t^3\rangle$ (§28.2). Find the points where it meets each coordinate plane ($x=0$, $y=0$, $z=0$), and explain why it never lies in a single plane.

A6. ⭐⭐ A particle follows $\mathbf{r}(t) = \langle 2\cos t,\; 2\sin t,\; t^2\rangle$. Describe its projection onto the $xy$-plane and onto the $xz$-plane. Contrast with the constant-rise helix, whose $z$-component is linear.

A7. ⭐⭐⭐ Find a single vector-valued function whose graph is the curve of intersection of the cylinder $x^2+y^2=4$ and the plane $z = x + 1$. (Hint: parametrize the cylinder first, then read $z$ off the plane.)

Part B — Component-Wise Calculus: Derivatives and Integrals (§28.3)

B1. ⭐ Differentiate $\mathbf{r}(t)=\langle t^3,\; \cos 2t,\; e^{-t}\rangle$.

B2. ⭐ Compute $\displaystyle\int \langle 6t,\; \sin t,\; \sec^2 t\rangle\,dt$. Remember the constant vector $\mathbf{C}$.

B3. ⭐ Evaluate $\displaystyle\int_0^1 \langle t^2,\; 4t^3,\; 1\rangle\,dt$ and state why the answer is a vector, not a function.

B4. ⭐⭐ For $\mathbf{r}(t)=\langle \ln t,\; t^2,\; \sqrt{t}\rangle$, find $\mathbf{r}'(t)$ and $\mathbf{r}''(t)$, noting the domain restriction.

B5. ⭐⭐ Solve the initial-value problem $\mathbf{r}'(t) = \langle 2t,\; e^t,\; \cos t\rangle$ with $\mathbf{r}(0) = \langle 1, 1, 0\rangle$.

B6. ⭐⭐ Let $\mathbf{u}(t) = \langle t, t^2, 1\rangle$ and $\mathbf{v}(t) = \langle 1, t, t^2\rangle$. Use the dot-product rule (§28.3) to compute $\frac{d}{dt}[\mathbf{u}\cdot\mathbf{v}]$ two ways — directly, and via $\mathbf{u}'\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{v}'$ — and confirm they agree.

B7. ⭐⭐⭐ With the same $\mathbf{u},\mathbf{v}$, compute $\frac{d}{dt}[\mathbf{u}\times\mathbf{v}]$ via the cross-product rule. Then state what goes wrong if you carelessly write $\mathbf{v}'\times\mathbf{u} + \mathbf{u}'\times\mathbf{v}$ instead (recall the sign trap in §28.3).

B8. ⭐⭐⭐ Prove the dot-product rule $\frac{d}{dt}[\mathbf{u}\cdot\mathbf{v}] = \mathbf{u}'\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{v}'$ by writing out the three-term sum and applying the ordinary scalar product rule slot by slot.

B9. ⭐⭐⭐ Show that if $\mathbf{r}(t)$ has constant length (i.e. $|\mathbf{r}(t)|$ is constant), then $\mathbf{r}(t)\cdot\mathbf{r}'(t) = 0$ for all $t$. (This is the constant-length fact from §28.3; you will reuse it for the unit normal in §28.9.)

Part C — Velocity, Speed, and Acceleration (§28.4)

C1. ⭐ A particle has position $\mathbf{r}(t)=\langle t^2,\; 2t,\; \ln t\rangle$. Find its velocity, speed, and acceleration at $t=1$.

C2. ⭐ For the helix $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$, verify that the speed is the constant $\sqrt 2$ and that velocity and acceleration are perpendicular at every $t$.

C3. ⭐⭐ A projectile has acceleration $\mathbf{a}(t)=\langle 0,0,-9.8\rangle$ (m/s²), initial velocity $\mathbf{v}(0)=\langle 20, 0, 15\rangle$ m/s, and initial position $\mathbf{r}(0)=\langle 0,0,2\rangle$ m. Find $\mathbf{r}(t)$ by integrating twice (§28.4).

