INDEX

Absolute value function, 19, 400

Acceleration, 110

average, 254

instantaneous, 254

Accumulated change, 357

calculating, 380

Acetaminophen

clearance of, 226–227

geometric decay of, 87–88

Addition law, 39

Aerobic rate, 289

Agnesi, M.G., 264

Algebraic equations, 141

Allee effect, 467–468

Allometric scaling, 44–45

Amplitude, 31

Ant biodiversity, 186–187. See also Derivatives as functions

Antiderivatives, 396

of acceleration, 353

for cos(ax), 350

definition, 348

evaluation theorem and, 379–380

general forms of, 349

initial value problem, 350–351

rules, 350

with slope fields, 353

stooping behavior of peregrine falcons, 354–355

of velocity, 353

Antidifferentiation, 348

Archimedes, 357

Arctic fin whale, sustainable exploitation of, 302–303

Area under curve, 357–358

estimating, 362–364

exact, under x² on the interval [0, 1], 364–365

finding, 347

mortality rates due to plague, 360–362

for nonnegative continuous function, 369

of various rates associated with disease, 360

Arithmetic, functional, 62

Arithmetic

mean, 540

sequence, 87

Arteriosclerosis, 230, 230f

Asymptotes

linear, 282–283

vertical, 279

Asymptotes and infinity

difference of infinities, 148

dose–response curves, 147–148

horizontal asymptotes, 145–148

infinite limits at infinity, 151–153

informal definition, 149

overview, 144

unabated population growth, 153

vertical asymptotes, 149–151

wolves eating moose, 150–151

Autonomous differential equation, 432, 476

first-order, 485

Autonomous models, 468

Average, 540

acceleration, 254

Mexico population growth, 108–109

rate of change, 108. See also Rates of change and tangent lines

velocity, 110, 112

Axon, 497

B

Back substitution, 605

Beetle, developmental rates of, 243

Bellows, T. S., 330

Bell-shaped curve, 223

Bell-shaped distributions

life expectancy function, 579–582

logistic distribution, 556–563

lognormal distribution, 567–572

normal distribution, 563–565

reproductive success, 582–585

survivorship function, 575–579

Bernoulli, Jacob, 53

Best-fitting line, 27

Beverton-Holt

dynamics, 94–95, 164, 171

model, 331

sockeye salmon dynamics, 164–166

Bicorn curve, 231

Bifurcation diagram, 493

for a model of an autocatalytic gene, 496–498

pitchfork bifurcation, 495

sudden disappearances of queen conchs, 493–495

Bifurcation theory, 490–491

Bighorn ram's horn, calculating length of a, 381

Binomial theorem, 200

Bins, 508

Biological switches, 495–500

Biopharmaceutics models, 456

Bird diversity in oak woodlands, histogram for, 509–510

Birth time quartiles, 521

Bison abundance, 241–242, 241f, 241t. See also Linear approximation

Bivariate data, 25

Blood-brain barrier, morphine across, 643–644

Blue sky catastrophe, 494

Body mass index (BMI), 219, 248

Bolt, U, 111, 254–256

Bounding property of integral, 375

Briere model, 275

C

Calculus, defined, 5–6

Calment, Jeanne-Louise, 520

Calorie(s)

in a candy bar, estimate of, 423–424

defined, 422

used while working out, estimating, 423

Carbon dioxide

concentrations in Hawaii, 29–30

data, fitting, 32–33

global extrema with, 294–295

output from electric power plants, 27–29

Carbon monoxide pollution, 221

Cardiac output, estimating, 420–421

Carlson's data for exponential growth model, 435–437

Carrying capacity, 438

Chain rule, 221–222

bell-shaped curve, 223

with graphs, 223–224

modeling problems using, 228–229

parasitoids, 222–223

Chebyshev, Pafnuty, 552

Chebyshev's inequality, 552–553

Circadian rhythms, 235–236

Circular tangents, 224–225

Cissoid of Diocles, 230

Clark, Colin W., 440

Clearance rate constant, determining, 458

Closed interval method, 12, 294

Cobwebbing, 95–97, 334

hump-shaped function, 97

solutions, 166, 167

square roots, 96

Codling moth, search period of, 295–296

Colson, J., 264

Comparison test, 532–533

Compartment models

blood-brain barrier, morphine across, 643–644

morphine-6-glucuronide dynamics from clinical studies, 644–645

Composite function, defined, 65

Composition limit law, 136

Compounded interest, continuously, 53–54

Compression, 60–61

Concave down, 252

Concave up, 252

Concavity, 252–253

intervals of, 279

Constant

harvesting, 440

of proportionality, 38

per capita growth rate model, 433–434

qualitative behavior of, 434–435

Continuity,

at a point, 135–136. See also Limit laws and continuity

laws, 137

of elementary functions theorem, 137

sequential limits, 155–156

Continuous

function, 365

defined, 382

random variable, 514–515

Contour

line, 596

map, 596–597

contour plot of Mount Eden, 599–600

of genetic circuit, 598–599

Convergence

of doubly improper integrals, 535–536

tests, 532–534

Convergent, 527–528

Cosine functions, 31

approximating, 258–259

derivatives of, 232–233 (See also Derivatives of trigonometric functions)

