Advanced Analytical Geometry Flashcards Preview

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Flashcards in Advanced Analytical Geometry Deck (18):
1

Basic elliptic paraboliod

Bowl shape
(x^2)/(a^2)+(y^2)/(b^2)=z

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

2

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

3

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

4

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

5

Hyperbolic paraboloid formula

Saddle shape
(x^2)/(a^2)-(y^2)/(b^2)=z

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

6

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)

7

Critical Points

Points at which fx(a,b,c)=fy(a,b,c)=0

8

Saddle Points

Critical point at which f(x,y)=f(a,b)

9

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

10

Check for extrema in surfaces using differtial equations

Need to look this up

11

Objective function

Function that describes the geometric surface in 3d coordinate plane

12

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

13

Lagrange Multiplier (for 2d plane: 2 variables)

G

14

Lagrange Multiplier (for 3d coordinate system)

G

15

Volume under 3d curvature

Volume is equal to the double integral of the multivariable function
V=∫∫f(x,y)dA

16

Volume between regions of two 3d curves

Volume is equal to the double integral of the multivariable function

V=∫∫(g(x,y)-f(x,y))dA

17

Average value of a function for a 3d plane within a region

(∫∫f(x,y)dA)/(RegionArea)

18

Basic ellipsoid formula

Egg shape
(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices