calc 3 exam 3

Area of a flat 2d region

integral from b to a of (top curve - bottom curve)

Type 1

curvy top and bottom, straight sides, dydx

Type 2

straight top and bottom, curvy sides, dxdy

Average value of a function

1/Area * double integral

Polar coordinates x =

rcostheta

Polar coordinates y =

rsintheta

Polar coordinates r² =

x² + y²

Polar coordinates tan theta =

y/x

Polar coordinates area differential

dA = rdrdtheta

Cylindrical coordinates x =

rcostheta

Cylindrical coordinates y =

rsintheta

Cylindrical coordinates z =

z

Cylindrical coordinates volume differential

dV = rdzdrdtheta

Spherical coordinates x =

psin0costheta

Spherical coordinates y =

psin0sintheta

Spherical coordinates z =

pcos0

Spherical coordinates p² =

x² + y² + z²

Clairaut’s Theorem

If fx and fy are continuous, fxy = fyx

Directional Derivative

Duf(x,y) = dot product of gradient and unit direction vector

Gradient Vector

fx(x,y)i + fy(x,y)j

Direction of max increase

gradient vector

Direction of max decrease

-gradient vector

Max value

magnitude of gradient vector

Min value

-magnitude of gradient vector

Tangent Plane

z = f(x0, y0) + fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0)

Implicit Differentiation

dz/dx = -Fx/Fz

Critical Points

Where fx = 0 and fy = 0

Second Derivative Test Discriminant

D(x,y) = fxxfyy - [fxy]²

Local Min

D>0 and fxx>0

Local Max

D>0 and fxx<0

Saddle Point

D<0

Inconclusive

D=0