Calculus Series by Howard Anton

[Greg Bernhardt]

• Author: Howard Anton
• Amazon Link:
  https://www.amazon.com/dp/0470647698/?tag=pfamazon01-20
  https://www.amazon.com/dp/0470647728/?tag=pfamazon01-20
  https://www.amazon.com/dp/0471153060/?tag=pfamazon01-20
  https://www.amazon.com/dp/0470183497/?tag=pfamazon01-20
  https://www.amazon.com/dp/0470183462/?tag=pfamazon01-20
• Prerequisities: High-School Mathematics

Table of Contents:

[LIST]
[*] Before Calculus
[LIST] 
[*] Functions
[*] New Functions from Old 
[*] Families of Functions
[*] Inverse Functions; Inverse Trigonometric Functions
[*] Exponential and Logarithmic Functions
[/LIST]
[*] Limits and Continuity
[LIST]
[*] Limits (An Intuitive Approach)
[*] Computing Limits
[*] Limits at Infinity; End Behavior of a Function
[*] Limits (Discussed More Rigorously) 
[*] Continuity
[*] Continuity of Trigonometric, Exponential, and Inverse Functions
[/LIST]
[*] The Derivative
[LIST]
[*] Tangent Lines and Rates of Change
[*] The Derivative Function
[*] Introduction to Techniques of Differentiation
[*] The Product and Quotient Rules
[*] Derivatives of Trigonometric Functions
[*] The Chain Rule
[/LIST]
[*] Topics in Differentiation
[LIST]
[*]Implicit Differentiation
[*] Derivatives of Logarithmic Functions
[*] Derivatives of Exponential and Inverse Trigonometric Functions 
[*] Related Rates
[*] Local Linear Approximation; Differentials
[*] L’Hôpital’s Rule; Indeterminate Forms
[/LIST]
[*] The derivative in Graphing and Applications
[LIST]
[*] Analysis of Functions I: Increase, Decrease, and Concavity
[*] Analysis of Functions II: Relative Extrema; Graphing Polynomials
[*] Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents
[*] Absolute Maxima and Minima
[*] Applied Maximum and Minimum Problems
[*] Rectilinear Motion
[*] Newton’s Method
[*] Rolle’s Theorem; Mean-Value Theorem
[/LIST]
[*] Integration
[LIST]
[*] An Overview of the Area Problem 
[*] The Indefinite Integral
[*] Integration by Substitution
[*] The Definition of Area as a Limit; Sigma Notation
[*] The Definite Integral
[*] The Fundamental Theorem of Calculus
[*] Rectilinear Motion Revisited Using Integration
[*] Average Value of a Function and its Applications 
[*] Evaluating Definite Integrals by Substitution
[*] Logarithmic and Other Functions Defined by Integrals
[/LIST]
[*] Applications of the Definite Integral in Geometry, Science, and Engineering
[LIST]
[*] Area Between Two Curves
[*] Volumes by Slicing; Disks and Washers
[*] Volumes by Cylindrical Shells
[*] Length of a Plane Curve
[*] Area of a Surface of Revolution
[*] Work
[*] Moments, Centers of Gravity, and Centroids
[*] Fluid Pressure and Force
[*] Hyperbolic Functions and Hanging Cables
[/LIST]
[*] Principles of Integral Evaluation
[LIST]
[*] An Overview of Integration Methods
[*] Integration by Parts
[*] Integrating Trigonometric Functions
[*] Trigonometric Substitutions
[*] Integrating Rational Functions by Partial Fractions
[*] Using Computer Algebra Systems and Tables of Integrals
[*] Numerical Integration; Simpson’s Rule
[*] Improper Integrals
[/LIST]
[*] Mathematical Modeling with Differential Equations
[LIST]
[*] Modeling with Differential Equations
[*] Separation of Variables
[*] Slope Fields; Euler’s Method
[*] First-Order Differential Equations and Applications
[/LIST]
[*] Infinite Series
[LIST]
