Biocalculus: Calculus for Life Sciences

898 Pages·2014·39.656 MB·English

by  James Stewart, Troy Day

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RefeRence page 1
e alGebra Geometry
c
n
e
er Arithmetic Operations Geometric Formulas
ef
p for r asb1cd−ab1ac ab 1 dc − adb1dbc Formulas for area A, circumference C, and volume V:
e Triangle Circle Sector of Circle
e and ke a1b c − ab 1 bc abc − ab 3 dc − abdc A −−1212abbh sin (cid:30) C A−−2(cid:31)(cid:31)rr2 A s−−12r(cid:30)r 2 (cid:30)s(cid:30) in radiansd
her d
ut
C Exponents and Radicals a h r s
¨ r
xmxn−xm1n xm −xm2n b ¨
xn r
1
sxmdn−xmn x2n−
xn
S D Sphere Cylinder Cone
x n xn
sxydn−xnyn y − yn V−4(cid:31)r3 V−(cid:31)r2h V−1(cid:31)r2h
3 3
x1yn−snx xmyn−snxm−ssnxdm A−4(cid:31)r2 A−(cid:31)rsr21h2
Îx snx r
snxy−snxsny n y − ny
s
r h
h
Factoring Special Polynomials r
x22y2−sx1ydsx2yd
x31y3−sx1ydsx22xy1y2d
x32y3−sx2ydsx21xy1y2d Distance and Midpoint Formulas
Distance between P1sx1, y1d and P2sx2, y2d:
Binomial Theorem
sx1yd2−x212xy1y2 sx2yd2−x222xy1y2 d−ssx22x1d21sy22y1d2
sx1yd3−x313x2y13xy21y3 S D
x 1x y 1y
sx2yd3−x323x2y13xy22y3 Midpoint of P1P2: 1 2 2, 1 2 2
nsn21d
sx1ydn−xn1nxn21y1 xn22y2
2
S D
n Lines
1…1 xn2kyk1…1nxyn211yn
k
SnD nsn21d … sn2k11d Slope of line through P1sx1, y1d and P2sx2, y2d:
where k − 1?2?3?…?k m− y22y1
x 2x
2 1
Quadratic Formula
Point-slope equation of line through P1sx1, y1d with slope m:
2b6sb224ac
If ax21bx1c−0, then x− 2a . y2y1−msx2x1d
Slope-intercept equation of line with slope m and y-intercept b:
Inequalities and Absolute Value
If a,b and b,c, then a,c. y−mx1b
If a,b, then a1c,b1c.
If a,b and c.0, then ca,cb. Circles
If a,b and c,0, then ca.cb.
Equation of the circle with center sh, kd and radius r:
If a.0, then
|x|−a means x−a or x−2a sx2hd21sy2kd2−r2
|x|,a means 2a,x,a
|x|.a means x.a or x,2a
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RefeRence page 2
tRigonometRy
Angle Measurement Fundamental Identities
(cid:30) radians−1808 1 1
(cid:30) 1808 r s csc (cid:31)− sin (cid:31) sec (cid:31)− cos (cid:31)
18− rad 1 rad−
180 (cid:30) ¨ sin (cid:31) cos (cid:31)
tan (cid:31)− cot (cid:31)−
s−r(cid:31) r cos (cid:31) sin (cid:31)
s(cid:31) in radiansd cot (cid:31)− 1 sin2(cid:31)1cos2(cid:31)−1
tan (cid:31)
11tan2(cid:31)−sec2(cid:31) 11cot2(cid:31)−csc2(cid:31)
Right Angle Trigonometry
sins2(cid:31)d−2sin (cid:31) coss2(cid:31)d−cos (cid:31)
opp hyp
sin (cid:31)− haydpj csc (cid:31)− ohpypp hyp opp tans2(cid:31)d−2tan (cid:31) sinS(cid:30)2 2(cid:31)D−cos (cid:31)
cos (cid:31)− sec (cid:31)−
hyp adj S D S D
¨ (cid:30) (cid:30)
opp adj cos 2(cid:31) −sin (cid:31) tan 2(cid:31) −cot (cid:31)
tan (cid:31)− cot (cid:31)− adj 2 2
adj opp
The Law of Sines
Trigonometric Functions B
sin (cid:31)− y csc (cid:31)− r y sin A − sin B − sin C a
r y a b c
x r (x, y) C
cos (cid:31)− sec (cid:31)− r c
r x
tan (cid:31)− y cot (cid:31)− x ¨ The Law of Cosines b
x y x
a2−b21c222bc cos A
Graphs of Trigonometric Functions b2−a21c222ac cos B
A
c2−a21b222ab cos C
y y y y=tan x
y=sin x y=cos x
1 1 Addition and Subtraction Formulas
π 2π 2π
x π 2πx π x sinsx1yd−sin x cos y1cos x sin y
_1 _1 sinsx2yd−sin x cos y2cos x sin y
cossx1yd−cos x cos y2sin x sin y
y y=csc x y y=sec x y y=cot x cossx2yd−cos x cos y1sin x sin y
tan x1tan y
tansx1yd−
1 1 12tan x tan y
tan x2tan y
π 2πx π 2πx π 2πx tansx2yd− 11tan x tan y
_1 _1
Double-Angle Formulas
sin 2x−2 sin x cos x
Trigonometric Functions of Important Angles
