Change, made visible

Sources and simplifications

Every reference cited on this site, and every place a figure idealizes the real system.

References

  1. Gilbert Strang and Edwin “Jed” Herman (2016). Calculus Volume 1. OpenStax, Rice University. textbook Free, peer-reviewed. Used here: §2.2 The Limit of a Function, §2.5 The Precise Definition of a Limit, §3.1 Defining the Derivative, §3.2 The Derivative as a Function, §3.3 Differentiation Rules, §3.6 The Chain Rule, §4.1 Related Rates, §5.1 Approximating Areas, §5.2 The Definite Integral, §5.3 The Fundamental Theorem of Calculus. (Limits)
  2. Steven R. Williams (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education 22(3), 219–236. doi:10.2307/749075. primary (Limits)
  3. David Tall and Shlomo Vinner (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics 12, 151–169. doi:10.1007/BF00305619. primary (Limits)
  4. J. J. O’Connor and E. F. Robertson. George Berkeley (biography). MacTutor History of Mathematics, University of St Andrews. reference (Limits)
  5. Judith V. Grabiner (1978). The origins of Cauchy’s theory of the derivative. Historia Mathematica 5, 379–409. doi:10.1016/0315-0860(78)90208-2. scholarship (Limits)
  6. Judith V. Grabiner (1983). Who gave you the epsilon? Cauchy and the origins of rigorous calculus. The American Mathematical Monthly 90(3), 185–194. doi:10.1080/00029890.1983.11971185. scholarship (Limits)
  7. J. J. O’Connor and E. F. Robertson. Karl Theodor Wilhelm Weierstrass (biography). MacTutor History of Mathematics, University of St Andrews. reference (Limits)
  8. Irene Biza, Constantinos Christou and Theodossios Zachariades (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean geometry to analysis. Research in Mathematics Education 10(1), 53–70. doi:10.1080/14794800801916457. primary (The derivative)
  9. J. J. O’Connor and E. F. Robertson. Isaac Newton (biography). MacTutor History of Mathematics, University of St Andrews. reference (The derivative)
  10. J. J. O’Connor and E. F. Robertson. Gottfried Wilhelm von Leibniz (biography). MacTutor History of Mathematics, University of St Andrews. reference (The derivative)
  11. James Demmel (2002). Math 128a lecture notes: numerical differentiation. University of California, Berkeley. institutional (The derivative)
  12. C. J. F. Ridders (1982). Accurate computation of F′(x) and F′(x)F″(x). Advances in Engineering Software 4(2), 75–76. doi:10.1016/S0141-1195(82)80057-0. primary (The derivative)
  13. A. Orton (1983). Students’ understanding of differentiation. Educational Studies in Mathematics 14, 235–250. doi:10.1007/BF00410540. primary (The derivative)
  14. Department of Engineering, University of Almería. Machine and Mechanism Theory, Lesson 8: gear trains. University of Almería. institutional (The chain rule)
  15. J. M. Clark, F. Cordero, J. Cottrill, B. Czarnocha, D. J. DeVries, D. St. John, G. Tolias and D. Vidakovic (1997). Constructing a schema: the case of the chain rule. Journal of Mathematical Behavior 16(4), 345–364. doi:10.1016/S0732-3123(97)90012-2. primary (The chain rule)
  16. Gilbert Strang and Edwin “Jed” Herman (2016). Calculus Volume 3, §4.5 The Chain Rule. OpenStax, Rice University. textbook (The chain rule)
  17. A. Orton (1983). Students’ understanding of integration. Educational Studies in Mathematics 14, 1–18. doi:10.1007/BF00704699. primary (Riemann sums)
  18. Dong-Hai Nguyen and N. Sanjay Rebello (2011). Students’ understanding and application of the area under the curve concept in physics problems. Physical Review Special Topics: Physics Education Research 7, 010112. doi:10.1103/PhysRevSTPER.7.010112. primary (Riemann sums)
  19. Gilbert Strang and Edwin “Jed” Herman (2016). Calculus Volume 2. OpenStax, Rice University. textbook §3.6 Numerical Integration: midpoint, trapezoidal and Simpson’s rules and their error bounds. (Riemann sums)
  20. J. J. O’Connor and E. F. Robertson (1996). The rise of calculus. MacTutor History of Mathematics, University of St Andrews. reference (Riemann sums)
  21. Archimedes; translated and edited by T. L. Heath (1897). The Works of Archimedes (Quadrature of the Parabola, Propositions 17 and 24). Cambridge University Press. primary (Riemann sums)
  22. J. N. Lyness (1969). Notes on the adaptive Simpson quadrature routine. Journal of the ACM 16(3), 483–495. doi:10.1145/321526.321537. primary (Riemann sums)
  23. R. C. Churchill (2006). Liouville’s Theorem on Integration in Terms of Elementary Functions (Corollary 5.4). Kolchin Seminar on Differential Algebra, Hunter College and the Graduate Center, CUNY. reference Lecture notes giving Rosenlicht’s algebraic proof (Amer. Math. Monthly 79 (1972) 963–972) of Liouville’s 1835 theorem; Corollary 5.4 shows e^(x²) has no elementary primitive, and the same argument (a′ − 2ax = 1 has no rational solution a) covers e^(−x²). (Riemann sums)
  24. Patrick W. Thompson (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics 26, 229–274. doi:10.1007/BF01273664. primary (The fundamental theorem)
  25. Michael Spivak (2008). Calculus, 4th edition (Chapter 5, Limits; Chapter 12, Inverse Functions; Chapter 18, The Logarithm and Exponential Functions). Publish or Perish. ISBN 978-0-914098-91-1. textbook (The epsilon-delta definition of a limit)
  26. Tom M. Apostol (1967). Calculus, Volume 1, 2nd edition (Chapter 1, the integral as a limit of step-function sums; Chapter 6, the logarithm, the exponential and the inverse trigonometric functions). John Wiley & Sons. ISBN 978-0-471-00005-1. textbook (Derivative of ln x)

Simplifications