[*] Sequences
[*] Monotone Sequences
[*] Infinite Series
[*] Convergence Tests
[*] The Comparison, Ratio, and Root Tests
[*] Alternating Series; Absolute and Conditional Convergence
[*] Maclaurin and Taylor Polynomials
[*] Maclaurin and Taylor Series; Power Series
[*] Convergence of Taylor Series
[*] Differentiating and Integrating Power Series; Modeling with Taylor Series
[/LIST]
[*] Parametric and Polar Curves; Conic Sections
[LIST]
[*] Parametric Equations; Tangent Lines and Arc Length for Parametric Curves
[*] Polar Coordinates
[*] Tangent Lines, Arc Length, and Area for Polar Curves
[*] Conic Sections
[*] Rotation of Axes; Second-Degree Equations
[*] Conic Sections in Polar Coordinates
[/LIST]
[*] Three-dimensional Space; Vectors
[LIST]
[*] Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 
[*] Vectors
[*] Dot Product; Projections
[*] Cross Product
[*] Parametric Equations of Lines
[*] Planes in 3-Space
[*] Quadric Surfaces
[*] Cylindrical and Spherical Coordinates
[/LIST]
[*] Vector-Valued Functions
[LIST]
[*] Introduction to Vector-Valued Functions
[*] Calculus of Vector-Valued Functions
[*] Change of Parameter; Arc Length
[*] Unit Tangent, Normal, and Binormal Vectors
[*] Curvature
[*] Motion Along a Curve
[*] Kepler’s Laws of Planetary Motion
[/LIST]
[*] Partial Derivatives
[LIST]
[*] Functions of Two or More Variables
[*] Limits and Continuity
[*] Partial Derivatives
[*] Differentiability, Differentials, and Local Linearity
[*] The Chain Rule
[*] Directional Derivatives and Gradients
[*] Tangent Planes and Normal Vectors
[*] Maxima and Minima of Functions of Two Variables
[*] Lagrange Multipliers
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Double Integrals
[*] Double Integrals over Nonrectangular Regions
[*] Double Integrals in Polar Coordinates
[*] Surface Area; Parametric Surfaces
[*] Triple Integrals
[*] Triple Integrals in Cylindrical and Spherical Coordinates
[*] Change of Variables in Multiple Integrals; Jacobians
[*] Centers of Gravity Using Multiple Integrals
[/LIST]
[*] Topics in Vector Calculus
[LIST]
[*] Vector Fields
[*] Line Integrals
[*] Independence of Path; Conservative Vector Fields
[*] Green’s Theorem
[*] Surface Integrals
[*] Applications of Surface Integrals; Flux
[*] The Divergence Theorem
[*] Stokes’ Theorem
[/LIST]
[*] Appendices
[LIST]
[*] Graphing Functions Using Calculators and Computer Algebra Systems
[*] Trigonometry Review
[*] Solving Polynomial Equations
[*] Selected Proofs
[/LIST]
[*] Answers to Odd-Numbered Exercises
[*] Index
[*] Web Appendices (Available for download at [url]www.wiley.com/college/anton[/url] or at [url]www.howardanton.com[/url] and in WileyPLUS.)
[LIST]
[*] Real Numbers, Intervals, and Inequalities
[*] Absolute Value
[*] Coordinate Planes, Lines, and Linear Functions
[*] Distance, Circles, and Quadratic Equations
[*] Early Parametric Equations Option
[*] Mathematical Models
[*] The Discriminant
[*] Second-Order Linear Homogeneous Differential Equations
[/LIST]
[*] Web Projects: Expanding the Calculus Horizon (online only) (Available for download at [url]www.wiley.com/college/anton[/url] or at [url]www.howardanton.com[/url] and in WileyPLUS)
[LIST]
[*] Blammo The Human Cannonball
[*] Comet Collision
[*] Hurricane Modeling
[*] Iteration and Dynamical Systems
[*] Railroad Design
[*] Robotics
[/LIST]
[/LIST]

