Calculus Without Tears
CWT - College Edition (CE) is all three volumes of Calculus Without Tears combined into one volume. Chapter 1 CE corresponds to CWT - Vol. 1, Chapter 2 CE corresponds ot CWT - Vol 2, and each remaining chapter in CWT - CE is (almost) identical to a chapter in CWT - Vol. 3. Thus the approach is exactly the same, but the very slow development and repetition of the first two volumes has been eliminated for students who already have had some exposure to calculus.
This is the book to buy if you've had some calculus, and want to really understand the motivations and proofs of the theorems of analytic calculus. However, now I'm recommending that you go through Mathematical Modeling and Computational Calculus first to really see how calculus is used to model and solve physics and engineering problems.
The Elephant in the Room
The academic community has long recognized that calculus education needs to be improved. In the 1990s the NSF spent millions funding various 'calculus reform' programs that came to naught. Now they are at it again, and the NSF is spending millions on various programs to improve STEM (Science, Technology, Engineering, and Math) education, and still they haven't made any significant changes to the current curriculum.
The fact is if you're now taking calculus in high school or college you are not learning how calculus is used to analyze and study physical phenomena (which starts with differential equations), and, you are not learning the type of calculus used ninety-nine percent of the time to solve scientific and engineering problems (that is computational calculus).
The reasons for this are not hard to understand, differential equations aren't taught because they are too difficult to solve analytically. Computational calculus is not taught because fifty years ago there were no computers and computational calculus requires too many calculations to be done by hand, and calculus is taught today the same way it was taught fifty years ago.
The solution is simple, teach computational calculus first, as it is almost trivially easy. And, teach differential equations from the start, as they are trivially easy to solve using computational calculus. Calculus immediately becomes easy to understand, the calculus taught is the calculus used to solve most scientific and engineering problems, and the curriculum is motivated by solving representative problems from physics and engineering. That is the way Calculus Without Tears does it.
CWT Gets the Basics Right
The basis of calculus is, believe it or not, the formula distance equals velocity times time. In the context of constant velocity motion, differentiation (calculating velocity) is given by the formula velocity = distance / time. Integration (calculating area or distance) and solving differential equations are accomplished using the formula distance = velocity * time. The Fundamental Theorem of Calculus for constant velocity motion is the formula distance = velocity * time ! The calculus of constant velocity motion is trivially easy, and, it is easily extended to cover non-constant velocity motion. CWT is the only calculus book taking this approach.
CWT is Motivated with Examples from Mechanics and Circuit Theory
The basic paradigm in physics and engineering analysis starts with a differential equation model of a process being studied, followed by solving the differential equations. The secondary school math curriculum is unmotivated because differential equations are never covered and as a result the student cannot formulate, much less solve, the physics problems that should motivate calculus and the entire math curriculum. CWT begins formulating and solving physics problems with differential equations in the first chapter of this volume and they are an integral and motivating part of the rest of the book. Differential equations can be terrifically difficult to solve analytically, that's why the standard curriculum avoids them. But, they are easy to solve computationally, and that's the way CWT does it.
CWT is Hands On and Corresponds to Real Engineering Calculus
Functions are to calculus as numbers are to arithmetic, and it is very helpful to be able to manipulate functions easily. CWT facilitates this by introducing MATLAB/FREEMAT and using it as an adjunct throughout the text. Real world calculus is about solving differential equations, and, usually they are solved numerically rather than analytically. It's so easy to solve differential equations numerically it's almost embarassing. You don't need to wait until your third or fourth college course. The secret is that arcane formula emit * yticolev = ecnatsid (read backwards).
CWT is Easy
Because CWT gets the basics right, it is easy to develop the princples of calculus. Calculus proofs in the standard texts are, for mathematicians, very clever, concise, and elegant, Unfortunately they are for students nearly incomprehensible. For example, check your calculus text, or google, or check Wikipedia, for a proof of the product rule, or Taylor's theorem for polynomial approximation, or the Fundamental Theorem of Calculus. In each case the proof you find will be for all practical purposes impossible to understand; but, as the song goes, it ain't necessarily so, the ideas underlying each of these principles are easy to the point of being almost obvious, as is demonstrated in CWT. The CWT approaches to the product rule, Taylor's theorem, and the FTOC are given in some detail in the synopsis of Vol. 3 linked to the left, just so that you can make this comparison.
CWT Puts the Big Picture in Focus
With CWT you will understand the purpose of calculus, which is to solve physics problems which are expressed as differential equations; amazingly that is not part of standard calculus courses. You will understand how easy it is to solve differential equations numerically, that too is not part of the standard courses. And, you will feel comfortable with calculus instead of, as is usually the case with the standard courses, feeling that calculus remains a huge mystery even after you got an 'A' in the course.
Final Thoughts - A New Perspective on Calculus
What's going on here, why is so much effort required to navigate the current calculus curriculum, and why is there so little reward? That is, why are so few interesting and important problems presented and solved in calculus courses? The culumiation of calculus is typically a course on differential equations that presents ad hoc techniques you'll never use to solve differential equations you'll never see again. Where is the payoff? The reality is that differential equation models of any but the most simple real systems cannot be solved, at all, analytically, so the promised land remains in the distance. There are two ways to realize the potential of calculus, the first is the classical study of linear systems, where functions are decomposed into sine and cosines, and the Laplace transform is used to analyze system performance. Fourier/Laplace theory is advanced math, and even then many real systems are not linear. The second way to realize the potential of calulus is to use computational calculus, as presented in the CWT books (and on this web site, e.g. in the airplane simulation), to analyze complex systems.