Engineering Calculus Notes 404

Engineering Calculus Notes 404 - These correspond to...

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392 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 3.10 Quadratic Curves and Surfaces In this section, we will use the Principal Axis Theorem to classify the curves ( resp . surfaces) which arise as the locus of an equation of degree two in two ( resp . three) variables. Quadratic Curves The general quadratic equation in x and y is Ax 2 + Bxy + Cy 2 + Dx + Ey = F. (3.35) If all three of the leading terms vanish, then this is the equation of a line. We will assume henceforth that at least one of A , B and C is nonzero. In § 2.1 we identiFed a number of equations of this form as “model equations” for the conic sections: Parabolas: the equations y = ax 2 (3.36) and its sister x = ay 2 (3.37) are model equations for a parabola with vertex at the origin, focus on the y -axis ( resp . x -axis) and horizontal ( resp . vertical) directrix.
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Unformatted text preview: These correspond to Equation ( 3.35 ) with B = 0, exactly one of A and C nonzero, and the linear (degree one) term corresponding to the “other” variable nonzero; you should check that moving the vertex from the origin results from allowing both D and E , and/or F , to be nonzero. Ellipses and Circles: the model equation x 2 a 2 + y 2 b 2 = 1 (3.38) for a circle (if a = b ) or an ellipse with axes parallel to the coordinate axes and center at the origin corresponds to B = 0, A , C and F of the same sign, and D = E = 0. Again you should check that moving the vertex results in the introduction of nonzero values for D and/or E and simultaneously raises the absolute value of F . However, when...
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