 

[MidgetDwarf]

I do not like this book. It is in the same league as Stewart, Larson, Newer Thomas clones. The layout of the book is full of diagrams that distract from the reading process.

I instead, would recommend an older version of thomas calculus with analytic geometry, serge lang calculus series, and 2nd ed stewart calculus. If one wants a modern Exposition that does in the style of a formal theorem/proof approach then Swokoswki Calculus is a good choice.

My personal preference is Thomas Calculus with analytic geometry 3rd ed in conjunction with Stewart 2nd.

 

[Student100]

I do not like this book. It is in the same league as Stewart, Larson, Newer Thomas clones. The layout of the book is full of diagrams that distract from the reading process.

I instead, would recommend an older version of thomas calculus with analytic geometry, serge lang calculus series, and 2nd ed stewart calculus. If one wants a modern Exposition that does in the style of a formal theorem/proof approach then Swokoswki Calculus is a good choice.

My personal preference is Thomas Calculus with analytic geometry 3rd ed in conjunction with Stewart 2nd.

Disagree, the latest Anton books are still the every man's intro to calculus. The book exists at a level between more rigorous texts, and garbage books like Stewart or Larson.

 

[Kulazo]

I am not an analyst in the field of mathematics and nor I would ever hope to be, but calculus is rather beautiful. Aaaaaand Stewart (and Anton, and everybody else in the same league) don't make it seem so.

 

[Kulazo]

That's all I wanted to say, I hope you understand.

 

[Kulazo]

Do you understand?

 

[Kulazo]

Well, what am I saying...Ofcourse you do!

 

[MidgetDwarf]

Disagree, the latest Anton books are still the every man's intro to calculus. The book exists at a level between more rigorous texts, and garbage books like Stewart or Larson.

It is the same as the Stewart, Larson books you criticize. Anton is hand wavy in his proofs, some paragraphs can be hard to read, not because the material is hard, rather Anton makes arguments that can be presented much clearer.

Of the typical freshman Calculus books, excluding books such as Spivak and Apostol, Stewart is not so bad, compared to books of the same ilk. Yes, Anton falls into this category. Books that are better are Thomas Calculus with Analytical Geometry 3 ed, Simmons Calculus, and Lang Calculus. Kaisler is also a wonderful book that uses infinitesimals.

Anton is not more rigorous than Stewart, that is a fact. It is a matter of preference of how easily one can read the authors sentences.

Anton Linear Algebra book also suffers from this problem. Some chapters are clear while others are not. The book falls apart in the last sections explaining what a linear transformation is. Explanation of spanning and change of basis is also obfuscated, thanks to pages of mindless ranting, for the sake to increase page count.

Do I like Stewart? Somewhat. Is it rigorous? Not really. However, some of his explanations are rather good. Take the section on Sequences and Series and Polar Coordinates. They are very clear. Something my favorite Calculus book (applied) Thomas 3rd ed is not great at explaining.

Anton is not bad, but his writing can be unclear at times. There are many other books that are much better.

 

[Kulazo]

Anton is not bad, but his writing can be unclear at times. There are many other books that are much better.

I'm searching for a first book on calculus and yet haven't decided on anything. But I have decided my second book on Calculus will be Courant but I'm very confused on the first book. Which one would you suggest? There is only the 9th edition of Thomas' Calculus & Analytic Geometry but there is also Simmons, Stewart, Grossman, Spivak, Courant, Anton, Kline, Adams and Apostol. From these, which one do you think I should read as a first book? I'm looking for a book that isn't very annoying (no distractions) and also answers the "Why?" rather than sticking with the "How?"and explains everything finely.

 

[MidgetDwarf]

I'm searching for a first book on calculus and yet haven't decided on anything. But I have decided my second book on Calculus will be Courant but I'm very confused on the first book. Which one would you suggest? There is only the 9th edition of Thomas' Calculus & Analytic Geometry but there is also Simmons, Stewart, Grossman, Spivak, Courant, Anton, Kline, Adams and Apostol. From these, which one do you think I should read as a first book? I'm looking for a book that isn't very annoying (no distractions) and also answers the "Why?" rather than sticking with the "How?"and explains everything finely.

I would recommend Thomas Calculus 3rd ed with Analitical Geometry. I do not want to go into detail, however i will say, that the newer editions of thomas are not the same as the old ones. They are a different book with the name thomas written on them. I would also get an old edition of Stewart as a supplement.

I would prefer simmons over stewart, however, simmons atleast the 1st ed, does not cover epsilon/delta and his sequence explanation is rather lacking as a first timer.

Stewart supports Thomas 3rd very well.

Look on amazon. There are many copies.

 

[MidgetDwarf]

I don't think courant, spivak, and apostle are suitable as an introduction, unless you have used rigorous math books throughout your education.

Ie, learned from authors such as, kiselev, gelfand, and others of the russian school of thought. Or kids who do not participate in mathematical activities like math counts, olympiad, osamo?(I think this one). Or that you are extremely intelligent and not just "smart".

