TuesEx1.) det[ 3x3 matrix with elements M[ij] ] write or check that
det(M) = det(M)
so I guess using a 3x3 matrix show that the determinant of the
transposed matrix and the determinant of the original
The Bolzano-Weierstrass Property and Compactness
Denitions: Let S be a subset of R. An upper bound of S is a number b such that x
for every x S . A least upper bound of S is a b such that b
b
b for ev
Math 347
Epsilonics IV: Sequences and Limits, II Solutions
A.J. Hildebrand
Practice Problems: Four Classic Proofs Solutions
1. Proof of the Monontone Convergence Theorem from the Completeness Axiom: U
Definitions: Convergent and divergent series
1. A series is an infinite sum, written
ak . It doesnt have to start at k = 0.
k =0
2. The nth partial sum of the series is Sn = a0 + a1 + a2 + . + an
3.
Nested Interval Theorem
Let cfw_[an , bn ] be a sequence of closed intervals such that n R, [an+1 , bn+1 ][an , bn ]. and
that limn bn an 0. Then by the Nested Interval Theorem,
In =
(1)
n=1
and is c
Nested Interval Theorem
Let cfw_[an , bn ] be a sequence of closed intervals such that n R, [an+1 , bn+1 ][an , bn ]. and
that limn bn an 0. Then by the Nested Interval Theorem,
In =
(1)
n=1
and is c
Proof of the MCT using the NIT
Nested Interval Theorem
Let cfw_[an , bn ] be a sequence of closed intervals such that n R, [an+1 , bn+1 ][an , bn ]. and
that limn bn an 0. Then by the Nested Interval
c 2012 Math Medics LLC. All rights reserved.
TRIGONOMETRIC IDENTITIES
Reciprocal identities
1
1
sin u =
cos u =
csc u
sec u
1
1
tan u =
cot u =
cot u
tan u
1
1
csc u =
sec u =
sin u
cos u
Pythagorea
Common Derivatives and Integrals
Derivatives
Basic Properties/Formulas/Rules d ( cf ( x ) ) = cf ( x ) , c is any constant. ( f ( x ) g ( x ) ) = f ( x ) g ( x ) dx dn d ( c ) = 0 , c is any constant.
Math 347
Epsilonics IV: Sequences and Limits, II Solutions
A.J. Hildebrand
Practice Problems: Four Classic Proofs Solutions
1. Proof of the Monontone Convergence Theorem from the Completeness Axiom: U
The Bolzano-Weierstrass Property and Compactness
Denitions: Let S be a subset of R. An upper bound of S is a number b such that x
for every x S . A least upper bound of S is a b such that b
b
b for ev