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REFERENCE PAGES
ALGEBRA GEOMETRY
e Arithmetic Operations Geometric Formulas
c
n
ere a(cid:2)b(cid:3)c(cid:3)(cid:2)ab(cid:3)ac a (cid:3) c (cid:2) ad(cid:3)bc Formulas for area A, circumference C, and volume V:
ef b d bd
p for r a(cid:3)c a c ab a d ad ATri(cid:2)an12gbleh C Air(cid:2)cle(cid:4)r2 ASe(cid:2)cto12rr o2(cid:5)f Circle
e (cid:2) (cid:3) (cid:2) (cid:10) (cid:2)
nd ke b b b dc b c bc (cid:2)12ab sin (cid:5) C(cid:2)2(cid:4)r s(cid:2)r(cid:5) (cid:2)(cid:5) in radians(cid:3)
a
e
her Exponents and Radicals a h r s
ut xm ¨ r
C xmxn(cid:2)xm(cid:3)n (cid:2)xm(cid:2)n b ¨
xn r
1
(cid:2)xm(cid:3)n(cid:2)xmn x(cid:2)n(cid:2)
xn
(cid:4) (cid:5)
x n xn Sphere Cylinder Cone
(cid:2)xy(cid:3)n(cid:2)xnyn (cid:2)
y yn V(cid:2)4(cid:4)r3 V(cid:2)(cid:4)r2h V(cid:2)1(cid:4)r2h
3 3
x1(cid:8)n(cid:2)snx xm(cid:8)n(cid:2)snxm(cid:2)(snx)m A(cid:2)4(cid:4)r2 A(cid:2)(cid:4)rsr2(cid:3)h2
(cid:7)
snxy(cid:2)snxsny n x (cid:2) snx r
y sny
r h
h
Factoring Special Polynomials
r
x2(cid:2)y2(cid:2)(cid:2)x(cid:3)y(cid:3)(cid:2)x(cid:2)y(cid:3)
x3(cid:3)y3(cid:2)(cid:2)x(cid:3)y(cid:3)(cid:2)x2(cid:2)xy(cid:3)y2(cid:3)
x3(cid:2)y3(cid:2)(cid:2)x(cid:2)y(cid:3)(cid:2)x2(cid:3)xy(cid:3)y2(cid:3)
Distance and Midpoint Formulas
Binomial Theorem
Distance between P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3):
1 1 1 2 2 2
(cid:2)x(cid:3)y(cid:3)2(cid:2)x2(cid:3)2xy(cid:3)y2 (cid:2)x(cid:2)y(cid:3)2(cid:2)x2(cid:2)2xy(cid:3)y2
(cid:2)x(cid:3)y(cid:3)3(cid:2)x3(cid:3)3x2y(cid:3)3xy2(cid:3)y3 d(cid:2)s(cid:2)x2(cid:2)x1(cid:3)2(cid:3)(cid:2)y2(cid:2)y1(cid:3)2
(cid:2)x(cid:2)y(cid:3)3(cid:2)x3(cid:2)3x2y(cid:3)3xy2(cid:2)y3 (cid:4) (cid:5)
(cid:2)x(cid:3)y(cid:3)n(cid:2)xn(cid:3)nxn(cid:2)1y(cid:3) n(cid:2)n(cid:2)1(cid:3) xn(cid:2)2y2 Midpoint of P1P2: x1(cid:3)2 x2, y1(cid:3)2 y2
2
(cid:4) (cid:5)
n
(cid:3)(cid:9)(cid:9)(cid:9)(cid:3) xn(cid:2)kyk(cid:3)(cid:9)(cid:9)(cid:9)(cid:3)nxyn(cid:2)1(cid:3)yn
k
(cid:4) (cid:5) Lines
n n(cid:2)n(cid:2)1(cid:3)(cid:9)(cid:9)(cid:9)(cid:2)n (cid:2)k(cid:3)1(cid:3)
where (cid:2) Slope of line through P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3):
k 1(cid:2)2(cid:2)3(cid:2)(cid:9)(cid:9)(cid:9)(cid:2)k 1 1 1 2 2 2
y (cid:2)y
Quadratic Formula m(cid:2) 2 1
x (cid:2)x
2 1
(cid:2)b(cid:8)sb2(cid:2)4ac
If ax2(cid:3)bx(cid:3)c(cid:2)0, then x(cid:2) .