 

[Student100]

It is the same as the Stewart, Larson books you criticize. Anton is hand wavy in his proofs, some paragraphs can be hard to read, not because the material is hard, rather Anton makes arguments that can be presented much clearer.

As does Stewart. At least Anton puts the majority of the important proofs in an appendix, so you can actually reference them to the theorems if you want to without Google. Anyway, Stewart is dense for the sake of being dense, without actually saying anything. Anton is more concise, his style of writing more enjoyable, and I feel the book makes a better introduction when compared to the other evils. Just compare the limit formalization and tell me Stewart does a better job than Anton.

That's not the point anyway, the point is on the bell curve of students that is introductory calculus, Anton will fall higher up on the curve than other de facto introduction books. All just an opinion.

Math majors will find the book boring.

Students inadequately prepared for math will find this book, or any, challenging.

The majority of students starting out in calculus would appreciate this text.

I just wanted to counterbalance your post, should someone fret they were stuck with a completely spotty book for their intro course.

 

[MidgetDwarf]

As does Stewart. At least Anton puts the majority of the important proofs in an appendix, so you can actually reference them to the theorems if you want to without Google. Anyway, Stewart is dense for the sake of being dense, without actually saying anything. Anton is more concise, his style of writing more enjoyable, and I feel the book makes a better introduction when compared to the other evils. Just compare the limit formalization and tell me Stewart does a better job than Anton.

That's not the point anyway, the point is on the bell curve of students that is introductory calculus, Anton will fall higher up on the curve than other de facto introduction books. All just an opinion.

Math majors will find the book boring.

Students inadequately prepared for math will find this book, or any, challenging.

The majority of students starting out in calculus would appreciate this text.

I just wanted to counterbalance your post, should someone fret they were stuck with a completely spotty book for their intro course.

Saying someones writing style is more enjoyable is subjective. I'll give a personal example. I enjoy reading James Joyce, my physics professor hates it. She enjoys Edgar Allan Poe, I think Poe is boring. Both are great writers. This example i gave is abstract, let me explain myself more concisely.

Anton is not more concise, he is extremely verbose. I have read 2 of his text and frankly I did not care to much of them. He is very sloppy with his proofs, his linear algebra text being the main culprit.

Finding a writers writing style is subjective as illustrated by my example.

Appendix of proofs? Laugh. Students should write every proof in order as it appears in a special notebook. For easy review and mastery.

If you want to talk about concise, thomas 3rd ed and Lang have Anton beat by miles. You are not going to find readable and explanatory proofs in a beginner calculus, that are better than Thomas, in any other book of this level. That is a fact.

Thomas in this edition even proves the most trivial theorems in a concise and well explained manner. Everything is proved before being introduced.

Stewart, Anton, and the other generic calculus books are much the same. Do you want to see an example of a window or a man shadow while walking for a related rates problems? This is usually the big difference. If you think Anton is a good book, then I hate to say it, you are not familiar with good math text.

Kaisler Calculus is even better than both stewart and anton. I like to bash stewart as the next guy does, but saying stewart is rubbish is going a bit far. It serves its purpose in introducing students to calculus. It is an introduction, the same as Anton. It should be followed by a more rigorous book i.e. courant, spivak, apostle, or even this russian book(forgot the name/dover republished).

 

[bacte2013]

Saying someones writing style is more enjoyable is subjective. I'll give a personal example. I enjoy reading James Joyce, my physics professor hates it. She enjoys Edgar Allan Poe, I think Poe is boring. Both are great writers. This example i gave is abstract, let me explain myself more concisely.

Anton is not more concise, he is extremely verbose. I have read 2 of his text and frankly I did not care to much of them. He is very sloppy with his proofs, his linear algebra text being the main culprit.

Finding a writers writing style is subjective as illustrated by my example.

Appendix of proofs? Laugh. Students should write every proof in order as it appears in a special notebook. For easy review and mastery.

If you want to talk about concise, thomas 3rd ed and Lang have Anton beat by miles. You are not going to find readable and explanatory proofs in a beginner calculus, that are better than Thomas, in any other book of this level. That is a fact.

Thomas in this edition even proves the most trivial theorems in a concise and well explained manner. Everything is proved before being introduced.

