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Calculus: notes and links to resources
http://gauss.vaniercollege.qc.ca/~kritchevski/calculus/
Eugene Kritchevski, Vanier College, last updated on September 13, 2017
DIFFERENTIAL CALCULUS
Calculus I Science 201-NYA-05 and Calculus I Commerce 201-103-VA
Algebra
worksheet on algebra
Geogebra applet - Pascal's triangle and expanding $(a+b)^n$
Tree diagram of a mathematical expression
REVIEW OF ALGEBRA - section freely available from stewartcalculus.com
REVIEW OF ANALYTIC GEOMETRY - section freely available from stewartcalculus.com
Functions
Geogebra applet - Unit circle, radians, sin and cos functions
Limits and Continuity
Geogebra applet - a piecewise function
Geogebra applet - left and right limits
Geogebra applet - proof of $\boxed{\lim_{x\to 0}\frac{\sin (x)}{x}=1}$
Geogebra applet - The precise definition of a limit
Derivatives
Geogebra applet - Power rule for positive integer exponents
Geogebra applet - Power rule for negative integer exponents
Geogebra applet - Mean value theorem (MVT)
Basic differential rules - notes (Wolfram CDF player required to view)
INTEGRAL CALCULUS (Calculus II Science 201-NYB-05 and Calculus II Commerce 201-203-VA)
Generalities on summation (sigma notation, properties of sums, telescoping sums, sums of powers of integers)
worksheet on summation
notes on summation
Riemann sums
worksheet on Riemann sums (Isaac Sabbah)
Geogebra Applet - area under a graph
Geogebra Applet - net area
Geogebra Applet - subdividing the interval
Geogebra Applet - right-endpoint Riemann sum for degree two polynomials
Area under a graph - introductory examples (needs CDF player to read the notes)
Fundamental Theorem of Calculus
Geogebra Applet - concept of the area function in FTC1
Geogebra Applet - proof of FTC1
Substitution
notes on substitution, with practice problems
video on substitution (Selwin Hollis)
worksheet on integration by substitution (Isaac Sabbah)
Integration by Parts
notes on integration by parts, with practice problems
video on integration by parts (Selwin Hollis)
worksheet on integration by parts (Isaac Sabbah)
Areas Between Curves
notes on areas between curves - including applets, practice problems and link to a video
Volumes
notes on volume by disks
Geogebra applet - disk method example: volume of a cone
Geogebra applet - disk method example: volume of a sphere
Geogebra applet - washers method
Geogebra applet - cylindrical shells example: volume of a Torus
Geogebra applet - cylindrical shells example: rotating $\set{0 \leq x \leq \sqrt{\pi}, 0 \leq y \leq \sin(x^2)}$ about the $y$-axis
video on volumes by slicing in general (Selwin Hollis)
video on volumes by disks/washers (Selwin Hollis)
video on volumes by cylindrical shells (Selwin Hollis)
section on on volumes by cylindrical shells freely available from stewartcalculus.com
worksheet on volumes (Isaac Sabbah)
Trigonometric Integrals
section on trigonometric integrals freely available from stewartcalculus.com
Trigonometric Substitution
notes on arsin and on the substitution for $\sqrt{r^2-y^2}$
section on trigonometric substitution freely available from stewartcalculus.com
video on trigonometric substitution (Selwin Hollis)
worksheet on trigonometric substitution (Isaac Sabbah)
Partial Fractions
Geogebra Applet - 3 dictinct linear factors
GeoGebra Applet - repeated linear factors
video on partial fractions (Selwin Hollis)
worksheet on partial fractions (Isaac Sabbah)
Differential Equations
GeoGebra Applet - first order differential equations
worksheet on differential equations (Isaac Sabbah)
Note for questions 4.(4) and 4.(5) in the worksheet: replace $dy$ with $y'$ in the question.
L'Hospital's Rule
notes and practice problems on l'Hospital's rule from Paul's Online Math Notes
Improper Integrals
worksheet on improper integrals (Isaac Sabbah)
Sequences
Introductory examples of sequences - notes
Geogebra applet - Web plot for a recursively defined sequence
video on Sequences - Part 1 (Selwin Hollis)
video on Sequences - Part 2 (Selwin Hollis)
Series
Watch the videos on series (Selwin Hollis) starting from video 49.
worksheet on series (Isaac Sabbah)
Review material
Strategy for Integration, section freely available from stewartcalculus.com
worksheet on integration review (Isaac Sabbah)
Appendices
proof of trigonometric identities
Other freely available resources on the web
A complete set of Video Lectures for Single-variable Calculus created by Selwin Hollis, Armstrong State University.
These videos (and a set of videos on differential equations) are also available for free on
iTunes U.
OpenStax math books
A list of open source textbooks (Calculus and other subjects) approved by the American Institute of Mathematics.
Paul's Online Math Notes.
Calculus notes used at the University of British Columbia, written by Joel Feldman and Andrew Rechnitzer.
Course materials from John Abbott College.
ADVANCED TOPICS (for FUN, special talks, summer research, etc)
Introduction to the math of the 1D vibrating string (Joint Honors Math-Physics)
Volume of a sphere in higher dimensions
Error in Lagrange interpolation and proof of Simpson's rule