Math 2270 Exam 1 Review

z = f(a,b) + δf/δx(a,b)(x-a) + δf/δy(a,b)(y-b)

z = f(a,b) + δf/δx(a,b)(x-a) + δf/δy(a,b)(y-b)

Find the Tangent Plane

L(x0,y0) = F(x0,y0) + δf/δx(a,b)(x-x0) + δf/δy(a,b)(y-y0)

Find tangent plane, plug in approxed values

Find the Linear Approximation

⛛f(x,y,z) / ||⛛f(x,y,z)||

Find the direction f increases most rapidly at (x,y,z)

||⛛f(x,y,z)||

How fast is f increasing?

ax + by + cz: <a,b,c>
OR
u × v
OR
⛛f(x0,y0,z0)

Find the Normal Vector

⛛f(x0,y0,z0) → a(x-x0) + b(y-y0) +c(z-z0) = 0

Find the plane at (x0,y0,z0)

||AB × AC||

Area of a Parallelogram with corners ABCD

1/2||AB × AC||

Area of a Triangle with corners ABC

Point: (x0,y0,z0)
Normal Vector: AB × AC
a(x-x0) + b(y-y0) +c(z-z0) = 0

Find the Plane containing ABC

|n ·  AD| / ||n||

Distance from point to plane

(||v × AC||)/||v||

Distance from point to line

r(t)=r0​+tv
r(0) is the point
v is the direction vector

Vector Equations

t = (x-x0)/a = (y-y0)/b = (z-z0)/c

Symmetric Equations

x(t) = x0 + ta,
y(t) = y0 + tb,
z(t) = z0 + tc

Parametric Equations

F(r(t)) = (δf/δx)(dx/dt) + (δf/δy)(dy/dt)

Rate of Change (chain rule)

⛛T(r(0))

Rate of Change in terms of ⛛T & R(t)

Cylinder

Surface when r=1

Cone

Surface when z=r

Paraboloid

Surface when z=r²

Sphere

Surface when P = 1

Cone

Surface when φ = π/4

The Plane z=0

Surface when φ = π/2

Half of the yz-plane

Surface when θ = π/2

x = rcos(θ)

Polar Coordinates: x =

y = rsin(θ)

Polar Coordinates: y =

r = √(x²+y²)

Polar Coordinates: r =

θ = arctan(y/x)

Polar Coordinates: θ =

r = Ρsinφ

Spherical Coordinates: r =

Ρ = √(x²+y²+z²)

Spherical Coordinates: Ρ =

φ = arccos(z/P)

Spherical Coordinates: φ =

θ = arctan(y/x)

Spherical Coordinates: θ =

x = rcosθ = Ρsinφcosθ

Spherical Coordinates: x =

y = rsinθ = Ρsinφsinθ

Spherical Coordinates: y =

z = Ρcosφ

Spherical Coordinates: z =

Use gradiant to find critical points
Parameterize boundary, plug into f
Solve d/dt = 0
Plug t into parameter
Solve for z, find max and min

Find Absolute Extrema (no lagrange)

Use gradiant to find critical points
⛛f(x,y,z) = λ⛛g(x,y,z)
Solve for points
Find max/min

Find Absolute Extrema (lagrange)

| δ²f/δx²(a,b) δ²f/δxδy(a,b) |
| δ²f/δyδx(a,b) δ²f/δy²(a,b) |

det(H(a,b)(f)) = δ²f/δx²(a,b) * δ²f/δy²(a,b) - (δ²f/δxδy(a,b))²

Hessian Matrix

Local Min

Hessian: Det(H(a,b)(f)) > 0 & δ²f/δx²(a,b) > 0

Local Max

Hessian: Det(H(a,b)(f)) > 0 & δ²f/δx²(a,b) < 0

Saddle Point

Hessian: Det(H(a,b)(f)) < 0

Inconclusive

Hessian: Det(H(a,b)(f)) = 0

1

cos(0)

0

sin(0)

(√2)/2

cos(π/4)

(√2)/2

sin(π/4)

0

cos(π/2)

1

sin(π/2)

-1

cos(π)

0

sin(π)

0

cos(3π/2)

-1

sin(3π/2)

v/||v||

Unit Vector

√(x² + y² + z²)

Magnitude

<a+x, b+y, c+z>

v + w

<a-x, b-y, c-z>

v - w

ax + by + cz

v · w Dot Product

||v||²

v · v

||v||||w||cos(θ)

v · w angle

w((v · w)/(||w||²))

projw(v)

v((v · w)/(||v||²))

projv(w)

i j k
v1 v2 v3
w1 w2 w3

v × w cross product

||v||||w||sin(θ)

||v × w||

u · (v × w)

Area of a Parallelapiped

s(t) = ∫(0,t)||v(u)||du

Arc Length

||v(t)|| (parameterized)

Speed

v(t)/||v(t)|| (parameterized)

Unit Tangent Vector

df/dt = (δf/δx * dx/dt) + (δf/δy * dy/dt) + (δf/δz * dz/dt) +

df/dt =

⛛f(x,y) = <δf/δx(x,y), δf/δy(x,y)>

Gradient ⛛f(x,y) =

Df(a,b)v = v/||v|| · ⛛f(x,y)

Directional Derivative Df(a,b)v Gradient

(1/||v||)Df(a,b)v

Directional Derivative Df(a,b)(v/||v||)

⛛f(x,y) = 0, solve for x,y,z.

Critical Points

⛛f(x0,y0) = <a,b>
f(a,b) + a(x-x0) + b(y-y0)

Tangent Plane at a point

⛛f(x,y,z) = λ⛛g(x,y,z)

Lagrange