Stewart, Anton, and the other generic calculus books are much the same. Do you want to see an example of a window or a man shadow while walking for a related rates problems? This is usually the big difference. If you think Anton is a good book, then I hate to say it, you are not familiar with good math text.

Kaisler Calculus is even better than both stewart and anton. I like to bash stewart as the next guy does, but saying stewart is rubbish is going a bit far. It serves its purpose in introducing students to calculus. It is an introduction, the same as Anton. It should be followed by a more rigorous book i.e. courant, spivak, apostle, or even this russian book(forgot the name/dover republished).

You seem to read numerous calculus textbooks? Just curious, did you read all of them front to back? Have you read "First Course in Calculus" by Serge Lang? I only read Lang, Stewart, Thomas (3rd), and Simmons, and I think Lang is the best in terms of clarity and conciseness.

 

[MidgetDwarf]

You seem to read numerous calculus textbooks? Just curious, did you read all of them front to back? Have you read "First Course in Calculus" by Serge Lang? I only read Lang, Stewart, Thomas (3rd), and Simmons, and I think Lang is the best in terms of clarity and conciseness.

Yes, front in back. I tend to 2 study with two books. Everytime I need to review Calculus, I used Thomas 3rd ed (my favorite beginner cal book), and different one. I liked Lang, yes it is very clear. I still prefer Thomas 3rd.

I find that the introduction to series and sequences is explained the best in Stewart and Simmons a close second.

 

[micromass]

I don't get what the point is of reading multiple calculus books. You should just move to analysis as quickly as you can. At most, an intro calculus book and then Spivak/Apostol/Nitecki is a good option. But reading Thomas AND Simmons AND Stewart AND Lang seems like a huge waste of time.

 

[bacte2013]

I don't get what the point is of reading multiple calculus books. You should just move to analysis as quickly as you can. At most, an intro calculus book and then Spivak/Apostol/Nitecki is a good option. But reading Thomas AND Simmons AND Stewart AND Lang seems like a huge waste of time.

I jump straight into Apostol's Mathematical Analysis and Pugh's Real Mathematical Analysis right after studying the Lang and Halmos' Naive Set Theory. To be honest, I think the "intermediate" books like Apostol (Calculus), Spivak, and Nitecki are not even necessary. Also regarding to those books, I found Mattuck's Introduction to Analysis much better them, both as self-studying and as supplements to books like Pugh and Rudin.

 

[micromass]

I jump straight into Apostol's Mathematical Analysis and Pugh's Real Mathematical Analysis right after studying the Lang and Halmos' Naive Set Theory. To be honest, I think the "intermediate" books like Apostol (Calculus), Spivak, and Nitecki are not even necessary. Also regarding to those books, I found Mattuck's Introduction to Analysis much better them.

The intermediate books are often much better written than the actual analysis books. Apostol's analysis is a kind of book that contains a lot of good info, but is really badly written. I think a good student would get more from an intermediate book than jumping straight to analysis.

 

[bacte2013]

The intermediate books are often much better written than the actual analysis books. Apostol's analysis is a kind of book that contains a lot of good info, but is really badly written. I think a good student would get more from an intermediate book than jumping straight to analysis.

Could you inform me why Apostol's Mathematical Analysis is a book with bad exposition? Also could you share me your idea why the intermediate books are better than the real analysis books? I did not completely read books like Spivak and Nitecki, but I thought the real analysis books like Tao, Mattuck, Garling, and Zorich are very good as a first introduction to the analysis and help students to learn without much difficulty.

 

[MidgetDwarf]

I don't get what the point is of reading multiple calculus books. You should just move to analysis as quickly as you can. At most, an intro calculus book and then Spivak/Apostol/Nitecki is a good option. But reading Thomas AND Simmons AND Stewart AND Lang seems like a huge waste of time.

Correct. However, I am not lucky enough like other people. I do not go to a University, I go to a community college were the standards are low. I do not watch television or go out much for that matter. Ofcourse, the time would have been better spent going to analysis. I barely gained somewhat of mathematical maturity after taking a Linear Algebra course. I am not mathematically inclined. You can say I have no talent, besides hard work.

Currently, I am reading: A proof book, Coddington Intro ODE, and a Courant: What is Mathematics.

 

[micromass]

I barely gained somewhat of mathematical maturity after taking a Linear Algebra course.

