Multivariable Calculus - BC Review and Vectors (copy)

Formula for an ellipse

How to find foci for ellipse

c² = a² - b² (value goes into major axis)

How to find eccentricity of an ellipse

E = c/a (distance between foci and center/distance between major vertex and center)

Area of an Ellipse

A = πab

Circumference of an Ellipse

Focus of a parabola

1/4p

Directrix of a Parabola

y = -1/4p

Length of Latus Rectum

4p

Parametric Second Derivative

Parametric Arc Length

Formula for Speed

How to express x in polar form

r * cos theta

How to express y in polar form

r * sin theta

r² =

x² + y²

How to convert Cartesian to theta

Discriminant of a Conic

B² - 4AC

Ax² + Bxy + Cy²

When the discriminant for a conic section is less than 0

ellipse (circle if A = C)

When the discriminant for a conic section is equal to 0

parabola

When the discriminant for a conic section is greater than 0

hyperbola

Polar Equation of a Conic section

directrix: x=d

Volume of a Parallelepiped given u, v, w (no magnitude)

Triple Scalar product

Volume of a Parallelepiped given u,v,w (with magnitude)

|v||w||u|costheta

cos theta =

a * b/||a||||b||

sin theta

a x b/||a||||b||

proj_ba

projecting a onto b

(a * b/||b||²)b

If Vectors PQ x PR = 0

points are colinear

If triple Scalar product = 0

vectors are coplanar

Area of a parallelogram

||v x w||

Area of a triangle

||v x w||/2

Vector Parametrization and Parametric Equations

r₀ + tv = <x₀,y₀,z₀> + t<a,b,c>

x = x₀ + at, etc

Different Parametrization of the Same Line

Check if vectors are parallel (scalar multiple)

Check if they have point in common, choose a point on the first line (r1), if the point lies on the second line (r2) there is a value of t that satisfies the second line equation equaling the point from the first line.

Determining Intersection of Two Lines

set lines 1 and 2 equal (r1 = r2)

distribute and have a system of linear equations (3 eq.,

solve for t1 and t2, plug back into equations to see if they intersect

Disk Method

Volume by Cross Section

A(x) = cross section area

Surface Area