
This ingenious, user-friendly introduction to calculus recounts adventures that take place in the mythical land of Carmorra. As the story's narrator meets Carmorra's citizens, they confront a series of practical problems, and their method of working out solutions employs calculus. As readers follow their adventures, they are introduced to calculating derivatives; finding maximum and minimum points with derivatives; determining derivatives of trigonometric functions; discovering and using integrals; working with logarithms, exponential functions, vectors, and Taylor series; using differential equations; and much more. This introduction to calculus presents exercises at the end of each chapter and gives their answers at the back of the book. Step-by-step worksheets with answers are included in the chapters. Computers are used for numerical integration and other tasks. The book also includes graphs, charts, and whimsical line illustrations. Barron's Easy Way books introduce a variety of academic and practical subjects to students and general readers in clear, understandable language. Ideal as self-teaching manuals for readers interested in learning a new career-related skill, these books have also found widespread classroom use as supplementary texts and brush-up test-preparation guides. Subject heads and key phrases that need to be learned are set in a second color.
https://www.worldofbooks.com/products/calculus-the-easy-way-...
https://www.thriftbooks.com/w/calculus-the-easy-way-easy-way...
1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract.
2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid.
3. Show that the technique also works on other different problems.
4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable.
5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity).
This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right?
This is infinitely better than the all too common higher level math textbook approach of:
1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance.
2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now).
3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.
Algebra the Easy Way - https://www.goodreads.com/book/show/6781.Algebra_the_Easy_Wa...
Trigonometry the Easy Way - https://www.goodreads.com/book/show/2077099.Trigonometry_the...
Calculus the Easy Way - https://www.goodreads.com/book/show/3074478-calculus-the-eas...
I read these in 5th grade or so and learned most of middle school and high school math just from those.
They're written as a fantasy novel in which the characters discover the relevant math concepts as part of the story. I know, you're thinking that's a silly gimmick and it can't possibly be any good, but it was actually done quite well. At least until the later parts of Calculus where the situations have to get pretty contrived.
Highly recommended, at least for precocious kids who like math and fantasy novels.
(Note: the other books in the "Easy Way" series do not follow the same style, and are just ordinary textbooks.)
Also, in a completely different direction, I haven't seen anyone mention Feynman yet, and that will definitely encourage a broader view of mathematics and science.
Or, to go another angle, you might consider things like "Thinking, Fast and Slow".
I had a similar experience with "Calculus the Easy Way", and its predecessors "Algebra the Easy Way" and "Trigonometry the Easy Way", all of which I'd highly recommend. They teach the topics as though being invented for the first time by a kingdom finding themselves in need of them.