Of course not. I know you're a big fan of Thomas, but books like that do not help somebody gain mathematical maturity. Proof books, Coddington, etc. do not help somebody gain mathematical maturity. It is the practice of going through challenging rigorous texts where somebody gains mathematical maturity. Now, I'm not a big fan of texts like Rudin where a lot of the motivation is missing, but somebody struggling REALLY hard to get through Rudin will have gained so much more maturity as somebody else having a fun but easy time with Coddington.

I am not mathematically inclined. You can say I have no talent, besides hard work.

Haha. Talent only gets you so far. You obviously have enough talent to make it as a mathematician. I'll say this: I don't think anybody is mathematically inclined. In my opinion, mathematics is very unnatural. Leave a child on his own, and he won't develop much mathematics. Children are forced in elementary school to do a lot of mathematics hoping that something will stick (and for most, it doesn't). Sure, you are not mathematically inclined, and neither am I. Mathematical knowledge comes from passion, hard work and challenging yourself. You clearly have all of those, so that should be enough to make it.

Do yourself a favor and start reading an analysis text. If you want or need it, I will help you get through it. I promise you, after a lot of sweating, cursing and suffering, you will love yourself for having done it. Feel free to PM me if you think you're up to the challenge.

 

[mathwonk]

I am afraid that after the last 40 or 50 years, Anton’s book has approached the fate of Thomas’s book. I.e. even if one once liked it (and I seem to recall I did not like it even in 1970 or so), there are now three authors’ names on the front. I suspect that Dr. Anton, who got his PhD in 1968, possibly after some absence from school, is now in his 70’s or 80’s and has little to do with the writing duties.

This book now costs 10 times more on Amazon than it is worth, starts out with a lengthy section on precalculus, is loaded with extra material to make it very laborious to wade through, and at crucial junctures, like the existence of extreme values, says things like “this proof is too difficult to include here”.

To test this last remark, I offer the following argument for your evaluation.

Assume f is continuous on the interval [0,1] and real valued. We claim there is a point of that interval where the value of f is at least as large as its value at every other point; i.e. there is some point c with 0 ≤ c ≤ 1, such that for every other point x with 0 ≤ x ≤ 1 we have f(x) ≤ f(c).

First let’s see where to look for c. If the theorem were true, then as a decimal, the point c would begin with some digit between 0 and 9, i.e. c would be in one of the 10 intervals [0,.1], [.1,.2],...,[.9,1], and hence the maximum of f on that interval would be at least as large as its maximum on every other such interval. So choose one of these intervals on which f is maximized. I.e. rank the intervals so that one interval is ≥ another, iff for every point of the second interval, there is some point of the first where f has at least as large a value. (I.e., rank each subinterval by the size of the smallest number larger than all the values of f there, possibly infinity.)

This the key statement, and is presumably why the proof is considered “hard”.

Since this ranking involves comparing the values of f at infinitely many points, it cannot be carried out in practice, but must be either true or false for any given pair of intervals. Since only finitely many subintervals exist, one (or more) of them is ranked highest in this way.

Thus if our point c exists, it must lie in the highest ranked of these 10 intervals. E.g. if the highest ranked interval is [.3,.4] then the decimal expansion of c would begin as .3. Now continue subdividing the interval into hundredths, thousandths,...

We obtain a sequence of intervals and finite expansions such that at every stage, e.g. at .341987265, we know that the values f assumes between .341987265 and .341987266, equal or exceed its values everywhere else in [0,1].

Now this sequence, if imagined continuing to infinity, defines a single decimal c = .341987265... This point c belongs to every one of the finite intervals considered on which f is maximal. We claim f(c) ≥ f(x) for every other x. We prove this by contradiction. I.e. if there were some x for which f(x) > f(c), then at some stage of our construction, c and x would belong to different subintervals, and by definition of continuity, if those subintervals are small enough, then at ALL points of the interval containing c, f would have value less than f(x). ( Review the definition of continuity if need be, to see why this is so.)

But by our construction, c always belongs to a subinterval where the values of f are maximal, so this is a contradiction.

Is this proof really too difficult to include in an 800 page tome of stuff supposedly explaining calculus? Even if understanding this proof takes an hour, wouldn’t that time be well spent? Most of it e.g. would probably be used reviewing the meaning of “continuous”, or the use of quantified statements. I learned this argument from a colleague at UGA, Danny Krashen, and he apparently uses it successfully in his elementary non honors calculus course.

