Course

Calculus

Limits, continuity, derivatives, their applications and integrals: the classic path of single-variable calculus, from the idea of approximation to the fundamental theorem.

Recommended before this course: Set theory (1 concept comes from that course and is tagged)

38concepts

5topics

492exercises

≈ 22 hof study approx.

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Course path

1
Limits 8 concepts
Limit of a function · One-sided limits · Limit laws · Indeterminate forms · Squeeze theorem · Special trigonometric limits · Limits at infinity · Asymptotes
2
Continuity 5 concepts
Continuous function · Intermediate value theorem · Extreme value theorem · Types of discontinuity · Composition and inverse from Set theory
3
Derivatives 9 concepts
Derivative · Tangent line · Basic differentiation rules · Product rule · Quotient rule · Chain rule · Derivatives of trigonometric functions · Derivatives of exponentials and logarithms · Implicit differentiation
4
Applications of derivatives 8 concepts
Rolle's theorem · Mean value theorem · Critical points · Local maxima and minima · Concavity and inflection points · Optimization problems · Related rates · L'Hôpital's rule
5
Integrals 8 concepts
Definite integral · Riemann sums · Fundamental theorem of calculus · Antiderivative · Indefinite integral · Integration by substitution · Integration by parts · Area between curves

Sources

James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015.
Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011.
Herbert B. Enderton, Elements of Set Theory, 1st edition, Academic Press, 1977.
Thomas Jech, Set Theory (The Third Millennium Edition), 3rd millennium edition, Springer, 2003.
John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2013.
Tom M. Apostol, Calculus, Vol. 1, 2.ª edición, John Wiley & Sons, 1967.