Cowie, Richard, 320

Crab harvests, estimating, 411–413

Critical value, 291

Cumulative distribution function (CDF)

birth time distribution, 516–517

of drug decay, 518–519

exponential, 518–519

Laplace, 537

for Mediterranean fruit fly, 519–520

properties, 517

of a random variable, 516

D

Data fitting, 25–38

carbon dioxide, 32–33

concentrations in Hawaii, 29–30

output from electric power plants, 27–29

fitting linear equations to data, 27–30

linear functions, 26–27

periodic and trigonometric functions, 30–31

periods and amplitudes, estimating, 31–32

slope of line, 26

Decay of beer froth, modeling, 52–53

Decisions and optimization. See also Differentiation, applications of

marginal value theorem, 319–320

Northwestern crows and whelks, 316–317

optimal decisions, 315–318

foraging and marginal value, 318–319

time for producing seeds, 317–318

time to harvest, 322–323

plantation, optimal rotation period for, 323–324

Decomposition of functions, 66

Definite integral, 9

and integration by parts, 397

approximating integrals with sums, 371

as a limit of a Riemann sum, 370

at a point, 375

computation using summation formula, 371–372

defined, 369

geometrical meaning of, 372–374, 384

of dx rule, 372–373

properties of, 373–375

signed area interpretation of, 373

substitution for, 390–391

sum as, 370

using signed areas, 373

Degree-days, 358

accumulation,

for seedless grapes, 383–384

for sweet corn, 359–360

calculating, using fundamental theorem, 383

of grapes to ripen after pollination, 374–375

under constant temperature, 358–359

Dendrites, 497

Dependent variables, 26, 592

Depolarized equilibrium, 484

Derivative, 5, 7–8

Derivative

at point, 171–172

absolute value functions, 178

algebric steps, 173–174

differentiability and continuity, 177–180

example, 172–173

instantaneous rates of change, 175–176

instantaneous velocity, 176

melting arctic sea ice, 177

multiple intersections, 174–175

slopes of tangent lines, 174

tangent line to parabola, 174

as a function

ant biodiversity, 186–187

defined, 182–183

derivatives and graphs, 190–192

estimation using table, 184–185

examples, 183–184, 186, 187–188, 191–192

f′, signs of, 190–191

foraging for food, 188–190

mean value theorem, 187–188

notational alternatives, 185–186

Derivatives of polynomials and exponentials

derivatives of sums, differences, and scalar multiples, 201

elementary differentiation rules, 201–203

example, 201–202

exponential depletion of resources, 206–207, 207f

exponentials, derivatives of, 204–206

fetal heart, growth of, 202–203

human immunodeficiency virus, clearance of, 205–206

natural exponential, 205

power rule, 200–201

weight lifting, 203

of a logarithm. See also Implicit differentiation

clearance of acetaminophen, 226–227

modeling using chain rule, 228–229

power law for positive real numbers, 227

of trigonometric functions, 232–237

circadian rhythms, 235–236

periodic populations, 236

rate of change of carbon dioxide, 234–235

sine and cosine, 232, 232f

trigonometric limits, 233–234

rules for sine and cosine, 233

Determinant, 612–613

of a 2-by-2 matrix, 612

Developmental rate (percent completed per day), 243–244, 249

Developmental thresholds, 348

Difference equations, 87–97

asymptotic behavior of, 158

Beverton-Holt dynamics, generalized, 94–95

cobwebbing, 95–97

drug delivery, 89–91

equilibrium, finding, 93–94

geometric decay of acetaminophen, 87–88

implicit in taking repeated square roots, 88–89

lethal recessive genes, 91–93

Differences, 132

derivatives of, 201

Differential calculus, 5, 8, 347

Differential equations. see also linear models in biology

analytical approach to, 434

constant per capita growth rate model, 433–434

Euler's method, 469–472

exponential

growth model, 435–437

population growth and decay, 433

finding and plotting solutions, 448–449

Leibniz's notation, 185, 352

logistic growth, 437–440

phase lines, 476–482

slope fields, 464–469

solutions to, 445–447

solving a separable, 448

verifying

a function, 445–446

an implicit solution, 447

Weber-Fechner, 352

Differentiation, applications of

with carbon dioxide, 294–295

closed interval method, 294

codling moth, search period of, 295–296

critical points, 291

decisions and optimization, 315–324

dose-response curves, 285–286

equilibrium stability, 327

extrema, finding, 290

Fermat's theorem, 290–291

finding, 294

first derivative test, 291

global extrema, 293–299

graphing

families of functions, 284–285

using calculus, 278–284

graphs, properties of, 279

insect population dynamics, stability of, 330–332

lethal genes, 310–311

linear

asymptotes, 282–283

difference equations, stability of, 328

linearization and difference equations, 327–341

local extrema, 289–293

