Chapter 40 — Exercises

This is the last problem set in the book, and it is unlike the others. There is little new machinery here. Instead, these problems ask you to see the whole: to explain how calculus enabled a piece of the modern world, to combine techniques from chapters that were taught separately, to trace each of the four anchors to its destination, and to look forward to the courses that lie past the last page.

Work the conceptual prompts in full sentences — they are rehearsals for explaining calculus to someone else, which is the surest test of understanding. Work the computational problems by hand first, then check the logic against what the relevant chapter taught. Tiers run ⭐ (warm-up recall) to ⭐⭐⭐⭐ (genuine synthesis or forward-looking).

Selected answers (marked †) appear in appendices/answers-to-selected.md.

A. The Six Themes — Conceptual Prompts (⭐–⭐⭐⭐)

1. ⭐ † State, in one sentence each, the six recurring themes of this book (see §40.6). You should be able to do this from memory.

2. ⭐ Theme 1 is "calculus is the mathematics of change." Name the two complementary operations of calculus and say which one measures a rate of change and which one accumulates it.

3. ⭐⭐ † Theme 3 says the Fundamental Theorem of Calculus is the keystone. Write the plain FTC, $\int_a^b f'(x)\,dx = f(b)-f(a)$, and explain in two or three sentences why §40.3 calls every later integral theorem "the same theorem in different clothing."

4. ⭐⭐ Theme 6 is "approximation is the soul of calculus." Give three distinct examples from the book of an exact object being replaced by an approximation that converges to it. (Hint: limits in Chapter 3, linearization in Chapter 11, Taylor series in Chapter 23.)

5. ⭐⭐⭐ Theme 4 contrasts hand computation with machine computation. Using the SIR model of Chapter 19 as your example, explain precisely what hand analysis gives you that machine integration does not, and vice versa.

6. ⭐⭐⭐ † Theme 2 says geometry and algebra are inseparable. For each of (a) the derivative, (b) the definite integral, (c) the gradient, give both its geometric meaning and its algebraic definition, and explain why they are "the same understanding seen from two sides" (§40.6).

7. ⭐⭐⭐ Theme 5 says calculus appears in every quantitative field. Pick three different fields (e.g., biology, economics, electrical engineering) and name, for each, one specific equation or model from this book that lives in that field. Identify the chapter each came from.

B. The FTC Family — Computational Synthesis (⭐⭐–⭐⭐⭐)

8. ⭐⭐ † Verify the plain FTC (Chapter 14) directly. Let $f(x) = x^3$. Compute $f'(x)$, integrate it from $0$ to $2$, and confirm the result equals $f(2) - f(0)$.

9. ⭐⭐ The Fundamental Theorem for line integrals (Chapter 35) says $\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)$. Let $f(x,y) = x^2 y$ and let $C$ be any path from $A=(0,0)$ to $B=(2,3)$. Compute $\nabla f$, then evaluate the line integral using only the endpoints. (You should never have to parametrize the path.)

10. ⭐⭐⭐ The Divergence theorem (Chapter 37) says $\iint_{\partial E}\mathbf{F}\cdot d\mathbf{S} = \iiint_E (\nabla\cdot\mathbf{F})\,dV$. For $\mathbf{F} = \langle x, y, z\rangle$, compute $\nabla\cdot\mathbf{F}$, then use the theorem to evaluate the outward flux through the surface of the unit ball $E = \{x^2+y^2+z^2 \le 1\}$ without computing a single surface integral. (The volume of the unit ball is $\tfrac{4}{3}\pi$.)

11. ⭐⭐⭐ † Arrange these four statements in order of increasing dimension/generality, and for each name the chapter where you met it: Green's theorem; the plain FTC; the Divergence theorem; the generalized Stokes' theorem $\int_{\partial M}\omega = \int_M d\omega$ (Chapter 38). Then state in one sentence the single idea they all share.

12. ⭐⭐ Green's theorem (Chapter 35) lets you compute an area as a boundary integral: area $= \tfrac{1}{2}\oint_{\partial D}(x\,dy - y\,dx)$. Confirm this gives the right answer for the unit disk by parametrizing its boundary as $x=\cos t,\ y=\sin t$, $0\le t\le 2\pi$. (You should get $\pi$.)

C. Cumulative Computation Across Chapters (⭐⭐–⭐⭐⭐)

13. ⭐⭐ † A particle moves with velocity $v(t) = 3t^2 - 4t$ (Chapter 5: velocity is the derivative of position; Chapter 14: net displacement is the integral of velocity). Find the net displacement over $0\le t\le 3$.

14. ⭐⭐⭐ Combine optimization (Chapter 10) with integration (Chapter 18). A rectangle sits with its base on the $x$-axis and its top two corners on the parabola $y = 4 - x^2$. (a) Find the dimensions that maximize its area. (b) Separately, find the total area under the parabola between its $x$-intercepts. (c) What fraction of that area does the largest rectangle occupy?

