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Course Notes for Calculus I
Remkes Kooistra
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\(\def\vs{{\it vs. }} \def\cf{{\it cf. }} \def\viz{{\it viz. }} \def\ie{{\it i.e. }} \def\etc{{\it etc. }} \def\eg{{\it e.g. }} \def\etal{{\it et al .}} \def\via{{\it via }} \def\adhoc{{\it ad hoc }} \def\apriori{{\it apriori }} \def\Afrak{\mathfrak{A}} \def\Bfrak{\mathfrak{B}} \def\Cfrak{\mathfrak{C}} \def\Dfrak{\mathfrak{D}} \def\Efrak{\mathfrak{E}} \def\Ffrak{\mathfrak{F}} \def\Gfrak{\mathfrak{G}} \def\Hfrak{\mathfrak{H}} \def\Ifrak{\mathfrak{I}} \def\Jfrak{\mathfrak{J}} \def\Kfrak{\mathfrak{K}} \def\Lfrak{\mathfrak{L}} \def\Mfrak{\mathfrak{M}} \def\Nfrak{\mathfrak{N}} \def\Ofrak{\mathfrak{O}} \def\Pfrak{\mathfrak{P}} \def\Qfrak{\mathfrak{Q}} \def\Rfrak{\mathfrak{R}} \def\Sfrak{\mathfrak{S}} \def\Tfrak{\mathfrak{T}} \def\Ufrak{\mathfrak{U}} \def\Vfrak{\mathfrak{V}} \def\Wfrak{\mathfrak{W}} \def\Xfrak{\mathfrak{X}} \def\Yfrak{\mathfrak{Y}} \def\Zfrak{\mathfrak{Z}} \def\afrak{\mathfrak{a}} \def\bfrak{\mathfrak{b}} \def\cfrak{\mathfrak{c}} \def\dfrak{\mathfrak{d}} \def\efrak{\mathfrak{e}} \def\ffrak{\mathfrak{f}} \def\gfrak{\mathfrak{g}} \def\hfrak{\mathfrak{h}} \def\ifrak{\mathfrak{i}} \def\jfrak{\mathfrak{j}} \def\kfrak{\mathfrak{k}} \def\lfrak{\mathfrak{l}} \def\mfrak{\mathfrak{m}} \def\nfrak{\mathfrak{n}} \def\ofrak{\mathfrak{o}} \def\pfrak{\mathfrak{p}} \def\qfrak{\mathfrak{q}} \def\rfrak{\mathfrak{r}} \def\sfrak{\mathfrak{s}} \def\tfrak{\mathfrak{t}} \def\ufrak{\mathfrak{u}} \def\vfrak{\mathfrak{v}} \def\wfrak{\mathfrak{w}} \def\xfrak{\mathfrak{x}} \def\yfrak{\mathfrak{y}} \def\zfrak{\mathfrak{z}} \def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\Ap{A^\prime} \def\Bp{B^\prime} \def\Cp{C^\prime} \def\Dp{D^\prime} \def\Ep{E^\prime} \def\Fp{F^\prime} \def\Gp{G^\prime} \def\Hp{H^\prime} \def\Ip{I^\prime} \def\Jp{J^\prime} \def\Kp{K^\prime} \def\Lp{L^\prime} \def\Mp{M^\prime} \def\Mp{N^\prime} \def\Op{O^\prime} \def\Pp{P^\prime} \def\Qp{Q^\prime} \def\Rp{R^\prime} \def\Sp{S^\prime} \def\Tp{T^\prime} \def\Up{U^\prime} \def\Vp{V^\prime} \def\Wp{W^\prime} \def\Xp{X^\prime} \def\Yp{Y^\prime} \def\Zp{Z^\prime} \def\ap{a^\prime} \def\bp{b^\prime} \def\cp{c^\prime} \def\dprime{d^\prime} \def\ep{e^\prime} \def\fp{f^\prime} \def\gp{g^\prime} \def\hp{h^\prime} \def\ip{i^\prime} \def\jp{j^\prime} \def\kp{k^\prime} \def\lp{l^\prime} \def\mp{m^\prime} \def\np{n^\prime} \def\op{o^\prime} \def\pp{p^\prime} \def\qp{q^\prime} \def\rp{r^\prime} \def\sp{s^\prime} \def\tp{t^\prime} \def\up{u^\prime} \def\vp{v^\prime} \def\wp{w^\prime} \def\xp{x^\prime} \def\yp{y^\prime} \def\zp{z^\prime} \def\App{A^{\prime\prime}} \def\Bpp{B^{\prime\prime}} \def\Cpp{C^{\prime\prime}} \def\Dpp{D^{\prime\prime}} \def\Epp{E^{\prime\prime}} \def\Fpp{F^{\prime\prime}} \def\Gpp{G^{\prime