logistic equation, 328

marginal value theorem, 319–320

maximum economic yield, 300–302

monotone difference equations, stability of, 332–333

Newton's method, 335–337, 336f

northwestern crows and whelks, 316–317

open interval method, 296–297

optimal

decisions, 315–318

foraging and marginal value, 318–324

time for producing seeds, 317–318

time to harvest, 322–323

optimization in biology, 299–307

plantation, optimal rotation period for, 323–324

regression, 307–311

residual sum-of-squares, 308

second derivative test, 293

sickle cell alleles, fate of, 334–335

stability, graphical approach to, 333–334

stability via linearization, 329–330

stock-recruitment curves, 286–287

sustainable exploitation of Arctic fin whale, 302–303

thermodilution, 292–293

tumor growth, 305–306

tumor regrowth, time to, 337–338

tylenol in blood stream, 283–284

vascular branching, 306–307

“W” shaped curve, 280–282

Dirichlet function, 15

Discrete

logistic equation, 328

logistic model, dynamics of, 162–164

random variable, 510

Distributive

law over division, 39

law over multiplication, 39

Divergent, 527–528

Domain of function, 16, 24, 592

Dominance property of integral, 375, 406

Doomsday prediction, 450–451

Dose-response curves, 147–148, 285–286

Doubling time for yeast, 437

Doubly improper integrals

convergence of, 535–536

Doubly infinite integrals, 534

Dove strategy, 480

Drug delivery, 89–91

dx rule, 372–373

Dye dilution, estimating, 421

E

e, see Euler's number

Eigenvalues, 591, 619–620, 619–625

using two-step method, 625–626

Eigenvectors, 591, 619–620, 619–625

using two-step method, 625–626

Einstein, Albert, 507

Elasticity, 247–248. See also Linear approximation

of metabolic rates, 247

Elementary differentiation rules, 201. See also Derivatives of polynomials and exponentials

example, 201–202

fetal heart, growth of, 202–203, 202f

weight lifting, 203

Elementary functions, 404

Epsilon-delta definition of limit, 127

example, 128

Equilibria, 476

classifying, 482–484

in phase-plane, 661

Equilibrium

finding, 93–94

solutions, 439

stability, 327

Error analysis, 245–246. See also Linear approximation

metabolic rate estimation, 246

and sensitivity, 246

Error

function, 404, 405–406

propagation, 246

Escape function, 222–223, 271

Essential discontinuity, 136

Euler, Leonhard, 53

Euler's method, 432, 469–472

effect of the choice of h, 471–472

for two time steps h, 471

Euler's number e, 53, 54

Euler's number e, 53, 54

exponential decay, 51–52

and exponential functions, 48–49, 48t

future value, 54

Malthus's estimate, 49–50

Malthus's law of misery, 51

Naïve approach to solving exponential equation, 55

versus polynomial growth, 50–51

Evaluation theorem, 379–380, 381, 406

Evolutionary game theory, 479

Excitable systems, 665–668

Exhaustion, method of, 357

Exponential

CDF, 518–519, 531

decay, 51–52

depletion of resources, 206–207

equations, 76–77

growth

continuously compounded interest, 53–54

decay of beer froth, modeling, 52–53

growth model, 435

Carlson's data, 435–437

doubling time for yeast, 437

model, 4

Exponential versus arithmetic growth, 267

Exponentials and power functions, 79–80

Exponentials, derivatives of, 204–205. See also Derivatives of polynomials and exponentials

exponential depletion of resources, 206–207

human immunodeficiency virus, clearance of, 205–206

natural exponential, 205

F

Fastest humans, 111–112

Fermat's theorem, 290–291

Fetal echocardiogram, 202

Fetal heart, growth of, 202–203

Fibonacci and growth rate of rabbits, 168–169

Fire ant damages, estimation of, 418–419

First derivative test, 291

First derivatives, 261

First element, 605

First-order

approximation for solution, 484–485

autonomous differential equation, 485

ordinary differential equations, 432

First percentile, 520

FitzHugh-Nagumo model, 665

Floor function, 124

Folium of Descartes, 230

Force of infection, 671

Forensic medicine, 460–461

Fourier series representation, 232

Friend's rule, 24

Frost, B. W., 125

Function(s), 13–15. See also Real numbers and functions

classifying, 19–20

increasing and decreasing, 19

piecewise-defined, 18

Function building

adding, 62

combining functions, 62

composing, 65, 65f

decomposing functions, 66

dividing, 62

functional arithmetic, 62

horizontal shifts, 58

Michaelis-Menten glucose uptake rate, 63–64

multiplying, 62

polynomials, 62

rational functions, 62

reflections, 59–60

shifting graphs, 59

short-tailed shrews exploiting cocoons, 66–67

stretching and compressing, 60–61

substracting, 62

tidal movements, modeling, 61–62

vertical shifts, 58

Function of two variables, 592, 594–597

graph of, 595

surface plots of the functions, 595–596

Functional response, 66

Fundamental theorem of calculus (FTC)

