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Formulas for Midterm 2 Position vector/Curve/Path
r (t )
Velocity vector/Derivative
r (t )
Speed
x (t )2
r (t )
Integral
r (t ) dt
y (t )2 x (t ) dt ,
(x (t ),y (t ), z (t )) (x (t ),y (t ), z (t ))
z (t )2 y (t ) dt ,
z (t ) dt
c 1,c 2,c 3
Length of a curve/path/arc length/distance travelled b
Length of curve from a to b:
a
r (t ) dt
Definition of the limit lim
f (x , y ) (x ,y ) (a ,b)
if and only if for all f (x ,y )
> 0 there exists
whenever (x
a )2
(y b)2
> 0 such that and (x , y ) (a ,b)
Definition of continuity f (x ,y )
is continuous at a point (a ,b) in the domain of f if and only if f (x , y ) lim (x ,y ) (a ,b)
f (a ,b)
Proving limits Using squeeze theorem, polar co-ordinates, and the definition.
Showing limits do not exist Using lines
y
mx or
polar co-ordinates.
Definition of partial derivatives The partial derivatives of f at (a ,b) are f
f x (a ,b)
x f
f y (a ,b)
To compute
y
f x treat y as
(a ,b) (a ,b)
lim
h
f (a
0
lim
h
f (a ,b
h ,b) f (a ,b) h h ) f (a ,b)
0
h
a constant and do ordinary differentiation.
Definition of Differentiability and Tangent Planes A function f is differentiable at (a ,b) if and only if f (x , y ) f (a ,b) f x (a ,b)(x a ) f y (a ,b)
lim
2
(x ,y ) (a ,b)
(x a )
2
exists.
(y b)
If the limit exists then the equation of the tangent plane to the surface z
f (x , y ) at
the point (a ,b) is z
f (a ,b)
f x (a ,b)(x a )
f y (a ,b)(y b)
Gradient vector The gradient of f at (a ,b) is f f (a ,b), (a ,b) x y
f (a ,b)
This is the direction of greatest increase of the surface
z
f (x , y )
Directional derivative Let
u
(u 1, u 2 ) be a unit vector. Then the directional derivative of f at
(a ,b) in the direction of u is D u f (a ,b)
It is the slope of the surface
If
is the angle between
h
z
u and
D u f (a ,b)
f (a
lim
hu 1,b
0
f (x , y ) in
hu 2 ) f (a ,b) h
the direction of
the gradient
f (a ,b) u
f (a ,b) ,
f (a ,b) cos
then
u .
Chain Rule If f (y 1,y 2,..., y n ) is a function and then for any
are a functions of (x 1,..., x m ) ,
y 1, y 2,...,y n
x i , f
f
y 1
f
y 2
f
x i
y 1
x i
y 2
x i
y n x i
In particular, if we have f (x ,y , z ) where f
f (t )
x
x (t )
x
f y
x (t ), y f
y (t )
z
y n
y (t ), z
z (t ) then
z (t )
Implicit Differentiation and Tangent Planes A surface
z
f (x ,y ) can
be represented by
the tangent plane at point
P
F (x ,y , z )
0
. The equation of
(a ,b,c ) is
F (a ,b,c ) (x , y , z )
F (a ,b,c ) (a ,b,c )
The partial derivatives of z are z
F x
z
F y
x
F z
y
F z
The gradient of surface
z
f (x ,y ) given f (a ,b)
by
F x F y , F z F z
F (x , y , z )
0
at point (a ,b) is