\prime}} \def\Hpp{H^{\prime\prime}} \def\Ipp{I^{\prime\prime}} \def\Jpp{J^{\prime\prime}} \def\Kpp{K^{\prime\prime}} \def\Lpp{L^{\prime\prime}} \def\Mpp{M^{\prime\prime}} \def\Mpp{N^{\prime\prime}} \def\Opp{O^{\prime\prime}} \def\Ppp{P^{\prime\prime}} \def\Qpp{Q^{\prime\prime}} \def\Rpp{R^{\prime\prime}} \def\Spp{S^{\prime\prime}} \def\Tpp{T^{\prime\prime}} \def\Upp{U^{\prime\prime}} \def\Vpp{V^{\prime\prime}} \def\Wpp{W^{\prime\prime}} \def\Xpp{X^{\prime\prime}} \def\Ypp{Y^{\prime\prime}} \def\Zpp{Z^{\prime\prime}} \def\app{a^{\prime\prime}} \def\bpp{b^{\prime\prime}} \def\cpp{c^{\prime\prime}} \def\dpp{d^{\prime\prime}} \def\epp{e^{\prime\prime}} \def\fpp{f^{\prime\prime}} \def\gpp{g^{\prime\prime}} \def\hpp{h^{\prime\prime}} \def\ipp{i^{\prime\prime}} \def\jpp{j^{\prime\prime}} \def\kpp{k^{\prime\prime}} \def\lpp{l^{\prime\prime}} \def\mpp{m^{\prime\prime}} \def\npp{n^{\prime\prime}} \def\opp{o^{\prime\prime}} \def\ppp{p^{\prime\prime}} \def\qpp{q^{\prime\prime}} \def\rpp{r^{\prime\prime}} \def\spp{s^{\prime\prime}} \def\tpp{t^{\prime\prime}} \def\upp{u^{\prime\prime}} \def\vpp{v^{\prime\prime}} \def\wpp{w^{\prime\prime}} \def\xpp{x^{\prime\prime}} \def\ypp{y^{\prime\prime}} \def\zpp{z^{\prime\prime}} \def\abar{\overline{a}} \def\bbar{\overline{b}} \def\cbar{\overline{c}} \def\dbar{\overline{d}} \def\ebar{\overline{e}} \def\fbar{\overline{f}} \def\gbar{\overline{g}} \def\ibar{\overline{i}} \def\jbar{\overline{j}} \def\kbar{\overline{k}} \def\lbar{\overline{l}} \def\mbar{\overline{m}} \def\nbar{\overline{n}} \def\obar{\overline{o}} \def\pbar{\overline{p}} \def\qbar{\overline{q}} \def\rbar{\overline{r}} \def\sbar{\overline{s}} \def\tbar{\overline{t}} \def\ubar{\overline{u}} \def\vbar{\overline{v}} \def\wbar{\overline{w}} \def\xbar{\overline{x}} \def\ybar{\overline{y}} \def\zbar{\overline{z}} \def\Abar{\overline{A}} \def\Bbar{\overline{B}} \def\Cbar{\overline{C}} \def\Dbar{\overline{D}} \def\Ebar{\overline{E}} \def\Fbar{\overline{F}} \def\Gbar{\overline{G}} \def\Hbar{\overline{H}} \def\Ibar{\overline{I}} \def\Jbar{\overline{J}} \def\Kbar{\overline{K}} \def\Lbar{\overline{L}} \def\Mbar{\overline{M}} \def\Nbar{\overline{N}} \def\Obar{\overline{O}} \def\Pbar{\overline{P}} \def\Qbar{\overline{Q}} \def\Rbar{\overline{R}} \def\Sbar{\overline{S}} \def\Tbar{\overline{T}} \def\Ubar{\overline{U}} \def\Vbar{\overline{V}} \def\Wbar{\overline{W}} \def\Xbar{\overline{X}} \def\Ybar{\overline{Y}} \def\Zbar{\overline{Z}} \def\aunder{\underline{a}} \def\bunder{\underline{b}} \def\cunder{\underline{c}} \def\dunder{\underline{d}} \def\eunder{\underline{e}} \def\funder{\underline{f}} \def\gunder{\underline{g}} \def\hunder{\underline{h}} \def\iunder{\underline{i}} \def\junder{\underline{j}} \def\kunder{\underline{k}} \def\lunder{\underline{l}} \def\munder{\underline{m}} \def\nunder{\underline{n}} \def\ounder{\underline{o