accumulated change, calculating, 380

calculation of accumulation of degree-days, 383–384

derivatives, 382–383

evaluation

notation, 380

theorem, 379–380

length of a bighorn ram's horn, calculating, 381

Fundamental theorem of integral calculus, 381–382

derivatives via, 382–383

Future value, 54. See also Exponential growth

G

Ganz, H., 212

General transitive property, 41

Genes, lethal, 310–311

Genetic circuit, contour plot of, 598–599

Genetic toggle switch for bacterium Escherichia coli, 668–670

Geometric sequence, 87

Geometric similarity, 40–41

Global extrema, 293–299. See also Differentiation, applications of

with carbon dioxide, 294–295

closed interval method, 294

codling moth, search period of, 295–296

finding, 294

open interval method, 296–297

Global warming, limiting, 141–142

Grant, W. B., 181

Graphing using calculus, 278–284. See also Differentiation, applications of

dose-response curves, 285–286

graphing families of functions, 284–285

graphs, properties of, 279

linear asymptotes, 282–283

stock-recruitment curves, 286–287

tylenol in blood stream, 283–284

“W” shaped curve, 280–282

Graphs

algebraic expressions to, 16

chain rule with, 223–224

derivatives and, 190–192 (See also Derivatives as functions)

shifting, 59

properties of

intervals of concavity, 279

intervals of increase and decrease, 279

vertical asymptotes, 279

x and y intercepts, 279

verbal descriptions to, 16–17

Growth rate model, constant per capita, 433–434

H

Haldane, John B. S., 40

Harvest, optimal time to, 322–323

Harvesting

constant, 440

effort variable, 440

proportional, 440

of queen conchs, 440–441, 491–492

Hawk–Dove game, 479, 486

dynamics of, 481–482

Hawk strategy, 480

Heaviside “cover-up” method, 399

Heisenberg's uncertainty principle, 507

Higher derivatives and approximations, 251–262

average acceleration, 254

concavity, 252–253

cosine, approximating, 258–259

examples, 254–256

instantaneous acceleration, 254

quadratic approximation, 258

rates of change, changing, 254

second derivatives, 251–252

second-order approximations, 257–260

sigmoidal decay in deaths, 253

Histograms

bird diversity in oak woodlands, 509–510

defined, 508

for percentage of body fat, 511–512

and probabilities, 508–522

roulette game, probability distribution of, 510–511

for species diversity, 511

survivorship, 519–520

for wheat yield, 564–565

for winnings in a game, 548–549

HIV levels in infected hosts, 455–456

Homogeneous linear differential equation, 648

Horizontal asymptote, 145. See also Asymptotes and infinity

dose-response curves, 147–148

finding, 145–147

and indeterminate form ∞⁰, 272–273

Horizontal line test, 72

Human immunodeficiency virus, clearance of, 205–206

Human population growth

data from 1000 AD until 2010 AD, 450

predatory impacts on, 593–594

United States population growth, calculation of, 391–392, 450

Hyperbolic secant function, 361

I

Impala population growth, model for, 620–622

Implicit differentiation, 224–229

circular tangents, 224–225

derivatives of logarithms, 226–229

limaçon of Pascal, 225

Implicitly defined solutions, 449–450

Improper integrals, 510

convergent, 527–528

divergent, 527–528

one-sided, 526–531

p-integrals, 528–529

two-sided, 534–537

Incidence

function, 209

rate, 8

Income tax rates, 18–19

Indefinite integrals, 384

substitution for, 386–387

Independent variables, 26, 592

Indeterminate form

∞⁰, 272–273

0^−∞, 271

0/0, 265, 267. See also l'Hôpital's rule

Infinite

integrals, doubly, 534

intervals, 12

limits at infinity, 151. See also Asymptotes and infinity

example, 152–153

unabated population growth, 153

∞/∞ indeterminate form, 265. See also l'Hôpital's rule

series, 510

Inflection point, 253

Infusion rate, determining, 456–458

Inhibitory neurotransmitters, 498

Inhomogeneous linear differential equation, 648

solutions to, 650–653

Initial value problem, 350–351

Instantaneous

acceleration, 254

per capita growth rate, 434

rate of change, 109, 175. See also Derivative at point; Rates of change and tangent lines