 

[micromass]

Mathwonk, that is a beautiful argument. I think it would be very suitable to prove this in a first analysis class, and then to use this proof to introduce the notion of completeness of $\mathbb{R}$.

 

[MidgetDwarf]

Of course not. I know you're a big fan of Thomas, but books like that do not help somebody gain mathematical maturity. Proof books, Coddington, etc. do not help somebody gain mathematical maturity. It is the practice of going through challenging rigorous texts where somebody gains mathematical maturity. Now, I'm not a big fan of texts like Rudin where a lot of the motivation is missing, but somebody struggling REALLY hard to get through Rudin will have gained so much more maturity as somebody else having a fun but easy time with Coddington.
Haha. Talent only gets you so far. You obviously have enough talent to make it as a mathematician. I'll say this: I don't think anybody is mathematically inclined. In my opinion, mathematics is very unnatural. Leave a child on his own, and he won't develop much mathematics. Children are forced in elementary school to do a lot of mathematics hoping that something will stick (and for most, it doesn't). Sure, you are not mathematically inclined, and neither am I. Mathematical knowledge comes from passion, hard work and challenging yourself. You clearly have all of those, so that should be enough to make it.

Do yourself a favor and start reading an analysis text. If you want or need it, I will help you get through it. I promise you, after a lot of sweating, cursing and suffering, you will love yourself for having done it. Feel free to PM me if you think you're up to the challenge.

Thanks, Micro. I will take you up on you're offer. It has to wait till December however. Just need to take 4 classes and I can finally transfer to university.
I'm going to take Calculus 3, Intro ODE, E n M honors using Purcell, and an English class. Hopefully you're offer is still on the table. The physics class seems it will be the hardest.

What Analysis book do you recommend? Rudin? Apostol? Or a better alternative. I have not taken a class were proving was required, except for a linear algebra class. This linear algebra class was a mixture of theory and computation. However, I try to follow every argument and replicate the proofs on my own.

 

[mathwonk]

Thank you micromass. MidgetDwarf, in line with micromass's recommendation for beginning analysis, may I suggest you try to understand that proof in #22 above? If you do so, then try this exercise:

A similar proof can obviously be given for the other basic topological result in calculus, the intermediate value theorem. Assume again f is continuous on [0,1] and that f(0) < 0 while f(1) > 0. We claim there is some point c in (0,1) where f(c) = 0. Again subdivide the interval into subintervals of length 1/10, and look at the values of f on the subdivision points in the sequence {.1, .2, ..., .9, 1}. Find the first point where f has a positive value and choose that as the right end point of your first subinterval. E.g. if the first point of this sequence where f is positive is .5, we take the subinterval [.4, .5], and let our decimal expansion begin with .4. Then f(.4) < 0 and f(.5) > 0, so we look for c in this subinterval.

Continuing in this way we imagine an infinite decimal expansion c = .4925711836... such that f is negative on every finite piece of this decimal, and positive on the finite decimal whose last digit is one larger. E.g. f(.4925711836) < 0 and f(.4925711837) > 0. If at any point we find f = 0 at some endpoint of one of these subintervals, of course we stop since the theorem is proved. We claim that f(c) = 0.

Exercise: use continuity to find a contradiction to the assumption f(c) < 0 and also to f(c) > 0.

All other theorems in elementary calculus follow from these two results, and the basic definitions, so they are worth knowing.

 

[micromass]

Thanks, Micro. I will take you up on you're offer. It has to wait till December however. Just need to take 4 classes and I can finally transfer to university.

Of course.

What Analysis book do you recommend? Rudin? Apostol? Or a better alternative. I have not taken a class were proving was required, except for a linear algebra class. This linear algebra class was a mixture of theory and computation. However, I try to follow every argument and replicate the proofs on my own.

So first about proofs. Proving is a skill worth having, and it is actually surprisingly easy to learn, but only if you learn it the right way. The right way is absolutely not by doing proof books (which I think are pretty useless). It is getting into a field (discrete math or abstract algebra are the easiest candidates) and start proving things yourself. The crucial thing is to have somebody close to you who can criticize every part of your proof. And then I mean: truly destroy your proof. This will happen several times, but after a while you'll get the hang of it and you'll be able to do proofs pretty well. The rest is experience.

