Chapter 25 — Exercises

These problems build the full calculus of paths from Chapter 25: sketching and orienting curves (§25.2), eliminating the parameter (§25.2), the chain-rule slope $dy/dx = \dot y/\dot x$ (§25.3), the second derivative (§25.3), tangent lines and cusps (§25.3), arc length (§25.4), surface area of revolution (§25.5), projectile motion (§25.6), and the cycloid and its classical relatives (§25.7–25.8). Work them with pencil first; the Python in §25.9 is for verifying a hand result, never for replacing it.

A four-star (⭐⭐⭐⭐) problem asks you to combine several tools or push past a routine template. Keep answers exact where possible; decimal approximations are flagged.

Tier counts

Part A — Sketching, Orientation, and Eliminating the Parameter

A1. ⭐ Sketch $x = t$, $y = 2t - 1$ for $t \in [0, 3]$. Mark the orientation with an arrow and give the Cartesian equation.

A2. ⭐ For $x = 2\cos t$, $y = 2\sin t$, $t \in [0, 2\pi]$, eliminate the parameter and describe the curve, including its orientation.

A3. ⭐ Eliminate the parameter for $x = t + 1$, $y = t^2$. State the resulting Cartesian equation and any restriction on $x$ or $y$.

A4. ⭐ Eliminate the parameter for $x = 3\cos t$, $y = 5\sin t$. Identify the curve and its semi-axes.

A5. ⭐⭐ The curve $x = \sin t$, $y = \sin t$ traces only part of the line $y = x$. Eliminate the parameter, then state exactly which segment is traced and why. (Recall the §25.2 pitfall about lost domains.)

A6. ⭐⭐ For $x = e^t$, $y = e^{-t}$, eliminate the parameter to get a relation in $x$ and $y$, and state the range of $x$ actually covered.

A7. ⭐⭐ Two parametrizations, $x = \cos t,\ y = \sin t$ on $[0, 2\pi]$ and $x = \cos(3t),\ y = \sin(3t)$ on $[0, 2\pi]$, trace the same set of points. How many times does each circle the origin, and which way? Explain using the angle swept (compare the doubly-traced circle in §25.2).

A8. ⭐⭐ Eliminate the parameter for the astroid $x = \cos^3 t$, $y = \sin^3 t$ to recover $x^{2/3} + y^{2/3} = 1$. (Hint: cube-root each coordinate, then use $\cos^2 t + \sin^2 t = 1$.)

Part B — Slope, Tangent Lines, and the Second Derivative

B1. ⭐ For $x = t^2$, $y = t^3$, compute $dy/dx$ as a function of $t$.

B2. ⭐ For $x = 1 + t^2$, $y = t - t^3$, find $dy/dx$ at $t = 2$.

B3. ⭐ For $x = \cos t$, $y = \sin t$, find the slope of the tangent at $t = \pi/6$.

B4. ⭐⭐ For $x = t^3 - 3t$, $y = t^2 - 1$, find all values of $t$ where the tangent is horizontal and all where it is vertical (the §25.3 conditions).

B5. ⭐⭐ For the ellipse $x = 4\cos t$, $y = 3\sin t$, write the equation of the tangent line at $t = \pi/4$.

B6. ⭐⭐ For $x = t^2 + 1$, $y = t^3 - 3t$, compute $d^2y/dx^2$ as a function of $t$. (Use the §25.3 rule; do not write $\ddot y/\ddot x$.)

B7. ⭐⭐ A student computes $d^2y/dx^2 = \ddot y/\ddot x$ for $x = t^2$, $y = t^4$ and reports $d^2y/dx^2 = 12t^2/2 = 6t^2$. Find the correct $d^2y/dx^2$ and explain in one sentence why the student's method fails (the §25.3 pitfall).

B8. ⭐⭐⭐ The curve $x = t^2 - 1$, $y = t^3 - t$ crosses itself at the origin. Find the two values of $t$ giving $(0,0)$ and the two distinct tangent slopes there.

B9. ⭐⭐⭐ For $x = \cos t$, $y = \sin 2t$ on $[0, 2\pi]$, find every point with a horizontal tangent. (This is a figure-eight; expect several.)

