Chapter 9 — Further Reading

Annotated pointers for going deeper on extrema, the Mean Value Theorem, L'Hôpital's rule, and curve sketching. Each entry says what to read and why it complements this chapter. Section numbers below are the standard ones in the cited editions; the full cross-reference tables live in appendices/appendix-h-stewart-chapter-mapping.md and appendices/appendix-i-openstax-chapter-mapping.md.

Standard Coverage (closest parallels to this chapter)

  • Stewart, Calculus: Early Transcendentals (9th ed.), §4.1–4.5. The canonical treatment, section-for-section aligned with ours:
  • §4.1 Maximum and Minimum Values ↔ our §9.2–9.3 (absolute extrema, EVT, Fermat, the Closed Interval Method).
  • §4.2 The Mean Value Theorem ↔ our §9.7 (Rolle's theorem and the MVT, with the same secant-subtraction proof).
  • §4.3 How Derivatives Affect the Shape of a Graph ↔ our §9.4–9.6 (increasing/decreasing test, first and second derivative tests, concavity, inflection).
  • §4.4 Indeterminate Forms and L'Hospital's Rule ↔ our §9.8 (all forms, including the $0^0$/$1^\infty$/$\infty^0$ exponential cases).

  • §4.5 Summary of Curve Sketching ↔ our §9.9–9.10. Stewart's exercise bank here is enormous; work the odd-numbered curve sketches for drill.

  • OpenStax Calculus Volume 1 (Strang & Herman, free), §4.3–4.6 and §4.8.

  • §4.3 Maxima and Minima ↔ our §9.2–9.3.
  • §4.4 The Mean Value Theorem ↔ our §9.7.
  • §4.5 Derivatives and the Shape of a Graph ↔ our §9.4–9.6.
  • §4.6 Limits at Infinity and Asymptotes ↔ our §9.10 (horizontal and oblique asymptotes).
  • §4.8 L'Hôpital's Rule ↔ our §9.8. Free, openly licensed, with clean worked examples — the best zero-cost companion. Read it alongside a hard exercise, not before.

For the Curious (applications and intuition)

  • Gilbert Strang, Calculus (MIT, free PDF), Chapter 3 ("Applications of the Derivative"). Strang's prose is unusually intuitive about why the second derivative controls bending. His treatment of maxima/minima emphasizes the picture over the procedure — a good antidote if the sign charts feel mechanical.

  • Steven Strogatz, Infinite Powers (2019), chapters on optimization and the "vocabulary of change." No exercises, pure narrative — but it conveys why the extremum-and-curvature toolkit shows up everywhere from optics to economics. Read it for motivation between problem sets.

  • Morris Kline, Calculus: An Intuitive and Physical Approach (Dover, inexpensive). Old but excellent on the physical meaning of the MVT (average vs. instantaneous velocity) and on curve sketching by hand in the era before graphing software.

For the Rigor-Seeker (proofs and analysis)

  • Michael Spivak, Calculus (4th ed.), Chapters 7 and 11. Chapter 7 proves the Extreme Value Theorem from completeness; Chapter 11 develops the MVL/Rolle/Cauchy chain and L'Hôpital with full rigor. This is where the §9.2 "Math Major Sidebar" promise is cashed out — the topological proof that a continuous function on a compact set attains its extremes.

  • Walter Rudin, Principles of Mathematical Analysis (3rd ed.), §5.8–5.13. The MVT, the generalized (Cauchy) MVT, and L'Hôpital's rule proved at the level of a real-analysis course. §5.13 gives the clean proof that L'Hôpital follows from Cauchy's MVT — the connection flagged in our §9.7.3 sidebar and §9.8.

  • Tom Apostol, Calculus Volume 1, Chapter 4. A careful, theorem-first development of the derivative's applications, with attention to the exact hypotheses each result needs — useful for understanding why the MVT's differentiability-on-the-open-interval condition cannot be weakened.

Applications Across Fields (the §9.12 threads)

  • Rowland & Tozer, Clinical Pharmacokinetics and Pharmacodynamics (4th ed.). The Bateman absorption model and $t_{\max}$/$C_{\max}$ derivations of Case Study 1, in their clinical home.
  • Hal Varian, Intermediate Microeconomics (9th ed.). Diminishing marginal utility, concavity, and risk aversion (Case Study 2) developed for economists; pairs directly with the marginal-analysis material in §9.12.
  • OpenStax University Physics Vol. 1, Ch. 8. Potential-energy curves, equilibria as critical points, and stability via the second derivative — the physics application of §9.12 worked out.

Computational Companion

  • SymPy documentation — calculus and solvers modules. The functions used in §9.11: sympy.diff, sympy.solve, and sympy.limit. After finishing a sketch by hand, confirm critical points with sp.solve(sp.diff(f, x), x) and indeterminate limits with sp.limit(...) — the §9.11 workflow of hand first, machine to verify.
  • 3Blue1Brown, Essence of Calculus, the L'Hôpital and "implicit differentiation" episodes (YouTube, free). Visual intuition for why differentiating top and bottom resolves a $\tfrac00$ form — the tangent-line picture of §9.8 animated.

How to use this list. For exam preparation, drill Stewart §4.1–4.5 or OpenStax §4.3–4.8 exercises. For conceptual depth, read Strang or Strogatz. For proofs (the EVT, the full MVT chain, L'Hôpital from Cauchy's MVT), go to Spivak or Rudin. The field-specific texts show the same $f'$/$f''$ toolkit doing real work outside mathematics — the recurring theme that calculus appears in every quantitative field.