Integrals Yielding the Natural Logarithmic Function
and Exponential Function
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
1 / 16
Recall
Recall the derivative of the natural logarithmic function:
d
1
(ln x) = .
dx
x
Math 100 (In
Area of Plane Regions
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Area of Plane Regions
1 / 13
Outline
1
Finding the Area of a Bounded Region
2
Area Between Two Curves
Math 100 (Inst. of Mathematics)
Area of Plane Regions
2 /
Applications of Differential Equations
Mathematics 100
Institute of Mathematics
Math 100 (Inst. of Mathematics)
Applications of Differential Equations
1 / 16
Outline
1
Obtaining Equations of Curves given the Slope
2
Rectilinear Motion
3
Marginals
Math 100
l
Lesson 10 OrthonormalBases and the Gram-Schmidt Process Denitions. Let V be a vector space with inner product ( , ) and 14,11 6 V.
- 1, We say u is orthogonal to u (n _L 1) whenever (u, v) : 0,
Denitions. An inner roduct on a vector 5 ace V is a functio
invm mm-m"~#
Lesson 9 . I The Fundamental Subspaees
1'1 = 111 1 0 ,. b1 1 0
4r] 7 212 : b2 :5 /l 72 , : b2 :11 4 +12 2
215; + 3ch : ()3 2 3 V 113 2 3
\-/ V
A 12
Denition. Let A E MmX"(R). The column space of A, denoted by C(A) consists of all linear combi
cfw_it [)1 Woh'hwmi '
. dot in '- WWW
grow: 8; \/ 91am M 5w
. Lesson 7 Linear Independence and Spanning Sets 2. If it happens that V : span S, then S is called the spanning set for V, or V is spanned by S, that is, every
vector in V can be expressed as a
Remark 10. Every matrix A E MWHCR) with full column rank (p(A) : n) has the following properties:
1. There are no free variables in the linear system At : b;
2. All columns of A are pivot columns;
3. MA) = cfw_6;
4, Ar : b has a unique solution or the sys
3 ve ofo F5
Example 4. /7 R1 . N 5 Exercises. Do what is asked.
3 L I . . .
1. Is S : cfw_ [ 21] , [g] , [765] linearly independent subset of R2? Nb . / 1. Find a basrs for R3 that Includes the vectors (2,1,1) and (0,3,4).
7 . (L
j l cfw_3 Y C T 5 2. Let
o
i
l'
2. P2, the set of all polynomials of degree n S 2, is a subspace of 13
3. The set of all symmetric 2 X 2 matrices is a subspace of M2x2(R).
' Remark. We do not need to verify all properties of a vector space to conclude that a subset W of a. ve
Remark 6. Let M , N be matrices which are compatible for multipication. Then
1. row rank of MN 5 row rank of N and Ktul) : M)
1' . 2. column rank of [MN 3 column rank of M.
Theorem. Let A E MmMOR), 1, Q are invertible matrices of app'r'opate sizes.
1. row
\(oum
a1+a3 : 072a1+a2 ~6a3 =0, 3(12 ~ 12% = O.
The system is equivalent to
3; a w vaxol tom
1 0 1 a1 0
W ninth ~
We martian WWW u NJMOC m o mu)
Example 9. Let V be the set of ordered pairs (:5, y) of real numbers, with operations
($1,111) 69 ($2,212) = (
.
Lesson 6 Vector Spaces and Subspaces
Denition. A vector :2 in R", or an nvector is the directed line segment denoted by
Ti
502
z:(cci,z2,.,rn)orz: , \vhcrexi,x2,.i,:cnlR.
Zn
Denition. The norm (or magnitude) of z is Ilzll = 112+ 33; + ~ + x
Denition
\E [4 atoms a V
some SFV
rm 17v.r~: - .
Lesson 8 - Basis and Dimensions
Denition. A set of vectors S : cfw_111,712, . . . ,vk in a vector space V is called a basis for V if
spans V that is, span S: V and
is linearly independent
Example 1.
1. Consider the
i
l'
r
Lesson 2 Solutions of Linear Equations
Denitions.
. 1. An equation (11:51 + 11212 + + anzn : b, which expresses b in terms of the variables 11:5527 . . ,xn called
unknowns and constants a1,a2, . , , ,an is called a linear equation.
2. A s
74931
We 1
we 5
1- ,7.4.
See Kolmanls book for the proofs of the following properties:
Properties of the Determinant
1
2.
memes
\1
8.
9.
. detA : detAT
detA
det B = det A.
ows (or two columns) of A are equal, then det A : 0/
If A has a zero row (or a ze
.