cos 2x−cos2x2sin2x−2 cos2x21−122 sin2x
(cid:31) radians sin (cid:31) cos (cid:31) tan (cid:31) 2 tan x
tan 2x−
08 0 0 1 0 12tan2x
308 (cid:30)y6 1y2 s3y2 s3y3
458 (cid:30)y4 s2y2 s2y2 1 Half-Angle Formulas
608 (cid:30)y3 s3y2 1y2 s3 12cos 2x 11cos 2x
908 (cid:30)y2 1 0 — sin2x− 2 cos2x− 2
Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Biocalculus
Calculus for the Life Sciences
Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
About the Cover Images
Spotted owl populations are analyzed
The doubling time of a population of the
using matrix models (Exercise 8.5.22).
bacterium G. lamblia is determined in
Exercise 1.4.29.
The fitness of a garter snake is a function The Speedo LZR Racer reduces drag
of the degree of stripedness and the in the water, resulting in dramatically
number of reversals of direction while improved performance. The project
fleeing a predator (Exercise 9.1.7). on page 603 explains why.
The project on page 297 asks how birds In Example 9.4.2 we use the Chain
can minimize power and energy by Rule to discuss whether tuna biomass
flapping their wings versus gliding. is increasing or decreasing.
The population size of some species, like this
The optimal foraging time for bumblebees
sea urchin, can be measured by evaluating a
is determined in Example 4.4.2.
certain integral, as explored in Exercise 5.3.49.
The interaction between Daphnia
The vertical trajectory of zebra finches is
and their parasites is analyzed in
modeled by a quadratic function (Figure 1.2.8).
Case Study 2 (page xlvi).
The size of the gray-wolf population depends
Populations of blowflies are modeled
on the size of the food supply and the
by chaotic recursions (page 430).
number of competitors (Exercise 9.4.21).
The energy needed by an iguana to
Example 4.4.4 investigates the time
run is a function of two variables,
that loons spend foraging.
weight and speed (Exercise 9.2.47).
Dinosaur fossils can be dated using The area of a cross-section of a human
potassium-40 (Exercise 3.6.12). brain is estimated in Exercise 6.Review.5.
The project on page 222 illustrates how The project on page 479 determines
mathematics can be used to minimize the critical vaccination coverage
red blood cell loss during surgery. required to eradicate a disease.
Natural killer cells attack pathogens and
Jellyfish locomotion is modeled by a are found in two states described by a pair
differential equation in Exercise 10.1.34. of differential equations developed in
Section 10.3.
In Example 4.2.6 a junco has a choice
The screw-worm fly was effectively
of habitats with different seed densities
eliminated using the sterile insect
and we determine the choice with
technique (Exercise 5.6.24).
the greatest energy reward.
The growth of a yeast population leads The project on page 467 investigates
naturally to the study of differential logarithmic spirals, such as those
equations (Section 7.1). found in the shell of a nautilus.
Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Biocalculus
Calculus for the Life Sciences
James Stewart
McMaster University and University of Toronto
Troy Day
Queen’s University
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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James Stewart, Troy Day
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To Dolph Schluter and Don Ludwig, for early inspiration
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