Download Calculus 3 Study Guide: Ch. 12-16 on 2&3 Variable Calculus and more Study notes Advanced Calculus in PDF only on Docsity!
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=
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 w v w ) i ( v w v w ) j ( v w v 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
ai b j ck
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
v v 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
z f ( , x y )
at the point ( a,b ):
x y
z f a b f a b x a f a b y b
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 y f a b f a b x a f a b y b
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
D f x y f x y f 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.
