A youtube Calculus Workbook (Part II)

With the explosion of resources available on the internet, virtually anything can be learned on your own, using free online resources. Or can it, really? If you are looking for instructional videos to learn Calculus, you will probably have to sort through thousands of hits, navigate through videos of inconsistent quality and format, jump from one instructor to another, all this without written guidance.

This free e-book is a guide through a play-list of Calculus instructional videos. The play-list and the book are divided into 16 thematic learning modules. The format, level of details and rigor, and progression of topics are consistent with a semester long college level Calculus II course, the first volume covering the equivalent of a Calculus I course. The continuity of style should help you learn the material more consistently than jumping around the many options available on the internet. The book further provides simple summary of videos, written definitions and statements, worked out examples – even though fully step by step solutions are to be found in the videos – and an index.

The present book is a guide to instructional videos, and as such can be used for self study, or as a textbook for a Calculus course following the flipped classroom model.

An essential companion to this book is the exercise manual Exercises for A youtube Calculus Workbook Part II: a flipped classroom model, which also outlines and discusses the structure for a flipped classroom course based on this material.

For future reference, the play list of all the videos is available at:

www.youtube.com/playlist? list=PLm168eGEcBjnS6ecJflh7BTDaUB6jShIL.

If you need to review any part of Calculus I, please refer to the first youtube workbook, whose associated play-list is available at:

www.youtube.com/playlist? list=PL265CB737C01F8961.

In particular, undefined notions or Theorems we may refer to that are not stated in the present book can be found in the first volume.

I hope that only few errors are left in this book, but some are bound to remain. I welcome feedback and comments at calculusvideos@gmail.com.

Preface

1. M1: Natural Logarithm and Exponential
   1. Natural Logarithm: definition and logarithm laws
   2. Calculus of Logarithms
   3. Logarithmic Differentiation
   4. One-to-one functions and inverse functions
   5. Finding inverse functions
   6. Calculus of inverse functions
   7. Natural Exponential: definition and properties
   8. Derivatives and integrals with exponentials
   9. Exponential and logarithmic equations and inequalities
2. M2: More transcendental functions
   1. General exponential functions
   2. General logarithm functions
   3. Inverse trig functions: arcsine
   4. Inverse trig functions: other inverse trig functions
   5. Inverse trig functions: derivative and integrals
   6. Hyperbolic functions
   7. Inverse hyperbolic functions
3. M3: Rule of De l’Hospital
   1. Rule of De L’Hospital: statement and proof
   2. Rule of de l’Hospital: examples (quotients)
   3. Rule of De L’Hospital: indeterminate products
   4. Rule of De L’Hospital: indeterminate powers
4. M4: Integration review and Integration by parts
   1. Review of Integration: basics and completing the square
   2. Review of Integration: trig formulas and manipulating fractions
   3. Integration by parts: indefinite integrals
   4. Integration by parts: definite integrals
   5. Integration by parts: one more example
5. M5: Trigonometric integrals and trigonometric substitutions
   1. Powers of sine and cosine
   2. Products of sine and cosine
   3. (co)secant, (co)tangent and their powers
   4. Trig substitutions
6. M6: Partial Fractions
   1. Partial fractions: generalities; long division
   2. only non-repeated linear factors
   3. with repeated linear factors
   4. with irreducible quadratic factors
   5. with repeated irreducible quadratic factors
7. M7: Improper Integrals
   1. Improper integrals of type I
   2. Improper integrals of type II
   3. Comparison for improper integrals
8. M8: Parametric Curves
   1. Introduction to parametric curves
   2. Tangent lines to parametric curves
   3. Symmetry; concavity
   4. plane areas
   5. arc length
   6. Surface area of surface of revolutions
9. M9: Polar Curves
   1. Polar coordinates
   2. Polar regions and polar curves
   3. tangent lines to polar curves
   4. arc length for polar curves
   5. area enclosed by a sector of a polar curve
10. M10: Sequences and Series
    1. Sequences
    2. limit of sequences
    3. abstract properties of sequences
    4. limit of sequences defined inductively
    5. fixed points and limits of sequences defined inductively
    6. Series
    7. Series: a criterion for divergence
    8. Geometric Series
    9. Telescoping sums
11. M11: Integral Test and Comparison Test
    1. Integral Test
    2. p-series
    3. Estimating the sum
    4. Direct Comparison Test
    5. Limit Comparison Test
    6. Estimating sums revisited
12. M12: Alternating Series Test
    1. Alternating Series Test
    2. Absolute and conditional convergence
    3. Estimating sums with the Alternating Series Test
13. M13: Ratio and Root Tests
    1. Ratio Test (Statement and proof)
    2. Ratio Test: examples
    3. Root Test
    4. Strategies to test series for convergence (M14)
14. M15: Power Series and Taylor Series
    1. Power series
    2. Intervals of convergence
    3. Representation of functions as power series
    4. term-by-term differentiation and integration of power series
    5. more power series representations
    6. Power series and sums of numerical series
    7. Taylor and MacLaurin series
    8. Examples of Taylor Series
    9. Convergence of Taylor Series
    10. More examples of Taylor Series
15. M16: Applications of power series
    1. Power series and sums of numerical series
    2. Estimating integrals
    3. Calculating limits
    4. More power series: products
    5. More power series: Binomial series
16. Notations
17. Index
18. Endnotes