MATH 1920 Lecture Notes - Lecture 5: Paraboloid, Quadratic Equation, Spherical Coordinate System

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pearswan514

1 Sep 2017

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Definition: a quadratic surface is determined by a quadratic equation in x, y, z: Ax2 + by2 + cz2 + dxy + eyz + fxz + ax + by + cz + d = 0. The surface is in standard position if it is given as : Ax2 + by2 + cz2 + d = 0. Equation of a parabola: a(x - h)2 + k = (y - k)2. Equation of an ellipse: x2/a2 + y2/b2 = 1. Equation of a hyperbola: x2/a2 - y2/b2 = 1 or y2/a2 - x2/b2 = 1. Six types of quadratic surfaces in three dimensions: Equation: x2/a2 + y2/b2 + z2/c2 = 1. Horizontal trace (substitute z = k (constant)): ellipse. Vertical trace (substitute x or y = k): ellipse. Sphere with radius 1 when a = b = c. Equation: x2/a2 + y2/b2 - z2/c2 = 1. Equation: -(x2/a2) - y2/b2 + z2/c2 = 1. Equation: -(x2/a2) - y2/b2 + z2/c2 = 0.

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First, create 3 equations of the form ax+by+cz=d , where a, b, c, and d are constants (integers between – 5 and 5). For example, x + 2y – 2= -1 . Perform row operations on your system to obtain a row-echelon form and the solution. Go to the 3D calculator website GeoGebra and enter each of the equations. After you have completed this first task, choose one of the following to complete your discussion post. 1. Reflect on what the graphs are suggesting for one equation, two equations, and three equations, and describe your observations. Think about the equation as a function f of x and y, for example, x + 2y +1 = 2 in the example above. Geogebra automatically interprets this way, that is, like z=f(x, y)=x+2y+1 , it isolates z in the equation. 2. What did the graphs show when you entered the second equation? 3. Give a simple description of the system x=0 y=0 z=0 x = 0 can be seen as the constant function x=g(y, z)=0y+0z=0 . Of course, you can use GeoGebra to “observe” the system. 4. Give an example with 2 equations as simple as possible with 3 variables (at least 1 being non-linear; keeping z to the one power on both equations) and describe the potential of GeoGebra to study nonlinear systems.