13.2 DIFFERENTIATION FORMULAS
Example 3
where r = 0
Let r be a differentiable vector function of t and set r = r . Show that
d r
1
= 3
dt r
r
(13.2.4)
SOLUTION
775
r
dr
dt
r .
This is a little tricky:
=
1 dr
1 dr
2 r
r dt
r dt
=
d r
dt r
dr
1 2 dr
r r
r
13.1 VECTOR FUNCTIONS
and
b
(13.1.10)
b
f (t) dt
f (t) dt.
a
a
The proof of (13.1.9) is left as an exercise (Exercise 56). Here we prove (13.1.10).
It is an important inequality.
PROOF
b
Set r =
f (t) dt and note that
a
r
2
b
= r qr=r q
f (t) dt
a
b
=
a
Math 2433 Week 9
15.7 - Maxima and Minima with Side Conditions
Method of Lagrange:
Using vector notation, we will take:
to be a
curve that lies entirely in U and has at each point a nonzero tangent
vector r t - So:
And suppose g is a continuously differen
Math 2433 www.math.uh.edu/~sbranton
14.1 Examples of real-valued functions of two and three variables
(read this section!)
Finding domain and range:
1.
2.
f ( x, y ) xy
f ( x, y )
1
x y
f ( x, y ) 9 x 2 4 y 2
3.
z
f ( x, y , z ) 2
4.
x y2
5. Express each
Math 2433 www.math.uh.edu/~sbranton
14.4 Partial Derivatives
The partial derivative of f with respect to x is the function fx obtained by
differentiating
f with respect to x, treating y as a constant.
The partial derivative of f with respect to y is the f