quadraic surface {cal3}

Question:

• Paramaterize 5 different lines (only one can be linear) that pass thru the point (4,3) but cannot be the origion of the point.

Quadrtic Surface

• Know: Conic sections Formulas/shapes: (to put into anki cards w/ function form + graph view)
• Ellipsoid:
  • ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2+{\left(\frac{z}{c}\right)}^2=1$
• cone
  • ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2={\left(\frac{z}{c}\right)}^2$
• hyberboloid of 1-sheet
  • ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2-{\left(\frac{z}{c}\right)}^2=1$
• hyberboloid of 2-sheet
  • $-{\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2+{\left(\frac{z}{c}\right)}^2=1$
• elliptic parabolid
  • ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=\frac{z}{c}$
• hyperbolic parabolid
  • ${\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2=\frac{z}{c}$

Level Curves:

• f(x,y) = C (C = const)
• f(x,y) = x^2 + y^2,
  • What quadrant surface is put into function form? Eliptic parabolid
• Circles
  • level curves
  • x^2 + y

eg. Let $x^2+y^2=C$, plot when c=1, show how LC changes when c increases.