C4. ⭐⭐ For the projectile of C3, at what time does it reach maximum height, and what is that height? (Maximize the $z$-component.)

C5. ⭐⭐ A particle moves with $\mathbf{v}(t) = \langle e^t, -e^{-t}, \sqrt 2\rangle$. Find its speed and show it has a minimum; find where.

C6. ⭐⭐⭐ A point moves along $\mathbf{r}(t) = \langle t - \sin t,\; 1 - \cos t,\; t\rangle$. Find the times in $[0, 2\pi]$ where the speed is smallest, and interpret the motion there.

C7. ⭐⭐⭐ Show that if a particle moves with constant speed, then its acceleration is always perpendicular to its velocity (use the constant-length fact on $\mathbf{v}$). Then explain why the converse holds: if $\mathbf{v}\cdot\mathbf{a}=0$ always, the speed is constant.

Part D — Arc Length and Unit-Speed Parametrization (§28.5–28.6)

D1. ⭐ Find the arc length of the helix $\mathbf{r}(t)=\langle 3\cos t,\; 3\sin t,\; 4t\rangle$ for $0\le t\le 2\pi$.

D2. ⭐ Find the arc length of the segment $\mathbf{r}(t)=\langle 1+2t,\; 3-t,\; 2t\rangle$, $0\le t\le 3$, and confirm it equals the straight-line distance between the endpoints.

D3. ⭐⭐ For $\mathbf{r}(t) = \langle t,\; \tfrac{t^2}{2},\; \tfrac{t^3}{6}\rangle$ on $0\le t\le 1$, show that the integrand under the arc-length root is a perfect square, and evaluate $L$ exactly.

D4. ⭐⭐ Compute the unit tangent vector $\mathbf{T}(t)$ for $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ (§28.6).

D5. ⭐⭐ Reparametrize the line $\mathbf{r}(t)=\langle 1, 2, 3\rangle + t\langle 2, -1, 2\rangle$ by arc length measured from $t=0$.

D6. ⭐⭐⭐ Reparametrize the helix $\mathbf{r}(t)=\langle 3\cos t, 3\sin t, 4t\rangle$ by arc length, then verify the result has speed $1$.

D7. ⭐⭐⭐ Using $s(t) = \int_a^t |\mathbf{r}'(u)|\,du$ and FTC Part 1 (Chapter 14), explain why $ds/dt$ equals the speed. Then explain why most curves (e.g. the ellipse) cannot be reparametrized by arc length in closed form, referencing the non-elementary integrals of Chapter 17.

Part E — Curvature (§28.7–28.8)

E1. ⭐ Find the curvature of a circle of radius $7$ and confirm it equals $1/7$ everywhere.

E2. ⭐ Use the plane-curve formula $\kappa(x) = \frac{|f''(x)|}{(1+f'(x)^2)^{3/2}}$ (§28.8) to find the curvature of $y = x^3$ at $x=1$.

E3. ⭐⭐ Find the curvature of the parabola $y = x^2$ at its vertex and at $x=2$. Where does the curve bend most sharply, and why does that match the geometry?

E4. ⭐⭐ Use $\kappa = \frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3}$ to find the curvature of the helix $\mathbf{r}(t)=\langle 3\cos t, 3\sin t, 4t\rangle$. Check against the general-helix value $\kappa = a/(a^2+b^2)$ from §28.8.

E5. ⭐⭐ Find the radius of curvature of $y = \ln x$ at $x=1$, and state the radius of the osculating circle there.

E6. ⭐⭐⭐ Find the point on $y = e^x$ where the curvature is maximum (set $\kappa'(x)=0$).

E7. ⭐⭐⭐ For the twisted cubic $\mathbf{r}(t)=\langle t, t^2, t^3\rangle$, compute $\kappa(t)$ in closed form using the cross-product formula, and confirm the curvature is largest at $t=0$ (matching the numerical result of §28.10).