2a Point-slope equation of line through P(cid:2)x, y(cid:3)with slope m:
1 1 1
Inequalities and Absolute Value y(cid:2)y (cid:2)m(cid:2)x(cid:2)x(cid:3)
1 1
If a(cid:6)band b(cid:6)c, then a(cid:6)c.
Slope-intercept equation of line with slope mand y-intercept b:
If a(cid:6)b, then a(cid:3)c(cid:6)b(cid:3)c.
If a(cid:6)band c(cid:7)0, then ca(cid:6)cb. y(cid:2)mx(cid:3)b
If a(cid:6)band c(cid:6)0, then ca(cid:7)cb.
If a(cid:7)0, then
Circles
(cid:6)x(cid:6)(cid:2)a means x(cid:2)a or x(cid:2)(cid:2)a
(cid:6)x(cid:6)(cid:6)a means (cid:2)a(cid:6)x(cid:6)a Equation of the circle with center (cid:2)h, k(cid:3)and radius r:
(cid:6)x(cid:6)(cid:7)a means x(cid:7)a or x(cid:6)(cid:2)a (cid:2)x(cid:2)h(cid:3)2(cid:3)(cid:2)y(cid:2)k(cid:3)2(cid:2)r2
1
REFERENCE PAGES
TRIGONOMETRY
Angle Measurement Fundamental Identities
(cid:4) radians(cid:2)180(cid:11) 1 1
s csc (cid:5)(cid:2) sec (cid:5)(cid:2)
r sin (cid:5) cos (cid:5)
(cid:4) 180(cid:11)
1(cid:11)(cid:2) rad 1 rad(cid:2) ¨
180 (cid:4) sin (cid:5) cos (cid:5)
s(cid:2)r(cid:5) r tan (cid:5)(cid:2) cos (cid:5) cot (cid:5)(cid:2) sin (cid:5)
(cid:2)(cid:5) in radians(cid:3) cot (cid:5)(cid:2) 1 sin2(cid:5)(cid:3)cos2(cid:5)(cid:2)1
tan (cid:5)
Right Angle Trigonometry 1(cid:3)tan2(cid:5)(cid:2)sec2(cid:5) 1(cid:3)cot2(cid:5)(cid:2)csc2(cid:5)
opp hyp
sin (cid:5)(cid:2) hyp csc (cid:5)(cid:2) opp hyp sin(cid:2)(cid:2)(cid:5)(cid:3)(cid:2)(cid:2)sin (cid:5) cos(cid:2)(cid:2)(cid:5)(cid:3)(cid:2)cos (cid:5)
opp (cid:4) (cid:5)
adj hyp (cid:4)
cos (cid:5)(cid:2) sec (cid:5)(cid:2) ¨ tan(cid:2)(cid:2)(cid:5)(cid:3)(cid:2)(cid:2)tan (cid:5) sin (cid:2)(cid:5) (cid:2)cos (cid:5)
hyp adj 2
adj (cid:4) (cid:5) (cid:4) (cid:5)
opp adj
tan (cid:5)(cid:2) cot (cid:5)(cid:2) (cid:4) (cid:4)
adj opp cos (cid:2)(cid:5) (cid:2)sin (cid:5) tan (cid:2)(cid:5) (cid:2)cot (cid:5)
2 2
Trigonometric Functions
The Law of Sines B
sin (cid:5)(cid:2) y csc (cid:5)(cid:2) r y
r y sin A (cid:2) sin B (cid:2) sin C a
(x, y) a b c
x r
cos (cid:5)(cid:2) sec (cid:5)(cid:2) r C
r x c
tan (cid:5)(cid:2) xy cot (cid:5)(cid:2) yx ¨ x The Law of Cosines b
a2(cid:2)b2(cid:3)c2(cid:2)2bc cos A
Graphs of Trigonometric Functions b2(cid:2)a2(cid:3)c2(cid:2)2ac cos B
y y y y=tan x c2(cid:2)a2(cid:3)b2(cid:2)2ab cos C A
y=sin x y=cos x
1 1
π 2π 2π Addition and Subtraction Formulas
x π 2πx π x sin(cid:2)x(cid:3)y(cid:3)(cid:2)sin x cos y(cid:3)cos x sin y
_1 _1 sin(cid:2)x(cid:2)y(cid:3)(cid:2)sin x cos y(cid:2)cos x sin y
cos(cid:2)x(cid:3)y(cid:3)(cid:2)cos x cos y(cid:2)sin x sin y
y y=csc x y y=sec x y y=cot x cos(cid:2)x(cid:2)y(cid:3)(cid:2)cos x cos y(cid:3)sin x sin y
tan x(cid:3)tan y
tan(cid:2)x(cid:3)y(cid:3)(cid:2)
1 1 1(cid:2)tan x tan y
tan x(cid:2)tan y
π 2πx π 2πx π 2πx tan(cid:2)x(cid:2)y(cid:3)(cid:2) 1(cid:3)tan x tan y
_1 _1
Double-Angle Formulas
sin 2x(cid:2)2 sin x cos x
Trigonometric Functions of Important Angles cos 2x(cid:2)cos2x(cid:2)sin2x(cid:2)2 cos2x(cid:2)1(cid:2)1(cid:2)2 sin2x
(cid:5) radians sin (cid:5) cos (cid:5) tan (cid:5) 2 tan x
tan 2x(cid:2)
0(cid:11) 0 0 1 0 1(cid:2)tan2x
30(cid:11) (cid:4)(cid:8)6 1(cid:8)2 s3(cid:8)2 s3(cid:8)3
45(cid:11) (cid:4)(cid:8)4 s2(cid:8)2 s2(cid:8)2 1 Half-Angle Formulas
60(cid:11) (cid:4)(cid:8)3 s3(cid:8)2 1(cid:8)2 s3 1(cid:2)cos 2x 1(cid:3)cos 2x