of carbon dioxide, 109–110

as derivative, 176

of a function, 347

Instantaneous velocity, 110

of falling object, 112–113

Integers, 11

Integral calculus, 5, 347

application, 347

approximating integrals with sums, 371

computation using summation formula, 371–372

difference rule, 374

dx rule, 372–373

scalar rule, 374

signed area interpretation of integrals, 373

Integrand, 369

Integration, 8–9

applications of, 417–426

combining substitution with, 397

of integrand, 395

interval of, 369

lower limits of, 369

numerical, 403–404

by parts, 394–396

with definite integral, 397

rate of airflow into lungs, estimation of, 392

of a rational function, 399–400

by substitution, 388–389

United States population growth, calculation of, 391–392

upper limits of, 369

using technology, 390

variable of, 369

Intermediate value theorem, 138–139. See also Limit laws and continuity

bisection method, 140

Interval of integration, 369

Intervals

of concavity, 279

of increase and decrease, 279

of real numbers, 12

Intrinsic growth rate, 434, 438

Inverse functions, 71–75. See also Logarithms

finding, 73–74

graphing, 74–75

horizontal line test, 72

one-to-one function, 71–72

using definitions of, 73

Irrational numbers, 11, 12

J

Jump discontinuity, 136

K

KISS principle, 433

L

Lake pollution, determining, 458–460, 469

Laplace, Pierre-Simon, 507

Laplace distribution, 536–537

Laws

of exponents, 39–40

of logarithms, 76

Left endpoint approximation, 404–405

using technology, 405–406

Left endpoint rule, 404, 406–407, 412

Leibniz, G. W., 107

Leontief input-output theory, 615

Lethal

recessive genes, 91–93

recessives revisited, 159–160

Level curve, 596

L'Hôpital's rule, 265–270

example, 266–267

exponential versus arithmetic growth, 267

∞/∞ indeterminate form, 265

logistic growth function and indeterminate form wvw, 268–269

theorem, 266

0/0 indeterminate form, 265

Life expectancy function, 579–582

of A. sarcophagus, 580

men vs women, 580–581

Life expectancy theorem, 580

Limaçon of Pascal, 225. See also Derivatives of logarithms

Limit laws and continuity

bisection method, 140–141

compositional limits, 136

continuity at a point, 135–136

continuity laws, 137

continuity of elementary functions theorem, 137

example, 135–136, 140–141

global warming, limiting, 141–142

intermediate value theorem, 138–139

properties of limit, 131–132

quick limits, 137–138

Squeeze theorem, 134

use of, 132–133

Limit(s), 5

computational dilemma, 122–123

definition, 118, 126–127

epsilon-delta proof of limit statement, 127, 128

failures, 121–122

finding, 119–120

floor function, 124–125

formal definition, 126–127

informal definition, 118

limit of constant, 128

nonexistence of, 158

of difference equations, 161 (See also Sequential limits)

of form 0⁰, 270–271

of polynomials, 132

of sequence, finding, 158–159

one-sided, 124

type I functional response, 125–126

Linear approximation, 239

bison abundance, 241–242

elasticity, 247–248

error analysis, 245–246

metabolic rate estimation, 246

metabolic rates, elasticity of, 247

sensitivity, error analysis and, 246

sin x, 243

spider mite destroyer, developmental rates of, 243–245

tangent line, approximating, 240–245

Linear asymptotes, 282–283

Linear differential equation, 454

compartment models, 642–645

with constant input, 648–650

homogeneous, 648

inhomogeneous, 648

solutions of two-dimensional systems of, 639

solving, 639–641

stable state distribution and long term behavior, 646–648

tumor growth modeled by, 637–639, 641–642

Linear equation, 562

to data, fitting, 27–30

no solutions vs infinitely many solutions, 610–611

Linear function, 25, 26–27, 113, 228. See also Data fitting

Linear model, 4

Linear models in biology

mixing models, 454–460

Newton's law of cooling, 460–461

organismal growth, 461–462

Linear multivariate model, 603–604

Linear regression, 25

on logarithmic scale, 80–81

Linear regression analysis of chirping, 597–598

Linearization, 240, 484–487

of first-order autonomous differential equation, 485

Hawk–Dove game, 486

of membrane voltage model, 487

Linearization and difference equations. See also Differentiation, applications of

equilibrium stability, 327

insect population dynamics, stability of, 330–332

linear difference equations, stability of, 328

logistic equation, 328, 328f

monotone difference equations, stability of, 332–333

Newton's method, 335–337

sickle cell alleles, fate of, 334–335

stability, graphical approach to, 333–334

stability via linearization, 329–330

tumor regrowth, time to, 337–338

Local extrema, 289–293. See also Differentiation, applications of

critical points, 291

Fermat's theorem, 290–291

finding, 290

first derivative test, 291

second derivative test, 293

thermodilution, 292–293

Local maximum, 289, 293, 601

Local minimum, 289, 293, 601

Logarithmic scale, 78

Logarithms, 75–80, 271, 400. See also Inverse functions

derivatives of (See Derivatives of logarithms)

