Kathryn Clark
Math 329 01
S. Rodman
25 January 2017
Theorem 20
Theorem 20: Considering only finite subsets of L, no linearly independent set has more points
than a spanning set.
We must show that no l
Kathryn Clark
S. Rodman
Math 329-01
21 November 2016
Theorem 3: For any two points v and w such that v w, there exists a set of points describing a
line passing through both v and w.
v = (v1, v2, v3).
Kathryn Clark
Math 329-01
S. Rodman
18 November 2016
Theorem 2:
A. For any ordered triple v = (v1, v2, v3) R there exists an ordered triple x = (x1, x2, x3)
such that when x is added to v the result
Kathryn Clark
Theorem One
16 November 2016
Let v = (v1, v2, v3) where v1, v2, and v3 are arbitrary real numbers. The norm of v is given by the
formula:
|v| = v12 + v22 + v32
We must show that |v| = v1
Kathryn Clark
Math 329 01
S. Rodman
23 January 2017
Theorem 19
Theorem 19: (Replacement Lemma) Suppose that S spans L, Q L and P is a point of S such
that when Q is written as a linear combination of
Kathryn Clark
Math 329 01
S. Rodman
28 November 2016
Theorem 5
Theorem 5: If (s) and (t) are two bases for , then the process of taking the list of
coefficients needed to describe any point v using th
Kathryn Clark
Math 329 01
S. Rodman
30 November 2016
Theorem 6
For any point Q in a linear space L, there exists a point X such that Q+X=0. In addition, for any
given Q this point is unique.
We must s
374
Chapter Seven INTEGRATION
Substitution
Getting an integrand into the right form to use a table of integrals involves substitution and a variety
of algebraic techniques.
et sin(5t + 7) dt.
Example
8.2 APPLICATIONS TO GEOMETRY
427
Arc Length of a Parametric Curve
A particle moving along a curve in the plane given by the parametric equations x = f (t), y = g(t),
where t is time, has speed given b
4.7 LHOPITALS RULE, GROWTH, AND DOMINANCE
247
(d) We have
1
1
1 1
= = .
x sin x
0 0
Limits like this can often be calculated by adding the fractions:
sin x x
1
1
= lim
,
lim
x0
x0 x
sin x
x sin x
giv
360
Chapter Seven INTEGRATION
Exercises and Problems for Section 7.1
Exercises
1. Use substitution to express each of the following inteb
grals as a multiple of a (1/w) dw for some a and b.
Then evalu
368
Chapter Seven INTEGRATION
On the right side we have an integral similar to the original one, with the sine replaced by a cosine.
Using integration by parts on that integral in the same way gives
1
7.6 IMPROPER INTEGRALS
6
0
401
1
dx = 3(4)1/3 + 3(2)1/3 = 8.542.
(x 4)2/3
Finally, there is a question of what to do when an integral is improper at both endpoints. In this
case, we just break the int
8.1 AREAS AND VOLUMES
419
Exercises and Problems for Section 8.1
Exercises
y
1. (a) Write a Riemann sum approximating the area of the
region in Figure 8.13, using vertical strips as shown.
(b) Evaluat
384
Chapter Seven INTEGRATION
Since x + 1/2 = ( 3/2) tan , we have = arctan(2/ 3)x + 1/ 3), so
!
1
2
1
2
+ C.
dx = arctan x +
x2 + x + 1
3
3
3
Alternatively, using a computer algebra system gives7
2