Proof books do more harm than good too in my opinion. Why do I say this, because every time somebody works through a proof book, I can tell in their style (and that's not a good sign). Proof books might show you the right techniques, but it does not show how to actually write down your proofs. Writing down proofs is as important as finding the proof. And writing down a good proof can be harder than finding the proof. Again, somebody who criticizes the style of your proofs is essential here.

Now for analysis. I can't recommend a specific book without knowing you better. There is a huge number of analysis books out there, and they can be very different. Some people like Rudin, others (like me) don't. It's not that Rudin is a bad book, it's that it doesn't fit their style. So if you want to find a good analysis book for you, then you'll need to examine your style better. For example, Rudin is really bare bones, it offers the results and very slick proofs but nothing more. There are other books which are more verbose and which offer more motivation. Then there are books which take the historical approach to analysis. There are books which focus more on the foundational issues. There are books which are more philosophical. There are books which are more geometrical. Etc. If you want to find an analysis book that suits you well, you'll need to find out what you are looking for in a book (aside from it teaching you a decent amount of analysis).

 

[bacte2013]

Of course.
So first about proofs. Proving is a skill worth having, and it is actually surprisingly easy to learn, but only if you learn it the right way. The right way is absolutely not by doing proof books (which I think are pretty useless). It is getting into a field (discrete math or abstract algebra are the easiest candidates) and start proving things yourself. The crucial thing is to have somebody close to you who can criticize every part of your proof. And then I mean: truly destroy your proof. This will happen several times, but after a while you'll get the hang of it and you'll be able to do proofs pretty well. The rest is experience.

Proof books do more harm than good too in my opinion. Why do I say this, because every time somebody works through a proof book, I can tell in their style (and that's not a good sign). Proof books might show you the right techniques, but it does not show how to actually write down your proofs. Writing down proofs is as important as finding the proof. And writing down a good proof can be harder than finding the proof. Again, somebody who criticizes the style of your proofs is essential here.

Now for analysis. I can't recommend a specific book without knowing you better. There is a huge number of analysis books out there, and they can be very different. Some people like Rudin, others (like me) don't. It's not that Rudin is a bad book, it's that it doesn't fit their style. So if you want to find a good analysis book for you, then you'll need to examine your style better. For example, Rudin is really bare bones, it offers the results and very slick proofs but nothing more. There are other books which are more verbose and which offer more motivation. Then there are books which take the historical approach to analysis. There are books which focus more on the foundational issues. There are books which are more philosophical. There are books which are more geometrical. Etc. If you want to find an analysis book that suits you well, you'll need to find out what you are looking for in a book (aside from it teaching you a decent amount of analysis).

I agree with Professor micromass. I actually acquired the proof skill through a "proof-teaching book", which was a waste of time because I learned the general guideline for writing various proofs, but I failed to produce my own proofs that have different algorithmic flavor than the proof book. I think Terrence Tao's Analysis I or Bloch's Real Numbers and Real Analysis are two great books to learn and sharpen the proof skills.

As for the analysis books, I think Apostol, Pugh, and Terrence Tao are excellent for learning the introductory analysis. I still do not understand the claim Professor micromass made about Apostol's Mathematical Analysis. Although the proofs in that book are not elegant as ones in Rudin and the problem sets are very easy, his exposition is very pedagogical and clear to understand.

 

[verty]

How about this proof for the extreme value theorem?

Let $N = \{c \in [0,1): \forall x \in [0,c], \; \exists y \in (c,1], \; f(x) \leq f(y)\}$.

1. N is an interval $[0, a)$ or $[0, a]$.
1.1. Suppose N is a closed interval, then there is an f(y) large enough beyond $a$ but not beyond any $a + \epsilon$, absurd.
1.2. Suppose N is an open interval, then there isn't an f(y) large enough beyond $a$ but there is beyond any $a - \epsilon$. But then $f(a)$ must be a maximum.

A similar comprehension will show there is a minimum. QED.

This proof actually fails but it illustrates an interesting point, it is very hard to tell where it fails (try). Therefore I think there is a good argument that rigor, an example being Mathwonk's proof above, should not appear in a first calculus course. It isn't that a proof is plausible, it's that there are plausible arguments that don't commute. And it should be left to a proof course so that students can tell the difference.