Semester 1 Review

Unit vector =v

v

iffu v u cv

cosu v u v

2proj ora

a b a bb a a

a aa

a

b

projab

is orthogonal to andu v u v

A line that contains the point P(x0,y0,z0) and direction vector :, ,a b c

0

0

0

x x at

y y bt

z z ct

0 0 0x x y y z z

a b c

parametric symmetric

The equation of the plane containing point P(x0,y0,z0) with normal vector ,, ,n a b c

0 0 0 0a x x b y y c z z

Ex. Let θ be the angle between the planes 5y + 6z = 2 and 2x + 8y – 3z = 1. Find cos θ.

Sketching 3-D

Ex. Sketch 2 2 29 9 9x y z

cos

sin

x r

y r

z z

2 2 2

tan yx

x y r

z z

Cylind Rect Rect Cylind

sin cos

sin sin

cos

x

y

z

2 2 2 2

2 2 2

tan

cos

yx

x y z

z

x y z

Spher Rect Rect Spher

Parameterizing curves:

Derivative and integral of vector functions

Ex. Find the parametric equations of the line tangent to at (1,4,4).

, ,r t f t g t h t

2 2cos ,4 ,4t tr t t e e

Ex. Find if and sin i cos j 16 kr t t t t

r t

0 i j 5kr

If r(t) = displacement vector, then

r(t) = velocity v(t)

r(t) = acceleration a(t)

|r(t)| = speed v(t)

unit tangent vectorr t

T tr t

��������������

2 2 2

arc length of on ,

t

a

s t f u g u h u du

r a b

3 curvature of

r t r tr

r t

First and second partial derivatives

Implicit differentiation:

Total Differential: dz = fxdx + fydy

x

z

Fz

x F

y

z

Fz

y F

Let w = f (x,y), with x = g(t) and y = h(t):

dw w dx w dy

dt x dt y dt

, , , ,x yf x y f x y f x y

Gradient is orthogonal to level curve of f

Directional derivative of f in the direction of unit vector u is , ,uD f x y f x y u

Ex. Find the parametric equations of the line tangent to the curve of intersection of and at (-1,1,2).2 2z x y

2 2 24 9x y z

Critical point = where fx and fy are both zero, or where one is undefined

d = fxx(a,b) fyy(a,b) – [ fxy(a,b)]2

If d > 0 and fxx(a,b) < 0, then (a,b) is a rel. max.

If d > 0 and fxx(a,b) > 0, then (a,b) is a rel. min.

If d < 0, then (a,b) is a saddle point.

If d = 0, then the test fails.

Ex. Find the critical points of and classify them as relative maximum, relative minimum,

or saddle point.

2 2, 6 20 96 5 8f x y x y x y

Ex. Use Lagrange multipliers to find the maximum value of subject to the constraint , 2 2f x y x xy y

2 80x y

Multiple Integrals – changing order

Ex. 2

1 12

0

cosy

y x dxdy

R

dA dydx rdrd ,

R Q

V f x y dA dV

Ex. Find the volume bounded by x2 + y2 = 64, under the plane z = y, and in the first octant.

,R

m x y dA where ρ is density

2 21 x y

R

S f f dA Surface area:

Jacobian:

,

,

x xu vy yu v

x y

u v

Ex. Evaluate , where R is the region bounded

by

3 2R

x y dA0, 2, 2 0, and 2 3x y x y x y x y

Bring your textbooks and ID cards on the first day of next semester so that we can exchanged them