B10. ⭐⭐⭐ Show that the astroid $x = a\cos^3 t$, $y = a\sin^3 t$ has $\dot x = \dot y = 0$ at $t = 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}$, confirming the four cusps (§25.8), and simplify $dy/dx$ at a generic $t$ to $-\tan t$.

Part C — Arc Length

C1. ⭐ Set up (do not evaluate) the arc-length integral for $x = t^2$, $y = t^3$ on $[0, 2]$.

C2. ⭐ Find the arc length of the circle $x = 5\cos t$, $y = 5\sin t$, $t \in [0, 2\pi]$, and confirm it equals $2\pi R$.

C3. ⭐⭐ Find the exact arc length of $x = t$, $y = \tfrac{2}{3}t^{3/2}$ on $[0, 3]$. (This reduces to a clean elementary integral.)

C4. ⭐⭐ The line $x = 1 + 3t$, $y = 2 - 4t$ runs on $t \in [0, 2]$. Use the arc-length integral and check the answer against the straight-line distance formula.

C5. ⭐⭐⭐ Find the arc length of one arch of the cycloid $x = r(t - \sin t)$, $y = r(1 - \cos t)$ and confirm the result is $8r$ (§25.7). Show each half-angle step.

C6. ⭐⭐⭐ Set up the perimeter integral for the ellipse $x = 4\cos t$, $y = 2\sin t$. Explain why it has no elementary antiderivative (the elliptic-integral obstruction of §25.4), then evaluate Ramanujan's approximation $L \approx \pi[3(a+b) - \sqrt{(3a+b)(a+3b)}]$ for $a=4$, $b=2$ as a decimal.

Part D — Surface Area of Revolution

D1. ⭐ Write the surface-area integral (about the $x$-axis) for revolving $x = t$, $y = t^2$, $t \in [0, 1]$. Do not evaluate. (Use the §25.5 formula.)

D2. ⭐⭐ Revolve the semicircle $x = R\cos t$, $y = R\sin t$, $t \in [0, \pi]$, about the $x$-axis and confirm the sphere's surface area $4\pi R^2$.

D3. ⭐⭐⭐ A profile curve $x = t$, $y = \sqrt{t}$ on $[0, 4]$ is revolved about the $x$-axis to model a turned lampshade. Set up the surface-area integral and simplify the radicand to the form $1 + \tfrac{1}{4t}$.

D4. ⭐⭐⭐ Revolve the line segment $x = t$, $y = 2t$, $t \in [0, 3]$, about the $x$-axis (a cone). Evaluate the surface integral exactly and check it against the cone lateral-area formula $\pi r \ell$.

Part E — Cycloid and Classical Curves

E1. ⭐⭐ Find the area under one arch of the cycloid $x = r(t-\sin t)$, $y = r(1-\cos t)$ using $A = \int y\,\dot x\,dt$, and confirm it is $3\pi r^2$ (§25.7).

E2. ⭐⭐⭐ For the trochoid $x = 2t - 3\sin t$, $y = 2 - 3\cos t$ (a prolate trochoid, $d > r$ as in §25.8), find the $t$-values in $[0, 2\pi]$ where $\dot x = 0$. Interpret physically: these locate where the tracked point momentarily moves backward.

E3. ⭐⭐⭐ The cardioid $x = 2\cos t - \cos 2t$, $y = 2\sin t - \sin 2t$ (§25.8) has one cusp. Find the value of $t \in [0, 2\pi]$ where $\dot x = \dot y = 0$.

E4. ⭐⭐⭐⭐ Show that the astroid $x = a\cos^3 t$, $y = a\sin^3 t$ has total arc length $6a$. Compute $\sqrt{\dot x^2 + \dot y^2}$, exploit the symmetry of the four equal arcs, and integrate one arc over $[0, \pi/2]$.

Part F — Projectile Motion (Physics)

F1. ⭐ A projectile is launched at $v_0 = 30$ m/s, $\theta = 40^\circ$, $g = 9.8$ m/s². Write $x(t)$ and $y(t)$ using the §25.6 model.

F2. ⭐⭐ For the projectile in F1, find the time of flight and the range. (Decimal answers.)

F3. ⭐⭐ Show that launch angles $\theta$ and $90^\circ - \theta$ give the same range (§25.6), and verify numerically for $25^\circ$ and $65^\circ$ at $v_0 = 20$ m/s.