}} \def\punder{\underline{p}} \def\qunder{\underline{q}} \def\runder{\underline{r}} \def\sunder{\underline{s}} \def\tunder{\underline{t}} \def\uunder{\underline{u}} \def\vunder{\underline{v}} \def\wunder{\underline{w}} \def\xunder{\underline{x}} \def\yunder{\underline{y}} \def\zunder{\underline{z}} \def\Aunder{\underline{A}} \def\atilde{\widetilde{a}} \def\btilde{\widetilde{b}} \def\ctilde{\widetilde{c}} \def\dtilde{\widetilde{d}} \def\etilde{\widetilde{e}} \def\ftilde{\widetilde{f}} \def\gtilde{\widetilde{g}} \def\htilde{\widetilde{h}} \def\itilde{\widetilde{i}} \def\jtilde{\widetilde{j}} \def\ktilde{\widetilde{k}} \def\ltilde{\widetilde{l}} \def\mtilde{\widetilde{m}} \def\ntilde{\widetilde{n}} \def\otilde{\widetilde{o}} \def\ptilde{\widetilde{p}} \def\qtilde{\widetilde{q}} \def\rtilde{\widetilde{r}} \def\stilde{\widetilde{s}} \def\ttilde{\widetilde{t}} \def\utilde{\widetilde{u}} \def\vtilde{\widetilde{v}} \def\wtilde{\widetilde{w}} \def\xtilde{\widetilde{x}} \def\ytilde{\widetilde{y}} \def\ztilde{\widetilde{z}} \def\Atilde{\widetilde{A}} \def\Btilde{\widetilde{B}} \def\Ctilde{\widetilde{C}} \def\Dtilde{\widetilde{D}} \def\Etilde{\widetilde{E}} \def\Ftilde{\widetilde{F}} \def\Gtilde{\widetilde{G}} \def\Htilde{\widetilde{H}} \def\Itilde{\widetilde{I}} \def\Jtilde{\widetilde{J}} \def\Ktilde{\widetilde{K}} \def\Ltilde{\widetilde{L}} \def\Mtilde{\widetilde{M}} \def\Ntilde{\widetilde{N}} \def\Otilde{\widetilde{O}} \def\Ptilde{\widetilde{P}} \def\Qtilde{\widetilde{Q}} \def\Rtilde{\widetilde{R}} \def\Stilde{\widetilde{S}} \def\Ttilde{\widetilde{T}} \def\Utilde{\widetilde{U}} \def\Vtilde{\widetilde{V}} \def\Wtilde{\widetilde{W}} \def\Xtilde{\widetilde{X}} \def\Ytilde{\widetilde{Y}} \def\Ztilde{\widetilde{Z}} \def\Alphatilde{\widetilde{\Alpha}} \def\Betatilde{\widetilde{\Beta}} \def\Gammatilde{\widetilde{\Gamma}} \def\Deltatilde{\widetilde{\Delta}} \def\Epsilontilde{\widetilde{\Epsilon}} \def\Zetatilde{\widetilde{\Zeta}} \def\Etatilde{\widetilde{\Eta}} \def\Thetatilde{\widetilde{\Theta}} \def\Iotatilde{\widetilde{\Iota}} \def\Kappatilde{\widetilde{\Kappa}} \def\Lambdatilde{\widetilde{\Lamdba}} \def\Mutilde{\widetilde{\Mu}} \def\Nutilde{\widetilde{\Nu}} \def\Xitilde{\widetilde{\Xi}} \def\Omicrontilde{\widetilde{\Omicron}} \def\Pitilde{\widetilde{\Pi}} \def\Rhotilde{\widetilde{\Rho}} \def\Sigmatilde{\widetilde{\Sigma}} \def\Tautilde{\widetilde{\Tau}} \def\Upsilontilde{\widetilde{\Upsilon}} \def\Phitilde{\widetilde{\Phi}} \def\Chitilde{\widetilde{\Chi}} \def\Psitilde{\widetilde{\Psi}} \def\Omegatilde{\widetilde{\Omega}} \def\alphatilde{\widetilde{\alpha}} \def\betatilde{\widetilde{\beta}} \def\gammatilde{\widetilde{\gamma}} \def\deltatilde{\widetilde{\delta}} \def\epsilontilde{\widetilde{\epsilon}} \def\zetatilde{\widetilde{\zeta}} \def\etatilde{\widetilde{\eta}} \def\thetatilde{\widetilde{\theta}} \def\iotatilde