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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.
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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=
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
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
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
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.