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By Zhukavets N.M., Shestakov I.P.

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0; 1=4/ and the directrix is y D 1=4. 0; 0/. ) The essential results about the behavior of the graph don’t require calculus or the theory of conic sections, but do require results on the real numbers such as the intermediate value theorem. 4. 3. Focus and Directrix of the Parabola y D x 2 and similarly as x increases through all negative real numbers. 0; 0/ on the left side of the y-axis and goes again through all positive heights as it rises on the right side of the y-axis. Once we know the shape of the graph of y D x 2 , we can use it for the graphs of other monic quadratic polynomials.

Babylonians used cuneiform, a wedgelike script that evolved from pictographs, for their writing. They also used base 60 in writing numbers, a choice that persists today in our methods for measuring time (minutes, seconds) and angle (degrees, minutes, seconds). Surviving on tablets are examples of the mathematical problems that Babylonians posed and solved. Many reduce to solving quadratic equations. Our understanding of the mathematics on the cuneiform tablets owes much to the pioneering work of Otto Neugebauer, an Austrian scholar who studied mathematics at Munich and G¨ottingen in the 1920s.

Although the theorem is fundamental to the development of calculus, its proof is usually deferred to an advanced calculus or real analysis course. Since polynomials are continuous functions, the intermediate value theorem applies to them, yielding the following result. 9 (intermediate value theorem for polynomials). x/ be a polynomial. c/ D d . As an example of the intermediate value theorem in action, given a positive real number d that is not the square of an integer, we can find nonnegative integers a < b with a2 < d < b 2.

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A base of the free alternative superalgebra on one odd generator by Zhukavets N.M., Shestakov I.P.


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