exponential equations, 76–77

exponentials and power functions, 79–80

graphing, 76

half-life and doubling time, 77–78

laws of, 76

linear regression on logarithmic scale, 80–81

logarithmic scale, 78–79

Loggerhead abundances, model for projecting, 608–609

Logistic

distribution, 556–563

of disease spreads, 556–558

linear equation, 562

equation, 328, 438

qualitative analysis of, 439

function, 155

growth

function and indeterminate form 0/0, 267

model, 437–440

using Carlson data, 438, 439–440, 451–452

organismal spread, 561–562

PDF, 558–561

mean, 560–561

regression, 562–563

Log-log plots, 79

Lognormal distribution, 567–572

lognormal PDF, 569–570

strep throat incubation periods, 568–569

for survival of moths, 570–572

Lognormally distributed, 567

Lotka-Volterra competition equations, 662–664

Lower

developmental threshold, 358

limits of integration, 369

M

Malthus, Thomas, 437

Malthus's estimate, 49–50

Marginal value theorem, 319–320

Mathematical

bioeconomics, 440

modeling, 2–4

Matrices, 603–604

abundances of juvenile and adult loggerheads, models for, 632–633

application of models, 630–633

blood and tissue compartments, models of, 633

defined, 605–608

determinant, 612–613

of a 2-by-2 matrix, 612

feeding fishes, model for, 609–610

Fibonacci's rabbits, modeling of, 628–629

fraction of children at school, home, and hospital, equation for, 614–615

graphical interpretation of solutions, 613

impala population growth, model for, 620–622, 626–628

input-output models, 615–616

inverse of, 612–613

lead amount in blood, model for calculating, 630–632

multiplication by a vector, 616–617

net production of electricity, equation for, 615–616

projecting future loggerhead abundances, model for, 608–609

solving equations, 609–617, 612

Matrix

addition, 607

difference equations theorem, 626

vector multiplication, 607–608

Maximum economic yield, 300–302,

Mean

arithmetic, 540

“average” win in roulette game, 541–542

computation of, 540–541

for a continuous random variable, 542

for a discrete random variable, 541

distance from the center to where dart lands, 542–543

exponential, 543

of extinction rates, 544

logistic PDF, 560–561

of symmetrical PDF, 546

Mean-squared property, 551–552

Mean value theorem (MVT), 187. See also Derivatives as functions example, 191–192