15. ⭐⭐⭐ † Linear approximation (Chapter 11) and Taylor series (Chapter 23) are the same idea carried to different orders. For $f(x) = \sqrt{x}$ near $a = 100$: (a) use the linear (first-order Taylor) approximation to estimate $\sqrt{101}$; (b) write the second-order Taylor approximation and re-estimate; (c) comment on which is closer to the true value $10.0498\ldots$.

16. ⭐⭐⭐ A solid is generated by rotating the region under $y = e^{-x}$ from $x=0$ to $x=\infty$ about the $x$-axis. This braids Chapter 17 (improper integrals) with Chapter 18 (volumes of revolution). Using the disk method, the volume is $\pi\int_0^\infty e^{-2x}\,dx$. Evaluate it.

17. ⭐⭐ Switch coordinate systems to make a hard integral easy — the lesson of Part V and Chapter 33. Evaluate $\iint_D (x^2+y^2)\,dA$ over the unit disk by converting to polar coordinates (Chapter 26) and using the area element $dA = r\,dr\,d\theta$ (Chapter 33).

18. ⭐⭐⭐ † The Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$ is computed by the polar trick referenced in §40.5. Outline the four steps: (1) write the squared integral as a double integral (Chapter 32), (2) the coordinate change you make (Chapter 26), (3) the Jacobian factor that appears (Chapter 33), (4) why the resulting integral is now elementary. You need not carry out every line — name the moves and the chapters they come from.

D. Trace the Anchor — Reflective Prompts (⭐⭐–⭐⭐⭐⭐)

19. ⭐⭐ † Gradient descent. Trace this anchor through the book. (a) What did the one-variable derivative tell you about which way to step (Chapter 6)? (b) Write the multivariable update rule using the gradient (Chapter 30). (c) Name the algorithm, built from the Chapter 7 chain rule run backward, that computes the gradient efficiently in a neural network.

20. ⭐⭐⭐ Gradient descent, by hand. Minimize $f(x) = (x-3)^2$ using the update $x_{n+1} = x_n - \eta f'(x_n)$ with step size $\eta = 0.1$ and start $x_0 = 0$. Compute $x_1, x_2, x_3$ by hand and state the value the sequence is converging to. (You may write a short Python loop in your answer, but compute the first three steps by hand.)

21. ⭐⭐⭐ † The SIR model. The system is $\frac{dS}{dt} = -\beta S I$, $\frac{dI}{dt} = \beta S I - \gamma I$, $\frac{dR}{dt} = \gamma I$. (a) From the sign of $\frac{dI}{dt}$ at the start of an outbreak, derive the threshold condition for whether the infected fraction initially grows. (b) Show this gives the basic reproduction number $R_0 = \beta/\gamma$ (with $S\approx 1$). (c) In one sentence, connect $\frac{dI}{dt}$ (a rate) and the epidemic curve (an accumulation) to the two operations of calculus.

22. ⭐⭐⭐ The normal curve. Explain why $P(a \le X \le b) = \int_a^b \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,dx$ cannot be evaluated with an elementary antiderivative, yet is computed to arbitrary precision by every statistics package. Name the Chapter 23 tool that does it, and state the §40.5 "Common Pitfall" distinction between "no closed form" and "unsolvable."

23. ⭐⭐⭐⭐ † Euler's formula. (a) Write Euler's formula and Euler's identity. (b) Sketch how comparing the Taylor series (Chapter 23) of $e^{i\theta}$, $\cos\theta$, and $\sin\theta$ produces the formula — show enough terms of each series to make the match visible. (c) Name two modern technologies (§40.5) that rest on this formula, and the next subject (beyond this book) it opens the door to.

E. Where Calculus Goes Next — Forward-Looking (⭐⭐⭐⭐)

24. ⭐⭐⭐⭐ Section 40.7 maps four destinations: differential equations, real analysis, differential geometry, and PDEs. For each, write two sentences: one naming the central new question it answers, and one naming a specific result from this book that it generalizes or makes rigorous. (Hints: Chapter 19; Chapter 3's $\varepsilon$–$\delta$ limit; Chapter 38's generalized Stokes' theorem; Maxwell's and Schrödinger's equations from §40.4.)

25. ⭐⭐⭐⭐ † Section 40.10 lists the honest limits of calculus. Choose two of them — "most differential equations have no closed-form solution," "determinism does not mean predictability" (chaos), or "smoothness is an assumption" — and for each, explain in a short paragraph why it is a genuine limitation and which broader mathematical tool (numerical methods, chaos theory, stochastic calculus, discrete mathematics) responds to it.

26. ⭐⭐⭐⭐ The closing synthesis claims the modern world is "made of differential equations" (§40.4). Pick one of the four great theories — Newtonian mechanics ($\mathbf{F}=m\ddot{\mathbf{r}}$), Maxwell's electromagnetism, general relativity ($G_{\mu\nu} = \tfrac{8\pi G}{c^4}T_{\mu\nu}$), or quantum mechanics (Schrödinger's equation) — and write a half-page essay: identify every piece of calculus visible in its governing equation, name the chapters those pieces came from, and explain what real-world technology depends on solving it. This is the capstone reflection of the entire book; take your time with it.