90(cid:11) (cid:4)(cid:8)2 1 0 — sin2x(cid:2) 2 cos2x(cid:2) 2
2
Calculus
Concepts and Contexts | 4e
Calculus and the Architecture of Curves
The cover photograph shows the
DZ Bank in Berlin, designed and
built 1995–2001 by Frank Gehry
and Associates. The interior atrium
is dominated by a curvaceous four-
story stainless steel sculptural ehry
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shell that suggests a prehistoric O.
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cferereantucree s apnadc eh.ouses a central con- esy of Fra
The highly complex structures urt
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that Frank Gehry designs would be
impossible to build without the computer.
The CATIA software that his archi-
tects and engineers use to produce the
computer models is based on principles of hry
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calculus—fitting curves by matching tangent O. G
lines, making sure the curvature isn’t too nk
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large, and controlling parametric surfaces. of Fr
“Consequently,” says Gehry, “we have a lot esy
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of freedom. I can play with shapes.” Co
The process starts with Gehry’s initial
sketches, which are translated into a succes-
sion of physical models. (Hundreds of different
physical models were constructed during the design
of the building, first with basic wooden blocks and then
evolving into more sculptural forms.) Then an engineer
uses a digitizer to record the coordinates of a series of
points on a physical model. The digitized points are fed
into a computer and the CATIA software is used to link hry
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these points with smooth curves. (It joins curves so that O.
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their tangent lines coincide; you can use the same idea to a
design the shapes of letters in the Laboratory Project on esy of Fr
page 208 of this book.) The architect has considerable free- urt
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curve, its derivative, and its curvature. Then the curves are
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connected to each other by a parametric surface, The CATIA program was developed in France
and again the architect can do so in many possible by Dassault Systèmes, originally for designing
ways with the guidance of displays of the geometric airplanes, and was subsequently employed in
characteristics of the surface. the automotive industry. Frank Gehry, because of
The CATIA model is then used to produce his complex sculptural shapes, is the first to use
another physical model, which, in turn, suggests it in architecture. It helps him answer his ques-
modifications and leads to additional computer tion, “How wiggly can you get and still make a
and physical models. building?”
Calculus
Concepts and Contexts | 4e
James Stewart
McMaster University
and
University of Toronto
Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . UnitedKingdom . United States
Calculus: Concepts and Contexts, Fourth Edition © 2010, 2005Brooks/Cole, Cengage Learning
James Stewart
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