E8. ⭐⭐⭐⭐ Prove that a curve with $\kappa(t) = 0$ for all $t$ (and nonzero speed) must be a straight line. (Hint: $\kappa = 0$ forces $\mathbf{T}'=\mathbf{0}$, so $\mathbf{T}$ is a constant unit vector; integrate.)

Part F — Unit Normal and Components of Acceleration (§28.9)

F1. ⭐ A particle has $\mathbf{v}=\langle 2,1,2\rangle$ and $\mathbf{a}=\langle 1,1,1\rangle$ at some instant. Find the tangential component $a_T = \frac{\mathbf{v}\cdot\mathbf{a}}{|\mathbf{v}|}$.

F2. ⭐⭐ For the same $\mathbf{v},\mathbf{a}$ of F1, find the normal component $a_N = \frac{|\mathbf{v}\times\mathbf{a}|}{|\mathbf{v}|}$, and verify that $a_T^2 + a_N^2 = |\mathbf{a}|^2$.

F3. ⭐⭐ For the helix $\mathbf{r}(t)=\langle \cos t,\sin t,t\rangle$, find $\mathbf{T}(t)$, then $\mathbf{T}'(t)$, then $\mathbf{N}(t)$. Confirm $\mathbf{N}$ points horizontally toward the $z$-axis.

F4. ⭐⭐ Decompose the acceleration of the projectile $\mathbf{r}(t)=\langle 20t,\;0,\;2+15t-4.9t^2\rangle$ into tangential and normal components at $t=0$.

F5. ⭐⭐⭐ A car rounds a flat circular bend of radius $R=50$ m at speed $v=20$ m/s. Using $a_N = \kappa v^2$ with $\kappa = 1/R$, compute the lateral acceleration, express it as a fraction of $g=9.8$ m/s², and decide whether it sits in the comfortable $0.1$–$0.15\,g$ design range of §28.9.

F6. ⭐⭐⭐⭐ (Modeling — civil engineering.) A highway curve is designed for $30$ m/s with comfort requiring $a_N \le 0.12\,g$. Find the minimum radius of curvature for the sweeping part of the bend. Then explain, referencing the clothoid spiral of §28.9, why the ramp entry should not jump straight to this curvature but should ramp $\kappa$ up linearly from $0$. Which single-variable quantity is the driver's "jerk" related to?

Part G — Motion in Space and Modeling (§28.4, §28.11)

G1. ⭐⭐ (Aerospace.) A satellite's measured acceleration over a short burn is $\mathbf{a}(t)=\langle -0.5, 0, 0.2\rangle$ m/s². Given $\mathbf{v}(0)=\langle 7600, 0, 0\rangle$ m/s and $\mathbf{r}(0)=\langle 0,0,0\rangle$, find the velocity and position after $10$ s. (Inertial navigation: integrate measured acceleration — §28.4, §28.11.)

G2. ⭐⭐⭐ (Physics — orbits.) For a body obeying $\mathbf{r}'' = -\frac{GM}{|\mathbf{r}|^3}\mathbf{r}$, define $\mathbf{L} = \mathbf{r}\times\mathbf{r}'$. Using the cross-product rule, show $\mathbf{L}' = \mathbf{0}$, and explain why this proves orbits are planar. Connect this to the conic-section orbits of Chapter 27 and Kepler's Second Law (§28.11).

G3. ⭐⭐⭐⭐ (Physics — charged particle.) A particle in a uniform field $\mathbf{B}=\langle 0,0,B\rangle$ obeys $m\mathbf{r}'' = q\,\mathbf{r}'\times\mathbf{B}$. Show the $z$-component of velocity is constant and the $(x,y)$ motion is circular, so the path is a helix (§28.11). Without solving the ODE, argue from $\mathbf{F}\perp\mathbf{v}$ that the speed never changes.

Tier Count

Total: 47 problems across 7 lettered parts. Applied problems span at least three fields — civil engineering (F5, F6), aerospace (G1), and physics (G2, G3) — alongside the biology (DNA) and data-science modeling threads carried in §28.11.