4.1 Class Notes - April 17 Prelim 3 a week from today...

April 17
Prelim 3 a week from today covers
7.4 Definite integrals
8.2 Volumes, 8.4 Improper Integrals
8.1 Integration by parts
Ch. 11. Probability distributions
9.2 Partial derivatives
i.e., the material from HWK 7,8,9
Formula and review sheets on Blackboad
Partial Derivatives
Given ( , ) the
partial derviative
is computed by holding
constant and
differentiating the
resulting function of
f x y
f
x
y
x
∂
∂
f(x,y) = – x
2
– y
2
2
Set 1
( , 1) 1
2
2
y
f x x
f
x
x
f
y
y
= −
− = −
∂
= −
∂
∂
= −
∂
Example
x
2
2
Volume of a cone
( , )
3
2
3
3
r h
V r h
V rh
r
V r
h
π
π
π
=
∂
=
∂
∂
=
∂
Example
2
2 1
2
2
2
2 2
2
( , ) ( )
2 ( ) 1
( )
0 ( ) 1
( )
( )
x
f x y x x y
x y
f x x y x
x x y
f x y x
x x y
y x y
−
−
= = +
+
∂ ⋅ + − ⋅
=
∂ +
∂ ⋅ + − ⋅
= − + =
∂ +
Explaining ∂ f/ ∂ y
2
2
2
2
2
2
( , )
( )
3
(3, )
3
3
(3 )
x
f x y
x y
f x
y x y
f y
y
f
y y
=
+
∂
= −
∂ +
=
+
∂
= −
∂ +

Problem 13 on page 517
2 3
3
2 3
2 2
2 3
( , ) ln(1 3 )
1
6
1 3
1
9
1 3
f x y x y
f
xy
x x y
f
x y
y x y
= +
∂
= ⋅
∂ +
∂
= ⋅
∂ +
Problem 17
2
2 2
2
2
( , )
2
x y
x y x y
x y
f x y xe
f
e xe xy
x
f
xe x
y
=
∂
= + ⋅
∂
∂
= ⋅
∂
Higher order partials
2 3 2 2
3 2 2 2
2 2
2
( , ) 4 5 12
8 5 24 15 24
8 24 30 24
15 48
f x y x xy x y
f f
x y xy xy x y
x y
f f
y xy x
x x y y
f f
y xy
y x x y
= − +
∂ ∂