Measurement error, 239, 245

Median, 520

Memory formation, modeling, 498–500

Mendel, Gregor, 91

Mendelian inheritance principle, 91

Menten, Maud, 63

Messier, F., 150

Metabolic rates, elasticity of, 247

Mexico population growth, 108–109

Michaelis, Leonor, 63

Michaelis-Menten

function, 150

glucose uptake rate, 63–64

Midpoint

error bound, 408

Midpoint rule, 404, 407–409, 412

Mixing models in biology

biopharmaceutics models, 456

clearance rate constant, determining, 458

HIV levels in infected hosts, 455–456

infusion rate, determining, 456–458

lake pollution, determining, 458–460

pharmacokinetics models, 456

Monomorphic strategy, 482

Monotone convergence theorem, 164, 167. See also Sequential limits

Morphine-6-glucuronide dynamics from clinical studies, 644–645

Multiplication law, 39

Multivariate modeling

basic concepts, 591–594

domain of a function, 592–593

function of two variables, 592

measuring frequency of chirping, 597–598

predatory impacts on population growth, 593–594

range of a function, 592–593

real world examples, 597–600

Myers, Ran, 287

N

Naïve approach to solving exponential equation, 55

Natural

exponential, 205, 223

logarithm, derivative of, 226

numbers, 11

Negative binomial escape function, 271, 272

Net increase, 381

Neurons, 497

Neurotransmitters, 498

Newton, Isaac, 107

Newton-Raphson difference equation, 335

Newton's

law of cooling, 460–461

method, 335–338. See also Linearization and difference equations

nonconvergence of, 337–338

theorem, 336–337

tumor regrowth, time to, 337

Non-autonomous differential equation, 432

Nonexistence of limits, 158

Nonlinear systems, 656–673

applications in physiology and medicine, 664–673

genetic toggle switch for bacterium Escherichia coli, 668–670

phase-plane method, 658–662

of prey-predator interactions, 657–658

susceptible (S), exposed (E), infectious (I) model of common cold, 670–673

Nonnegativity property of integral, 375

Normal distribution, 563–565

of IQ score in socially disparate communities, 566–567

North American bison population, modeling (project), 277

Northwestern crows and whelks, 316–317

nth derivatives, 261

Nullcline(s), 465, 476

Numerical integration, 403–404

O

Occam's razor, 433

Olympic weightlifting, 43–44

One degree-day, 358

One-sided limits, 124. See also Limits

Open interval method, 12, 296–297

Optimal foraging and marginal value, 318–319

Optimization in biology, 299–307. See also Differentiation, applications of

and lethal genes, 310–311

maximum economic yield, 300–302

regression, 307–311

residual sum-of-squares, 308

summation, properties of, 308

sustainable exploitation of Arctic fin whale, 302–303

tumor growth, 305–306

vascular branching, 306–307

P

Parabola, 174

Parasitoids, 222–223

Pareto distribution, 529–531

Partial fractions, 398–399

Pascal, Blaise, 507, 541

Payoff matrix, 479

Pennings, Tim, 303

Per capita rates

birth rate, 433

clearance rate, 454

death rate, 433

growth, 213

growth rate, 438

Percent error, 247

Percentiles

birth time quartiles, 521

drug decay, chart for, 520–521

first, 520

median, 520

overweight baby, chart for, 521–522

third, 520

Peregrine falcons, stooping behavior of, 354–355

Periodic and trigonometric functions, 30–31. See also Data fitting

Periodic populations, 236

Pharmacokinetics models, 456

Phase lines, 476–482

for clonal population, 477–478

graph for, 478–480

for Hawk-Dove replicator equation, 480–481

steps to draw, 477

Phase-plane

analysis, 591

method, 658–662

for excitable systems, 665–668

of Lotka-Volterra competition equations, 662–664

step-by-step, 659–662

Photosynthesis, 15

Physiological time, 358

Pinching theorem, 133

p-intervals, 528–529

Pitchfork bifurcation, 495

Planktonic copepods, 125

Plantation, optimal rotation period for, 323–324

Point discontinuity, 136

Poiseuille's law, 23, 250

Polymorphic strategy, 482

Polynomial(s), 62

of degree n, 62

growth vs exponential, 50–51

fifth-order, 264

limits of, 132

Population resilience, 485–486

Power functions and scaling laws, 38–46

allometric scaling, 44–45

geometric similarity, 40–41

laws of exponents, 39–40

Olympic weightlifting, 43–44

properties of power functions, 38

proportionality, 40–41

Power law for positive real numbers, 227

Power rule, 200

example, 201

Power-to-root property, 41

Precalculus functions, 404

Predatory impacts on population growth, 593–594

Prevalence of the disease, 556

Principle of parsimony, 433, 437

Probability, 212, 219

density function (PDF)

area under, 514

birthtime distribution, 515–516

continuous random variable with, 514–515

cumulative distribution function (CDF), 516

defined, 513

exponential, 519

fundamental theorem of, 531

heights of fathers and sons, comparison of, 512–513

Laplace, 537

logistic, 558–561

lognormal, 569–570

from a nonnegative function, 513–514

of normal distribution, 563

Pareto, 546–547

symmetrical, 545–546

uniform, 516

distribution, 510

theory, 507

Product(s), 132, 137

Product rule, 209–211. See also Quotient rule

example, 211–212

per capita or intrinsic rate of growth, 213

survival rates, 212–213

Proportional harvesting, 440

Proportionality, 40

rules of, 41

Pythagorean theorem, 303

Q

Quadratic approximation, 258

Quadratic population function, 228

Qualitative analysis, 434

Queen conchs

harvesting of, 440–441, 491–492

sudden disappearances of, 493–495

Quick limits, 137–138

Quotient rule, 213–218. See also Product rule dose–response curve for patients, 217–218

