Chapter 11 — Further Reading

An annotated guide to where this chapter's material is treated elsewhere, with explicit section mappings to the two reference texts (Stewart and OpenStax) and a few classics. Use these to see a second derivation, find more exercises, or go deeper than the chapter does.

Standard Coverage

These sections cover the same ground as Chapter 11 and make the natural first stop for extra worked examples and problem sets.

  • Stewart, Calculus: Early Transcendentals (9th ed.), §3.10 — Linear Approximations and Differentials. The canonical treatment of $L(x)$, the differential $dy = f'(x)\,dx$, and error/relative-error propagation. Its sphere-volume and circle-area error examples mirror our §11.4–11.5; work its exercises alongside our Part B and Part C.
  • Stewart, §4.8 — Newton's Method. The standard derivation of $x_{n+1}=x_n-f(x_n)/f'(x_n)$, the geometry of sliding down the tangent, and the failure cases. Maps directly to our §11.6–11.9. Its exercises pair well with our Part D and Part E.
  • OpenStax Calculus Volume 1, §4.2 — Linear Approximations and Differentials. Free, slightly gentler parallel to Stewart §3.10; good for re-reading the differential and percentage-error vocabulary of our §11.4–11.5.
  • OpenStax Calculus Volume 1, §4.9 — Newton's Method. Free parallel to Stewart §4.8, including a clean statement of the iteration and worked square-root and cosine-fixed-point examples like our §11.7 and §11.10.

(Chapter-by-chapter crosswalks for the whole book live in appendices/appendix-h-stewart-chapter-mapping.md and appendices/appendix-i-openstax-chapter-mapping.md.)

Going Deeper — Convergence and Numerical Analysis

For the why behind quadratic convergence (§11.8) and the robust hybrid methods of §11.12.

  • Burden & Faires, Numerical Analysis (10th ed.), Ch. 2 — Solutions of Equations in One Variable. The standard undergraduate proof that Newton converges quadratically near a simple root, plus full treatments of bisection, the secant method, and fixed-point iteration. This is the rigorous home of our §11.8 and §11.12.
  • Trefethen & Bau, Numerical Linear Algebra. For readers heading toward data science: shows how Newton's method generalizes to systems and underlies optimization solvers (connecting to "Newton for optimization" in §11.10 and Chapter 31).
  • Press et al., Numerical Recipes, "Root Finding and Nonlinear Sets of Equations." Practical, code-first discussion of why production solvers (e.g. brentq) blend bisection with secant/inverse-quadratic interpolation rather than trusting raw Newton — the engineering behind our §11.12 warning.

Going Deeper — Error Analysis in Experiment

For the measurement side of the chapter (§11.5 and Case Study 2).

  • Taylor, An Introduction to Error Analysis (2nd ed., University Science Books). The standard undergraduate lab reference. Chapters 3–4 derive exactly the propagation rules we use, including the multivariable form $\sigma_z^2 \approx f_x^2\sigma_x^2 + f_y^2\sigma_y^2$ previewed in §11.11. Essential if you do any laboratory work.
  • Bevington & Robinson, Data Reduction and Error Analysis for the Physical Sciences. A more advanced companion, with the statistical underpinnings of why worst-case errors add and independent errors add in quadrature.

Computer Arithmetic (Case Study 1)

  • Goldberg, "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (ACM Computing Surveys, 1991). The classic essay on how reals are stored — the mantissa/exponent split that the square-root hardware exploits.
  • Muller et al., Handbook of Floating-Point Arithmetic (2nd ed.). Detailed treatment of how chips compute division and square root with a seed-plus-Newton cascade, with full error bounds.

Historical and Conceptual

  • Newton's original (1669) and Raphson's (1690) formulations, discussed in Katz, A History of Mathematics and in Stewart §4.8's historical margin notes. See the Historical Note in §11.6 — Newton's method predates the modern derivative-based form by decades.
  • Berlinski, A Tour of the Calculus. A literary, big-picture companion that situates linear approximation within the larger story of the subject; good non-technical reading to accompany §11.1 and §11.13.

Forward Pointers Within This Book

  • Chapter 23 — Power and Taylor Series. The full generalization of §11.11: linear approximation is the first-order Taylor polynomial; higher orders give arbitrary accuracy.
  • Chapter 29 — Functions of Several Variables. The tangent plane and multivariable differentials/error propagation hinted at in §11.11.
  • Chapter 31 — Optimization in Several Variables. Where "Newton's method for optimization" (§11.10) becomes a workhorse.
  • Chapter 12 — Antiderivatives. Where the differential notation $dx$ of §11.4 becomes the integration element.