By Lerner M. E., Repin O. A.

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Whenever you talk about infinity, you’re always on shaky ground. What would it mean to zoom in forever or an infinite number of times? You can’t do it; you’d never get there. We can imagine — sort of — what it’s like to zoom in forever, but there’s something a bit fishy about the idea — and thus the qualifications. com/extras/calculus.  . ✓ Algebra review: Richard Feynman, the great 20th century physicist, said (tongue-in-cheek) that calculus was the language that God spoke. Well, I don’t know about that, but I do know that algebra is the language of calculus.

So knowing how they work and understanding the fundamental connection between roots and powers is essential. And, of course, that’s what I’m about to tell you. Roots rule — make that, root rules √ √ Any root can be converted into a power, for example, 3 x = x 1∕3, x = x 1∕2, and √ 4 3 x = x 3∕4. So, if you get a problem with roots in it, you can just convert each root into a power and use the power rules instead to solve the problem (this is a very useful technique). Because you have this option, the following root rules are less important than the power rules, but you really should know them anyway: √ √ ✓ 0 = 0 and 1 = 1 But you knew that, right?

Here’s how: 9x 4 − 25 = (3x 2 )2 − (5)2 Now, because (this)2 − (that)2 = (this − that)(this + that), you can factor the problem: (3x 2 )2 − (5)2 = (3x 2 − 5)(3x 2 + 5) A difference of squares, a2 − b2, can be factored, but a sum of squares, cannot be factored. In other words, a2 + b2, like the numbers 7 and 13, is prime — you can’t break it up. , Sum and difference of cubes You might also want to memorize the factor rules for the sum and difference of cubes: a3 + b3 = (a + b)(a2 − ab + b2 ) a3 − b3 = (a − b)(a2 + ab + b2 ) Trying some trinomial factoring Remember regular old trinomial factoring from your algebra days?

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A boundary value problem for mixed-type equations in domains with multiply connected hyperbolicity subdomains by Lerner M. E., Repin O. A.


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