math test 2

Ellipse equation

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

parametric surfaces in space

1. parameterize function (vector)r (u,v) 2. find domain for variables u and v

Domain & Range? rational function: f(x,y)= (x^3-x+10)/(x^2+y^2-9) (rational function on top of rational function)

Domain: (x^2+y^2-9) CANNOT= 0 (denominator cannot be zero or world will explode) Range: R (rational function its self has range of R)

Domain & Range? natural log: h(x,y)= ln((y-x-10)/(x^2y^4+5))

Domain: (y-x-10) > 0 (the whole thing inside parenthesis cannot equal zero... you can't take the natural log of anything zero or below... the denominator will never be able to be zero or negative so you can simply state that the numerator has to be greater than zero.) Range: R

Domain & Range? exponential power: f(x,y,z)= e^(x^2yz-xzy-10x+z)

Domain: R Range: (0, infinity)

Domain & Range? circle: f(x,y)= 9-x^2-y^2 (identify it is a circle because both x and y are squared)

Domain: R Range: (-infinity, 9]

hyperbola (what does it look like?)

an open curve formed by a plane that cuts the base of a right circular cone

f(x, y, z) = sqrt(x^2 + y^2) −2z ellipsoids paraboloids hyperbolic paraboloids planes generalized cylinders cones

cones

hyperboloid 1sheet

one neg sign on right (on z)

f(x, y, z) = 5x^2 + y^2 − z ellipsoids paraboloids hyperbolic paraboloids planes generalized cylinders cones

paraboloids

f(x, y, z) = 2x − 3y + 4z ellipsoids paraboloids hyperbolic paraboloids planes generalized cylinders cones

planes

hyperbola equation

same as ellipse equation but - not + (x-h)^2/a^2 - (y-k)^2/b^2 = 1

hyperboloid 2sheets

two neg signs (not on z)

ellipsoid

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

cone equation

z=(minus) sqrt(x^2 + y^2) ... gives downward cone with vertex at origin

paraboloids

z=(plus/minus) (x^2/a^2 + y^2/b^2)+k (remember k is also value on taxis)