reciprocal rule, 214

whelks, 215

Quotients, 132, 137

R

Random mating and Mendelian inheritance principle, 91

Random variable, 510

Range of a function, 592

Rate of airflow into lungs, estimation of, 392

Rates of change, 7–8, 254

average

rate of change, 108–109

velocity, 110

fastest humans, 111–112

of carbon dioxide, 109–110, 234–235

instantaneous

rate of change, 109–110

velocity, 110

of falling object, 112–113

Mexico population growth, 108–109

rate of change of carbon dioxide, 109–110

tangent lines, 113–115

velocity, 110–113

Rational

function, 62, 132

integration of, 399–400

numbers, 11, 12

Real numbers, 11, 12

absolute value function, 19

algebraic expressions to graphs, 16

function, classification of, 19–20

functions, 13–15

income tax rates, 18–19

increasing and decreasing functions, 19

piecewise-defined functions, 18

verbal descriptions to graphs, 16–17

vertical line test, 17–18

words to algebraic representations, 15–16

Reciprocal rule, 214

Reflections, 59–60

Regression, 307–311. See also Optimization in biology

and lethal genes, 310–311

residual sum-of-squares, 308

summation, properties of, 308

Renewal function, 417–418

Replicator equation, 479

Reproductive number, 583

in the United States, 584

Reproductive success, 582–585

effect of measles epidemics, 585

of painted turtles, 583–584

of vole life, 582–583

Residual sum-of-squares, 308

Retirement account, calculating amount of money in, 419–420

Ricker stock-recruitment relationship, 274

Riemann, Georg Friedrich Bernhard, 365

Riemann sum, 370

defined, 365

as a limit of integral, 370

limits of, 365

with right end points, 371

summation formulas, 365

Riemann sums, 404

Right endpoint

approximation, 404–405

rule, 404, 412

Rolle's theorem, 194

Rossi, H., 262

Roulette game, probability distribution of, 510–511

S

Saddle node bifurcation, 494

Sample points, 364–365

Sandwich theorem, 133

Scalar

multiple, 201

multiplication, 607

Scaling, allometric, 44–45

Scaling laws. See Power functions and scaling laws

Secant line, 7, 113

Second

derivative, 251–252, 261

test, 293

element, 605

Second-order approximations, 257–260

Second-order chemical kinetics, 400–401

Semi-log plots, 79

Semistable equilibrium, 482

Sensitivity, 246, 260

Separation of variables, 445, 447–450

Sequences, 6, 85–86. See also Difference equations

Sequential continuity theorem, 157–158

Sequential limits

asymptotic behavior of difference equations, 158

Beverton-Holt sockeye salmon dynamics, 164–166

and continuity, 155–156

discrete logistic model, dynamics of, 162–164

disruptive selection, 166–168

example, 156–157

Fibonacci and growth rate of rabbits, 168–169

informal definition, 155–156

lethal recessives revisited, 159–160

limit of sequence, finding, 158–159

limits of difference equations, 161

Monotone convergence theorem, 164

nonexistence of limits, 158

sequential continuity theorem, 157–158

Set intersection, 62

Shifts

horizontal, 58

vertical, 58

Short-tailed shrews exploiting cocoons, 66–67

Sickle cell alleles, fate of, 334–335

Sigmoidal curve, 253

Sigmoidal decay in deaths, 253

Signed area, 373

Signed area interpretation of integrals, 373

Simpson error bound, 409

Simpson's rule, 404, 409–411, 421

crab harvests, estimating, 411–413

Sine, derivatives of, 232–233. See also Derivatives of trigonometric functions

Sinusoidal functions, 31

Slope fields, 464–469

autonomous, 468

purely time-dependent, 468–469

sketching solutions using, 465–467

Solution (s)

defined, 432

to differential equations, 445–447

equilibrium, 439

finding and plotting, 448–449

first-order approximation for, 484–485

to inhomogeneous linear differential equation, 650–653

matrices, graphical interpretation of, 613

of two-dimensional systems of linear differential equation, 639

Source-sink dynamics, 647–648

Spider mite destroyer, developmental rates of, 243–245

Splitting property of integral, 375–376

Squeeze theorem, 134. See also Limit laws and continuity

Stability

equilibrium, 327

graphical approach to, 333–334

of insect population dynamics, 330–332

of linear difference equations, 328

of monotone difference equations, 332–333

via linearization, 329–330

Stable equilibrium, 482

Stable stage distribution, 628

Stable state distribution theorem, 646–647

Standard deviation, 547, 565

computation of, 548

for a continuous random variable, 549

Standard normal distribution, 565–566

Step function, 124

Stinkbug species, developmental rate of, 348, 351

Stock-recruitment curves, 164, 286–287

Stooping behavior of peregrine falcons, 354–355

Stretching and compressing, 60–61

Substitution, integration by, 388, 397

with leftover x values, 389

with radical function, 388

with a trigonometric function, 389

using, with the help of technology, 390

Substitution for definite integral, 390–391

Subtraction law, 39

Sum, 132, 137

derivatives of, 201

of squared errors, 308

Summation, properties of, 308

Super-exponential growth, 450–451

Super-exponential model, 4

Survival

function, 417–418

of person at age t, computing, 398

rates, 212–213

renewal equation, 418

and renewal in a clinic, 417–418

Survivorship function, 575–579

of aging dinosaurs, 576

relationship between CDF and PDF, 576–577

Survivorship-mortality equation, 578

for humans in the United States, 578–579

mortality rates of dinosaurs, 577–578

Susceptible (S), exposed (E), infectious (I) model of common cold, 670–673

Sylvester, J., 231

Symmetrical PDF, 545–546

means of, 546

T

Tangent lines, 8, 113. See also Rates of change and tangent lines

approximating, 113–115, 240–245

for parabola, 115

to parabola, 174

slopes of, 174 (See also Derivative at point)

Tangents, circular, 224–225

Theoretical time of conception, 462

Thermodilution, 292–293, 421–422

Third

derivatives, 261

percentile, 520

Tidal movements, modeling, 61–62

Tidal volume, 392

Torricelli's trumpet, 533–534

Transcendental equations, 141

Transitive property, 41

Trapezoidal rule, 404, 412

Trigonometric functions, 237

derivatives of (See Derivatives of trigonometric functions)

Tumor

growth, 305–306

regrowth, time to, 337–338

Turning points on graph, 191

Two-sided limit, 124

Two-strategy replicator equation, 480

Tylenol in blood stream, 283–284

U

Uniform probability density function (PDF), 516

Union of intervals, 13

United States population growth, calculation of, 391–392, 450

Unstable equilibrium, 328, 482

Upper

development threshold, 358

limits of integration, 369, 370

V

Variable of integration, 369

Variance

computation of, 548

for a continuous random variable, 549

for discrete random variables, 547

integrating to get a, 550

Laplacian, 550–551

mean-squared property, 551–552

for Pareto distribution, 552

unbiased estimate of, 548

Vascular branching, 306–307

Vectors, 603–604

addition of, 616–617

defined, 605–608

matrix multiplication by, 616–617

scalar multiplication by, 616–617

Velocity, 110–111. See also Rates of change and tangent lines

average, 110

fastest humans, 111–112, 111t

instantaneous, 110

instantaneous velocity of falling object, 112–113

Vertical asymptotes, 279. See also Asymptotes and infinity

difference of infinities, 148

finding, 149–150

informal definition, 149

wolves eating moose, 150–151

Vertical line test, 17–18

Von Bertalanffy growth

curve, 155

equation, 461–462

W

Weber-Fechner

differential equation, 352

law, 352

Weight lifting, 203

Whelks, 215f

Wilson-Cowan model, 498

Work done

by a bear digging a trench, 425–426

by a constant force, 422–423

in digging a cellar, 424–425

“W” shaped curve, 280–282

X

x intercepts, 279

Y

y intercepts, 279

Z

Zach, Reto, 279

z direction, 594

z scores, 565