Advanced Analytical Geometry Flashcards
Basic elliptic paraboliod
Bowl shape
(x^2)/(a^2)+(y^2)/(b^2)=z
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
One-sheet Hyperboloid
Hour Glass
(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Two-sheet hyperboloid
Reverse facing bowls
-(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Elliptic cone formula
Two cone tips touching
(x^2)/(a^2)+(y^2)/(b^2)=(z^2)/(c^2)
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Hyperbolic paraboloid formula
Saddle shape
(x^2)/(a^2)-(y^2)/(b^2)=z
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Planes Tangent to a curved surface
0=f(x,y,z)(x-a)+f(x,y,z)(y-b)+f(x,y,z)(z-c)
Critical Points
Points at which fx(a,b,c)=fy(a,b,c)=0
Saddle Points
Critical point at which f(x,y)=f(a,b)
Absolute Extrema on a surface
1) Find all of the critical points
2) Solve for z
3) Pick the point at which the z-value is the highest or lowest
Check for extrema in surfaces using differtial equations
Need to look this up
Objective function
Function that describes the geometric surface in 3d coordinate plane
Constraint
Function in the xy plane that describes the region within the surface in which you will be working determines the boundary of your objective function
Lagrange Multiplier (for 2d plane: 2 variables)
G
Lagrange Multiplier (for 3d coordinate system)
G
Volume under 3d curvature
Volume is equal to the double integral of the multivariable function
V=∫∫f(x,y)dA