F4. ⭐⭐⭐ A ball must clear a $4$ m wall that stands $30$ m away. Launched at $v_0 = 22$ m/s, set up the condition $y = 4$ when $x = 30$ and reduce it to one equation in $\theta$ (eliminate $t$ as in §25.6). State how you would solve it numerically.

Part G — Applied and Synthesis (≥ two fields)

G1. ⭐⭐⭐ (Biology — animal tracking.) A foraging beetle follows $x = t - \sin t$, $y = 1 - \cos t$ for $t \in [0, 2\pi]$ (positions in cm, $t$ in minutes). Compute the total path length traveled (an arc-length integral, §25.4) and compare it to the straight-line displacement between start and end. Report the tortuosity ratio (path length ÷ displacement). (Connects to the biology track of the §25.10 Modeling Portfolio.)

G2. ⭐⭐⭐ (Engineering — CAD turning.) A bottle profile is $x = t$, $y = 2 + \cos t$ for $t \in [0, \pi]$ (cm). Set up the integral for the surface area generated by revolving the profile about the $x$-axis (§25.5), which a CAD kernel would integrate numerically to estimate coating mass.

G3. ⭐⭐⭐⭐ (Data science — optimizer trajectory.) A gradient-descent run traces $\theta_1(t) = e^{-t}\cos t$, $\theta_2(t) = e^{-t}\sin t$ for $t \in [0, 4]$ (a spiral into the minimum at the origin). Set up the arc-length integral measuring how far the optimizer "wandered" (the §25.10 data-science portfolio prompt), and find $dy/dx = d\theta_2/d\theta_1$ at $t = 0$ exactly.

G4. ⭐⭐⭐⭐ (Physics + optimization synthesis.) Generalize the §25.6 range formula to a projectile launched from height $h$ (so $y(t) = h + (v_0\sin\theta)t - \tfrac12 g t^2$). Show the optimal angle is no longer $45^\circ$ and derive that $\sin^2\theta_{\text{opt}} = \dfrac{v_0^2}{2v_0^2 + 2gh}$. Comment on why a higher launch favors a flatter angle. (Uses optimization from Chapter 10.)

Answers and Hints

Selected full solutions appear in appendices/answers-to-selected.md. A few orienting checks:

• A6: $xy = 1$, the branch with $x > 0$ (since $e^t > 0$).
• B4: Horizontal where $\dot y = 2t = 0 \Rightarrow t = 0$; vertical where $\dot x = 3t^2 - 3 = 0 \Rightarrow t = \pm 1$.
• B7: Correct answer $d^2y/dx^2 = \frac{1}{\dot x}\frac{d}{dt}(2t^2) = \frac{4t}{2t} = 2$ for $t \neq 0$; the student divided $\frac{d}{dt}(dy/dx)$ by $\ddot x$ instead of $\dot x$.
• C3: $\dot x = 1$, $\dot y = t^{1/2}$, integrand $\sqrt{1+t}$; $L = \tfrac{2}{3}\big[(1+t)^{3/2}\big]_0^3 = \tfrac{2}{3}(8 - 1) = \tfrac{14}{3}$.
• C6: $a=4$, $b=2$: $L \approx \pi[18 - \sqrt{14\cdot 10}] = \pi[18 - \sqrt{140}] \approx \pi(18 - 11.832) \approx 19.38$.
• E2: $\dot x = 2 - 3\cos t = 0 \Rightarrow \cos t = 2/3 \Rightarrow t \approx 0.841,\ 5.442$ rad.
• F2: $t_{\text{flight}} = \frac{2v_0\sin\theta}{g} = \frac{60\sin 40^\circ}{9.8} \approx 3.94$ s; $R = \frac{v_0^2\sin 80^\circ}{9.8} \approx \frac{900(0.985)}{9.8} \approx 90.4$ m.
• G1: This is one cycloid arch with $r=1$, so path length $= 8$ cm; displacement $= x(2\pi) - x(0) = 2\pi \approx 6.283$ cm; tortuosity $\approx 8/6.283 \approx 1.273$.
• G4: Maximize $R(\theta)$ with the height-$h$ landing time; the $45^\circ$ result is the special case $h = 0$.