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Math100 textbook
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- University Of Johannesburg (UJ)
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INTRODUCTORY ERNEST F. HAEUSSLER JR.MATHEMATICAL RICHARD S. PAUL
RICHARD J. WOOD
ANALYSIS
FOURTEENTH EDITION
FOR BUSINESS, ECONOMICS, AND
THE LIFE AND SOCIAL SCIENCES
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561590_MILL_MICRO_FM_ppi-xxvi.indd 2 24/11/14 5:26
, ALGEBRA
Algebraic Rules for Exponents Radicals
Real numbers
p
a0 D 1 n
a D a1=n p
p
aCbDbCa . a/n D a; n an D a .a > 0/
p
n
1 p m
ab D ba a n
D .a ¤ 0/ p
n
am D . n a/
p D am=n
an n p n
a C .b C c/ D .a C b/ C c n
am an D amCn rab D p a b
a.bc/ D .ab/c a n
a
a.b C c/ D ab C ac .am /n D amn n
Dp n
.ab/n D an bn p b bp
a.b c/ D ab ac m n
p
a D mn a
.a C b/c D ac C bc a n an
.a b/c D ac bc D n
b b
aC0Da
a m Factoring Formulas
a0D0 D am n
a1Da an
a C . a/ D 0 ab C ac D a.b C c/
. a/ D a a2 b2 D .a C b/.a b/
. 1/a D a Special Products a2 C 2ab C b2 D .a C b/2
a b D a C . b/ a2 2ab C b2 D .a b/2
a .b/ D a C b x.y C z/ D xy C xz a3 C b3 D .a C b/.a2 ab C b2 /
1 .x C a/.x C b/ D x2 C .a C b/x C ab a3 b3 D .a b/.a2 C ab C b2 /
a D1
a .x C a/2 D x2 C 2ax C a2
a 1 .x a/2 D x2 2ax C a2
Da .x C a/.x a/ D x2 a2 Straight Lines
b b
.x C a/3 D x3 C 3ax2 C 3a2 x C a3 y2 y1
. a/b D .ab/ D a. b/ .x a/3 D x3 3ax2 C 3a2 x a3 mD (slope formula)
x2 x1
. a/. b/ D ab y y1 D m.x x1 / (point-slope form)
y D mx C b (slope-intercept form)
a a Quadratic Formula x D constant (vertical line)
D
b b y D constant (horizontal line)
a a a If ax2 C bx C c D 0, where
D D a ¤ 0, then
b b b
p Absolute Value
a b aCb b ˙ b2 4ac
C D xD
c c c 2a
jabj D jaj jbj
a b a b ˇaˇ
D ˇ ˇ jaj
c c c ˇ ˇD
Inequalities b jbj
a c ac
D ja bj D jb aj
b d bd
If a < b, then a C c < b C c. jaj a jaj
a=b ad If a < b and c > 0, then ja C bj jaj C jbj (triangle inequality)
D
c=d bc ac < bc.
a ac If a < b and c > 0, then
D .c ¤ 0/ a. c/ > b. c/. Logarithms
b bc
Summation Formulas logb x D y if and only if x D by
Special Sums logb .mn/ D logb m C logb n
m
P
n P
n logb D logb m logb n
cai D c ai P
n n
iDm iDm 1Dn logb mr D r logb m
P
n P
n P
n iD1
.ai C bi / D ai C bi Pn
n.nC1/
logb 1 D 0
iDm iDm iDm iD 2 logb b D 1
P
n Pm
pCn iD1
logb br D r
ai D aiCm p
Pn
n.nC1/.2nC1/
iDm iDp
i2 D 6 blogb p D p .p > 0/
pP1 iD1
P
n P
n
Pn
n2 .nC1/2 loga m
ai C ai D ai i3 D logb m D
iDm iDp iDm 4 loga b
iD1
, FINITE MATHEMATICS
Business Relations Compound Interest Formulas
Interest D (principal)(rate)(time) S D P.1 C r/n
Total cost D variable cost C fixed cost P D S.1 C r/ n
total cost r n
Average cost per unit D re D 1 C 1
quantity n
Total revenue D (price per unit)(number of units sold)
Profit D total revenue total cost S D Pert
P D Se rt
re D er 1
Ordinary Annuity Formulas
Matrix Multiplication
1.1 C r/ n
ADR D Ran r (present value)
r n
X
.1 C r/n 1 .AB/ik D Aij Bjk D Ai1 B1k C Ai2 B2k C C Ain bnk
SDR D Rsn r (future value) jD1
r
.AB/T D BT AT
A 1 A D I D AA 1
Counting 1
.AB/ D B 1A 1
nŠ
n Pr D Probability
.nr/Š
nŠ
n Cr D #.E/
rŠ.n r/Š P.E/ D
C
n 0 C n C1 C C n Cn 1 C n Cn D 2n #.S/
n C0 D 1 D n Cn #.E \ F/
P.EjF/ D
D n Cr C n CrC1 #.F/
nC1 CrC1
P.E [ F/ D P.E/ C P.F/ P.E \ F/
P.E0 / D 1 P.E/
Properties of Events P.E \ F/ D P.E/P.FjE/ D P.F/P.EjF/
For E and F events for an experiment with sample space S
For X a discrete random variable with distribution f
E[EDE
E\EDE X
.E0 /0 D E f.x/ D 1
E [ E0 D S x X
E \ E0 D ; D .X/ D E.X/ D xf.x/
E[SDS x X
E\SDE Var.X/ D E..X /2 / D .x /2 f.x/
E[;DE p x
E\;D; D .X/ D Var.X/
E[FDF[E
E\FDF\E
.E [ F/0 D E0 \ F0 Binomial distribution
.E \ F/0 D E0 [ F0
E [ .F [ G/ D .E [ F/ [ G
E \ .F \ G/ D .E \ F/ \ G f.x/ D P.X D x/ D n Cx px qn x
E \ .F [ G/ D .E \ F/ [ .E \ G/ D np
p
E [ .F \ G/ D .E [ F/ \ .E [ G/ D npq
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