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31 sie 2016 · The following calculus notes are sorted by chapter and topic. They are in the form of PDF documents that can be printed or annotated by students for educational purposes. If you instead prefer an interactive slideshow, please click here. Formula Sheet: Calculus BC. trigonometric identities, unit circle; derivative rules, derivatives of common ...
Case 1. x-3>0 [This is equivalent to x>3.] Multiplying the given inequality (1) by the positive quantity x-3 preserves the inequality: * + 4<2;t-6, 4<x-6 [Subtract jr.], 10<x [Add 6.] Thus, when x>3, the given inequality holds when and only when x>10. Case 2. x-3<0 [This is equivalent to x<3]. Multiplying the given inequality (1) by the ...
Our last month will be combining the multivariate calculus with vector calculus and this culminates in several important theorems which tie all of Calculus III topics together into several beautiful and useful packages!
Let x, y and z be distinct points of S such that z 62xy. Then f x; y; z g is a noncollinear set. Proof. Suppose that L is a line containing the given three points. Since x and y are distinct, by (I-2) we know that L = xy. By our assumption on L it follows that z 2 L; however, this contradicts the hypothesis z 62xy.
Using any three non-collinear points. Just like any two non-collinear points determine a unique line, any three non-collinear points determine a unique plane. This method requires the use of the cross product and the previous technique. Suppose that we know the points A, B, and C all lie in a plane.
Not all fractions can be represented as decimal fractions. For instance, expanding 1 3 into a decimal fraction leads to an unending decimal fraction 1 3 = 0:333333333333333 It is impossible to write the complete decimal expansion of 1 3 because it contains in nitely many digits. But we can describe the expansion: each digit is a three.
I understand that if I take one point or any number of collinear points, then I can draw infinite planes just by rotating around the line that connects these points, but why do we need 3 non collinear points to define a plane, why not more?
