Calculus

Elementary

• Parametric equations

Differentiation

• Smooth functions
• Differentiation of inverse trigonometric functions
• Differentiation of hyperbolic functions
• Rolle’s theorem
• Mean value theorem
• Taylor’s theorem
  • Remainder term of Taylor’s theorem
• Cauchy’s mean value theorem
• Darboux’s intermediate value theorem
• L’Hôpital’s rule
• Fermat’s theorem

Integration

• Fundamental theorem of calculus
• Fubini’s theorem
• Green’s theorem

Applications of Integration

• Length of curves
• Line integrals of scalar fields
• Line integrals of vector fields

Series

• Properties of convergent series
• Geometric series $\displaystyle \sum ar^{n}$
• $p$-series and $p$-series test $\displaystyle \sum \frac{1}{n^{p}}$
• Harmonic series $\displaystyle \sum \frac{1}{n}$
  • Oresme’s proof of the divergence of the harmonic series
• Alternating Series $\displaystyle \sum (-1)^{n-1}b_{n}$
• Alternating harmonic series $\displaystyle \sum (-1)^{n-1}\frac{1}{n}$
  • Derivation of the series form of the natural logarithm and convergence of the alternating harmonic series
• Comprehensive summary of various series convergence tests in analysis
  • Divergence test
  • Integral test
  • Direct comparison test
  • Limit comparison test
  • Alternating series test
  • Ratio test
  • Root test
• Absolute Convergence and Conditional Convergence
• Taylor series and Maclaurin series
  • Conditions for a function and its Taylor series to be equal
  • Application of Taylor series
  • Taylor expansions of $e^{x}, \sin x, \cos x$
  • Series expansion of $\arctan x$
• Euler’s formula in calculus

References

• James Stewart, Daniel Clegg, and Saleem Watson, Calculus (early transcendentals, 9E)