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)