Lesson 1 Matrices Properties and Operations
Definitions. .
1. An matrix with m rows and TL columns (an m by 77. matrix) A is a rectangular array of numbers:
N N
(111 (112 1
~21 (1-12 - ' ' -21:
fl = . . i: [afjlmxn = [fl-ijl-
L .
Ilmi Hm: " ' "mu
'lhc m
I A < 3 V T
. ' . . 49
a , ,
"t
I];
. 2. Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column (111 (112 a1" aCI b1
found in Step 1. . I 6121 (122 02" $2 12 I I .
Denition. Let A : _ . . II : , Ib : .
Lesson 5 Cofactor Expansion and Applications of the Determinant 1 1 0
Example 3. Given A = 2 3 4 . Let's [ind rst all of its nine col'actors.
Denition. If A is a square matrix, then the (i,j)-minor of A denoted by MUM) (or simply Miil is defined to be 5 8
Therefore,
1 2 o 3 1
o 0 2 e
A Method for Finding the Inverse of a Nonsingular Matrix
1. Form the n X 271 augmented matrix [A : In].
4:1
1 3 :> ~1
~5 10 5
O 1 0 0
0 A? 1 0
1. 0 O 1
0 1
O 3
2 0
2
*2
~3
2. Using elementary row operations, transform this mat
. m wt
1 I 11 [zlr7KL'C/(KS
Example 8. Consider the homogeneous system 5 Find the value(s) of c for which the system
l ,
m+zg~3z3+2x4:0 1 1 73 2 ' avey = 1 C/ l
2x1 2m + 2:53 2 0 a 2 72 2 0 (coefcient matrix) SOV; Fj 01' W y = c I C N
. v1+c2~13=0 A1
a:
Lesson 3 Inverse of a Matrix
Denitions.
1. An n X n matrix A is said to be nonsingular or invertible if there exists a unique n X n matrix B such that
AB = BA ; I". The matrix B is called the inverse of A and denote it b i
2. If no such matrix B exists
i.
wmnuW ~ .-
Cramers Rule
Theorem, 1f Ax : b is a system of n linear' equations in n unknowns where .r :
detA 7 0, then the system has a unique solution given by
_ dot. A:
:t'
I detA '
where A, is the matrix obtained by replacing the entries in th
Lesson 4 Denition and Properties of Determinants
One application of determinants 18 to determine whether a linear system with n equations in n unknowns has a
unique solution We rst dene the concept of permutations before dening the detein),.inants
Denitio
Actually Isis, based on my research, the effects of student rankings in college
can be bad or good depending on a students performance. The possible
advantages could be 1) upon seeing his or her standing, a student can be
motivated to perform better to su
SSX%AG- 6 5, X'fG-(i'7 Am
a (d Gala/Ff
TGECM]
Sow-Hon 11 :7 0
1n mrpw, W be. 01st M 3 576") = cfw_X I-xz,i>, mueb
g %: IX' @,\j)e'D EX I<".)0>
9 Q4 (cm) 1)
D Boundwxcs:
(allelic) 3>l-)
M w
- a 5 c are [0. q
38:738. 24:93, l-K-ld> (19,3090
'3"qu = <I,o,-I
I. Triple Integrals
1. Evaluate the following triple integrals.
ZZZ
(a)
xyz 2 dV , where S = [0, 1] [1, 2] [0, 3]
Z SZ Z
(2x + y sin z) dV , where G = [1, 1] [0, 2] 0, 2
(b)
ZG
ZZ
(c)
xy sin yz dV , where S is bounded by the coordinate planes, x = , y =
2
Math 55 EXAM 2 October 22, 2016
Show all necessary solutions and box your nal answers. Use only black or blue non-erasable ink in writing.
Do not detach any page from your bluebooks. Follow directions carefully.
1. (8 pts.) Let 13(2, y, z) = (22:31, y3, :
MATHEMATICS 55
Third Examination
03 December 2016
Write legibly and box every nal answer. Use black or blue non-erasable ink only. Show all necessary
. solutions for parts I-VII.
. 3" .
I. Consider the sequence cfw_ (2n)! n=1. (4 paints)
1. Show that th
I. Sequences
1. Find the first five terms of the following sequences.
1
n
(a)
(d)
n
2n + 1 n=1
n=1
1
(1)n (n + 1)
(b)
2
(e)
n n=1
3n
n=0
n
(c)
(f)
n 3 n=3
n + 1 n=1
(k) cfw_f (n)n=1 , where f (n) = 1 when n is odd, f (n) =
n
n o
cos
6
n n=0
e
(h)
n