The Hitchhiker's Guide to Calculus (AMS/MAA Classroom Resource Materials, 57) Reprint Edition

My 12 year old loves mathematics with a passion and wants to know EVERYTHING his 12 year old brain can handle :). This includes calculus, trigonometry, etc. I had rigorous education in math thanks to my engineering degree but I had to "relearn" in order to make it relatable to him as well as to refresh my own basics. This book is a masterpiece in my opinion. It truly does take the reader on a journey and each chapter builds very nicely on the previous one. Bear in mind, it assumes that you have a foundation in algebra, numbers and the "language" of math, so definitely not for the totally uninitiated. But it really rewards you with an insight that is often missing when you have to study math for a grade :). It has helped me explain concepts like functions, limits, etc. to my son. Ultimately math is a tool and I want him to learn it well but with an awareness of its utility beyond just solving math problems.

Michael Spivak’s little book is an extremely pleasant tour on the main ideas of Differential and Integral Calculus, reading as a short tale that is paralleled to a trip to Europe as a hitchhiker. In essence, it presents the concepts of derivatives and integrals. The book assumes that you somewhat knows how to calculate and focus instead on the ideas, their interrelations and theoretical significance. How this book relates to Spivak’s larger book “Calculus” or other university books on the same subject? I believe that he wrote it for the pure fun of fiddling with the ideas, and at the same time as a panorama of the basics of the subject-matter. I read this book years after my Calculus course and was already well conversant with the author’s book on Calculus, so I saw many connections, because everything treated in this little book is presented in great rigour in the other, larger book, but with many details that need to be filled in. On the other side, The Hitchhiker’s Guide can be read in one or two hours, perhaps on a train or plane trip... you can refresh your mind and remember what was so pleasant when you learned Calculus. Beware that even in so short and simple work, Spivak ir pretty rigorous, so it is not exactly for the laymen, at least for a supposed layman that doesn’t want to think in a mathematical way. I enjoyed it a lot, indeed read it two times in less than a month.

In The Hitchhiker's Guide to Calculus, a brief supplement to a calculus text, Michael Spivak clearly explains some of the key concepts in differential and integral calculus. The book will be particularly useful to students in computationally oriented courses who wish to gain an understanding of the concepts that justify their computations and to instructors looking for ways to explain those concepts to their students. Spivak focuses on the concepts, assuming that the reader will get plenty of practice doing computations in his or her calculus course, making the book most useful to students taking calculus. To fully benefit from the book (and a calculus course), readers should have a firm understanding of algebra and trigonometry.After a brief review of how to find the slope-intercept and point-slope forms of the equation of a line, Spivak explains that a curve such as y = x^2 is approximately linear in a small enough neighborhood of a point, a point he illustrates by taking a magnifying lens to a point on the curve. Spivak demonstrates that the slope of the curve y = f(x) at a point (x, f(x)) can be approximated by the slope of the line through the points (x, f(x)) and (x + h, f(x + h)) for values of h whose absolute value is small and argues that the approximation can be improved by making |h| smaller. He defines the derivative as the limit as h approaches zero of the difference quotient [f(x + h) - f(x)]/[(x + h) - x] = [f(x + h) - f(x)]/h that comes from calculating these slopes. He calculates the derivative for the functions f(x) = x^2, f(x) = x^3, and f(x) = 1/x. Spivak then demonstrates why a function is defined to be continuous at a point if it is equal to its limit at the point. He also shows how to use the derivative to find an equation of the tangent line at a point and to find the instantaneous velocity of an object, which he points out was the original impetus for defining the derivative of a function. Up to this point of the text, the derivatives Spivak has calculated can be solved through algebraic manipulation. Next he finds the derivative of the sine function, which requires the reader to understand a geometric argument and analytic trigonometry. Spivak uses the derivative of the sine to find the derivative of the cosine. He also shows that for a differentiable function, the relative maxima and minima occur when the derivative is zero and uses this observation to find the minimum surface area of a cylinder with a fixed volume. Spivak justifies Rolle's Theorem and uses it to prove the Mean-Value Theorem. He then shows that if the derivative of a function is positive on an interval, it is increasing on that interval; that if the derivative is negative on an interval, it is decreasing on that interval; and that if the derivative is zero on an interval, then the function is constant on the interval.Spivak begins his discussion of integral calculus by showing how pi can be approximated by finding the area of a regular polygon that circumscribes a circle. From there, he shows that a similar argument involving rectangles can be used to approximate the area under the curve. He uses this argument to find the area under the curve y = x^2 over an interval, then generalizes the technique he uses in order to define the definite integral. Spivak demonstrates that the derivative of the (signed) area found by integration is the function itself, thereby justifying the Fundamental Theorem of Calculus. He also shows how the Fundamental Theorem of Calculus enables you to evaluate a definite integral. Finally, he shows how the definite integral can be used to find the mass of an object and to calculate volume, arc length, and surface area.There are a few exercises in the text, including a joke about the tendency of mathematicians to reduce problems to ones that they already know how to solve (hardly, the only instance of humor in the text). Solutions are provided for some of the exercises. The results of others are discussed later in the text.The book is not without its flaws. In discussing how to find the slope of a line, Spivak refers to the coordinates (x, y) of a point in the plane as the horizontal and vertical distances of the point from the origin (page 4). He should have said displacement since distances cannot be negative while the coordinates can be. In calculating the average speed of an object in free fall (on page 56), there is an error in which .01^2 is given as .001 rather than .0001. Also, he incorrectly states that he is calculating the weight of an object when he is actually calculating its mass (pages 114-115).My main regret is that this book is so brief. Spivak does a terrific job of explaining the concepts. That said, I think the reader would have benefited if Spivak had opted to illustrate more of the formulas he derives with examples that illustrate how to apply the formulas or included more topics. Of course, a well-prepared student who is willing to work hard and attempt difficult problems could benefit from working through Spivak's text Calculus, 4th edition, which includes the material Spivak omits here.

This book was poorly written and barely got into what calculus is about. It was hard to understand and the examples were crappy. I highly recommend using another book if you are trying to learn calculus or are trying to get a refresher.