A youtube Calculus Workbook (Part II)
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About the book
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Description
This book is a guide through a playlist of Calculus instructional videos. The format, level of details and rigor, and progression of topics are consistent with a semester long college level second Calculus course, or equivalently, together with the first workbook, an AP Calculus BC course. The book further provides simple summary of videos, written definitions and statements, worked out exampleseven though fully stepbystep solutions are to be found in the videos and an index. The playlist and the book are divided into 16 thematic learning modules. Exercises, some with and some without solutions, and sample tests with solutions are provided in a separate companion manual. The book can be used for self study, or as a textbook for a Calculus course following the “flipped classroom” model.
Preface
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 ebook is a guide through a playlist of Calculus instructional videos. The playlist 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 playlist 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.
Content
Preface
 M1: Natural Logarithm and Exponential
 Natural Logarithm: definition and logarithm laws
 Calculus of Logarithms
 Logarithmic Differentiation
 Onetoone functions and inverse functions
 Finding inverse functions
 Calculus of inverse functions
 Natural Exponential: definition and properties
 Derivatives and integrals with exponentials
 Exponential and logarithmic equations and inequalities
 M2: More transcendental functions
 General exponential functions
 General logarithm functions
 Inverse trig functions: arcsine
 Inverse trig functions: other inverse trig functions
 Inverse trig functions: derivative and integrals
 Hyperbolic functions
 Inverse hyperbolic functions
 M3: Rule of De l’Hospital
 Rule of De L’Hospital: statement and proof
 Rule of de l’Hospital: examples (quotients)
 Rule of De L’Hospital: indeterminate products
 Rule of De L’Hospital: indeterminate powers
 M4: Integration review and Integration by parts
 Review of Integration: basics and completing the square
 Review of Integration: trig formulas and manipulating fractions
 Integration by parts: indefinite integrals
 Integration by parts: definite integrals
 Integration by parts: one more example
 M5: Trigonometric integrals and trigonometric substitutions
 Powers of sine and cosine
 Products of sine and cosine
 (co)secant, (co)tangent and their powers
 Trig substitutions
 M6: Partial Fractions
 Partial fractions: generalities; long division
 only nonrepeated linear factors
 with repeated linear factors
 with irreducible quadratic factors
 with repeated irreducible quadratic factors
 M7: Improper Integrals
 Improper integrals of type I
 Improper integrals of type II
 Comparison for improper integrals
 M8: Parametric Curves
 Introduction to parametric curves
 Tangent lines to parametric curves
 Symmetry; concavity
 plane areas
 arc length
 Surface area of surface of revolutions
 M9: Polar Curves
 Polar coordinates
 Polar regions and polar curves
 tangent lines to polar curves
 arc length for polar curves
 area enclosed by a sector of a polar curve
 M10: Sequences and Series
 Sequences
 limit of sequences
 abstract properties of sequences
 limit of sequences defined inductively
 fixed points and limits of sequences defined inductively
 Series
 Series: a criterion for divergence
 Geometric Series
 Telescoping sums
 M11: Integral Test and Comparison Test
 Integral Test
 pseries
 Estimating the sum
 Direct Comparison Test
 Limit Comparison Test
 Estimating sums revisited
 M12: Alternating Series Test
 Alternating Series Test
 Absolute and conditional convergence
 Estimating sums with the Alternating Series Test
 M13: Ratio and Root Tests
 Ratio Test (Statement and proof)
 Ratio Test: examples
 Root Test
 Strategies to test series for convergence (M14)
 M15: Power Series and Taylor Series
 Power series
 Intervals of convergence
 Representation of functions as power series
 termbyterm differentiation and integration of power series
 more power series representations
 Power series and sums of numerical series
 Taylor and MacLaurin series
 Examples of Taylor Series
 Convergence of Taylor Series
 More examples of Taylor Series
 M16: Applications of power series
 Power series and sums of numerical series
 Estimating integrals
 Calculating limits
 More power series: products
 More power series: Binomial series
 Notations
 Index
 Endnotes