if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.” From that, considering that I didn't know much of the basic Algebra he was talking about which I was suppose to know before reading, this book can clearly explain and help us understand Calculus. (I think that deserves a four star.) PS: A kind reader has created an animated powerpoint slideshow that helps present this idea more visually (best viewed in PowerPoint, due to the animations). Thanks!)

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I think anyone can appreciate the core ideas of calculus. We don’t need to be writers to enjoy Shakespeare. To start out, see how far you can get by plugging in the roots of equations. Begin by getting a common denominator on the right side of the equation:Did you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because “Sure, it works in practice. But does it work in theory?”. Your first step in any problem that involves partial fractions is to recognize which case you’re dealing with so that you can solve the problem. One case where you can use partial fractions is with repeated quadratic factors.

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Setting up a system of equations is an alternative method for finding the value of unknowns when you’re working with partial fractions. It’s not as simple as plugging in the roots of factors, but it’s your only option when the root of a quadratic factor is imaginary. First, you’ve got to split up the integrand into a u and a dv so that it fits the formula. For this problem, choose ln(x) to be your u. Then, everything else is the dv, namely

Conclusion

A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but the intuition of growth is how it was discovered. The natural log can be seen as an integral, or the time needed to grow. Which explanations help beginners more? Because we expect it. Expectations play a huge part in what’s possible. So expect that calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what’s happening, and expand our brain along the way.

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Imagine teaching art like this: Kids, no fingerpainting in kindergarten. Instead, let’s study paint chemistry, the physics of light, and the anatomy of the eye. After 12 years of this, if the kids (now teenagers) don’t hate art already, they may begin to start coloring on their own. After all, they have the “rigorous, testable” fundamentals to start appreciating art. Right? A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell, or a wine bottle. This article, and the video, show you how to find its area. To solve the second integral, complete the square in the denominator: Divide the b term (6) by 2 and square it, and then represent the C term (13) as the sum of this and whatever’s left: Generally speaking, when a linear factor is raised to the nth power, add n partial fractions. For example, suppose that you want to integrate the following expression:

We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. Eat it.”

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What’s the surface area of a representative band? Well, if you cut the band and unroll it, you get sort of a long, narrow rectangle whose area, of course, is length times width. By the way, in the above explanation, you might be wondering why the width of the rectangular band is Here’s a great mnemonic device for how to choose the u (again, once you’ve selected your u, everything else is automatically the dv.For each quadratic factor in the denominator that’s raised to the third power, add three partial fractions in the following form: