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Calculus 3 Study Guide: Ch. 12-16 on 2&3 Variable Calculus, Study notes of Advanced Calculus

A study guide for calculus 3, covering topics such as distance between vectors, equations of spheres and planes, dot and cross products, limits and continuity, contour diagrams, and coordinate systems including polar, cylindrical, and spherical coordinates. Learn about vector components, projections, and taylor polynomials.

Typology: Study notes

Pre 2010

Uploaded on 12/06/2009

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Calculus 3 Finals Study Guide
Chapter 12
2 Variables
Distance between (x,y,z) and (a,b,c):
2 2 2
( ) ( ) ( )x a y b z c
Equation of sphere of radius r centered at (a,b,c):
2 2 2 2
( ) ( ) ( )x a y b z c r
If one variable is missing then the graph goes along the axis of the missing value
Ex. Z=x^2 looks like a parabola that travels along the Y-axis
Contour diagrams
3 Variables
Graphing
Limits/Continuity (take values along x=some value and y=some value, if they
differ then the limit does not exist. Ex x/(2x+y) along x=0 f(x)= 0 but along y=0
f(y)= 1/2 . since ½ does not equal 0, the limit DNE
Chapter 13
Dot product (determines angle between two vectors, two vectors are orthogonal if
dot=0. (SCALAR VALUE)
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Calculus 3 Finals Study Guide

Chapter 12

2 Variables

 Distance between ( x,y,z ) and ( a,b,c ):

2 2 2

( xa )  ( yb )  ( zc )

 Equation of sphere of radius r centered at ( a,b,c ):

2 2 2 2

( xa )  ( yb )  ( zc )  r

 If one variable is missing then the graph goes along the axis of the missing value

Ex. Z=x^2 looks like a parabola that travels along the Y-axis

 Contour diagrams

3 Variables

 Graphing

 Limits/Continuity (take values along x=some value and y=some value, if they

differ then the limit does not exist. Ex x/(2x+y) along x=0 f(x)= 0 but along y=

f(y)= 1/2. since ½ does not equal 0, the limit DNE

Chapter 13

 Dot product (determines angle between two vectors, two vectors are orthogonal if

dot=0. ( SCALAR VALUE)

 Cross product (must be 3-dimensional):

1 2 3

1 2 3

i j k

v w v v v

w w w

2 3 3 2 1 3 3 1 1 2 2 1

( v wv w ) i  ( v wv w ) j  ( v wv w ) k

Cross product produces a normal to the vector tot the plane

 Equation of plane through

0 0 0

( x , y , z )

with normal

aib jck

0 0 0

a x (  x )  b y (  y )  c z (  z )  0

 Resolving vectors into components parallel and perpendicular to unit vector

u

 Projection of

v

on to

u

parallel

vv u u

or Vparallel= [[v]]cos(theta)*u with u

being a unit vector

perp parallel

v   v v

Chapter 14

 Partial derivatives

 Tangent plane to surface

zf ( , x y )

at the point ( a,b ):

x y

zf a bf a b xaf a b yb

 Tangent plane approximation for f ( x,y ) near the point ( a,b )

If f is differentiable at ( a,b ), near ( a,b ),

x y

f x yf a bf a b xaf a b yb

0

P

is a critical point of f if

0

f ( P )

is either

or undefined

Second Derivative Test for functions of two variables:

Suppose

0 0

f ( x , y )  0

. Let

2

0 0 0 0 0 0

xx yy xy

Df x y f x yf x y

 If D > 0 and

0 0

xx

f x y

, then f has a local minimum at

0 0

( x , y )

 If D > 0 and

0 0

xx

f x y

, then f has a local maximum at

0 0

( x , y )

 If D < 0, then f has a saddle point at

0 0

( x , y )

 If D = 0, then anything can happen at

0 0

( x , y )

 Extreme Value Theorem: If f is a continuous function on a closed and bounded

region R , then f has a global maximum at some

0 0

( x , y )

in R and a global minimum at

some

1 1

( x , y )

in R.

Chapter 16

 Other coordinate systems

1. polar coordinates (r,)

conversions to/from Cartesian ( x,y ) coordinates

x = r cos, y = r sin

r² = x² + y², tan= y/x

*double integrals in polar coordinates

dA  r dr d 

2. cylindrical coordinates (r,  ,z)

r  0

   z  

conversions to/from rectangular ( x,y,z ) coordinates

x = r cos, y = r sin, z = z

r² = x² + y², tan= y/x, z = z

***** triple integrals in cylindrical coordinates

dV  r dr d  dz

3. spherical coordinates (  )

conversions to/from rectangular ( x,y,z ) coordinates

x =  sin  cos, y =  sin  sin, z =  cos 

 ² = x² + y² + z²

***** triple integrals in spherical coordinates

2

dV   sin d  d  d

 Iterated integrals of volume always have 1 has the function

 Double integrals can be reversed to make the integration easier.