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Nydus Open Learning

Mathematics

Mathematics · Avançado · Nível 1

Calculus I

The real numbers and completeness; functions, domains and inverses; limits with epsilon and delta, the limit laws, the squeeze theorem and continuity with the intermediate and extreme value theorems; the derivative from its definition and from the rules, including the chain rule, implicit differentiation and the exponential and logarithmic derivatives; related rates, linearisation, extrema, the mean value theorem, curve shape, optimisation and L'Hopital's rule; and the definite integral from Darboux sums through both parts of the fundamental theorem to substitution.

Ages 18+ 30 lessons about 10 hours in all

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As lições

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  1. 1 The real numbers and completeness
  2. 2 Absolute value, intervals and the triangle inequality
  3. 3 Functions: domain, image and composition
  4. 4 Injections, surjections and inverse functions
  5. 5 The limit of a function, and one-sided limits
  6. 6 The epsilon-delta definition of a limit
  7. 7 Constructing a delta
  8. 8 The limit laws, and the hypothesis they all share
  9. 9 The squeeze theorem and the sine limit
  10. 10 Continuity, the intermediate value theorem and the extreme value theorem
  11. 11 Limits at infinity, infinite limits and asymptotes
  12. 12 The derivative at a point
  13. 13 The derivative as a function, and where it fails
  14. 14 The power, sum, product and quotient rules
  15. 15 The chain rule
  16. 16 Derivatives of the trigonometric functions
  17. 17 Implicit differentiation and the derivative of an inverse
  18. 18 Exponential and logarithmic derivatives
  19. 19 Related rates
  20. 20 Linear approximation and differentials
  21. 21 Maxima, minima and critical points
  22. 22 Rolle's theorem and the mean value theorem
  23. 23 Monotonicity, concavity and the shape of a graph
  24. 24 Applied optimisation
  25. 25 L'Hopital's rule and the indeterminate forms
  26. 26 Antiderivatives
  27. 27 Area, Riemann sums and the Darboux criterion
  28. 28 The definite integral and its properties
  29. 29 The fundamental theorem of calculus
  30. 30 Integration by substitution