{\widetilde{\iota}} \def\kappatilde{\widetilde{\kappa}} \def\lambdatilde{\widetilde{\lamdba}} \def\mutilde{\widetilde{\mu}} \def\nutilde{\widetilde{\nu}} \def\xitilde{\widetilde{\xi}} \def\omicrontilde{\widetilde{\omicron}} \def\pitilde{\widetilde{\pi}} \def\rhotilde{\widetilde{\rho}} \def\sigmatilde{\widetilde{\sigma}} \def\tautilde{\widetilde{\tau}} \def\upsilontilde{\widetilde{\upsilon}} \def\phitilde{\widetilde{\phi}} \def\chitilde{\widetilde{\chi}} \def\psitilde{\widetilde{\psi}} \def\omegatilde{\widetilde{\omega}} \def\Alphabar{\bar{\Alpha}} \def\Betabar{\bar{\Beta}} \def\Gammabar{\bar{\Gamma}} \def\Deltabar{\bar{\Delta}} \def\Epsilonbar{\bar{\Epsilon}} \def\Zetabar{\bar{\Zeta}} \def\Etabar{\bar{\Eta}} \def\Thetabar{\bar{\Theta}} \def\Iotabar{\bar{\Iota}} \def\Kappabar{\bar{\Kappa}} \def\Lambdabar{\bar{\Lamdba}} \def\Mubar{\bar{\Mu}} \def\Nubar{\bar{\Nu}} \def\Xibar{\bar{\Xi}} \def\Omicronbar{\bar{\Omicron}} \def\Pibar{\bar{\Pi}} \def\Rhobar{\bar{\Rho}} \def\Sigmabar{\bar{\Sigma}} \def\Taubar{\bar{\Tau}} \def\Upsilonbar{\bar{\Upsilon}} \def\Phibar{\bar{\Phi}} \def\Chibar{\bar{\Chi}} \def\Psibar{\bar{\Psi}} \def\Omegabar{\bar{\Omega}} \def\alphabar{\bar{\alpha}} \def\betabar{\bar{\beta}} \def\gammabar{\bar{\gamma}} \def\deltabar{\bar{\delta}} \def\epsilonbar{\bar{\epsilon}} \def\zetabar{\bar{\zeta}} \def\etabar{\bar{\eta}} \def\thetabar{\bar{\theta}} \def\iotabar{\bar{\iota}} \def\kappabar{\bar{\kappa}} \def\lambdabar{\bar{\lamdba}} \def\mubar{\bar{\mu}} \def\nubar{\bar{\nu}} \def\xibar{\bar{\xi}} \def\omicronbar{\bar{\omicron}} \def\pibar{\bar{\pi}} \def\rhobar{\bar{\rho}} \def\sigmabar{\bar{\sigma}} \def\taubar{\bar{\tau}} \def\upsilonbar{\bar{\upsilon}} \def\phibar{\bar{\phi}} \def\chibar{\bar{\chi}} \def\psibar{\bar{\psi}} \def\omegabar{\bar{\omega}} \def\del{\partial} \def\delbar{\overline{\partial}} \def\Cech{\check{C}} \def\half{\frac{1}{2}} \def\defeq{\mathrel{\mathop:}=} \def\alg{\mathrm{alg}} \def\Alt{\mathrm{Alt}} \def\Amp{\mathrm{Amp}} \def\Arg{\mathrm{Arg}} \def\an{\mathrm{an}} \def\anti{\mathrm{anti}} \def\Ap{\mathrm{Ap}} \def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}} \def\arccosh{\mathrm{arccosh\hspace{0.07cm}}} \def\arctanh{\mathrm{arctanh\hspace{0.07cm}}} \def\arccsch{\mathrm{arccsch\hspace{0.07cm}}} \def\arcsech{\mathrm{arcsech\hspace{0.07cm}}} \def\arccoth{\mathrm{arccoth\hspace{0.07cm}}} \def\arccsc{\mathrm{arccsc\hspace{0.07cm}}} \def\arcsec{\mathrm{arcsec\hspace{0.07cm}}} \def\arccot{\mathrm{arccot\hspace{0.07cm}}} \def\arg{\mathrm{arg}} \def\BC{\mathrm{BC}} \def\Bel{\mathrm{Bel}} \def\calCH{\mathcal{CH}} \def\csch{\mathrm{csch}\hspace{0.07cm}} \def\CH{\mathrm{CH}} \def\ch{\mathrm{ch}} \def\closed{\mathrm{closed}} \def\codim{\mathrm{codim}} \def\coth{\mathrm{coth}\hspace{0.07cm}} \def\Coh{\mathfrak{Coh}} \def\Coker{\mathrm{Coker}} \def\Cone{\mathrm{Cone}} \def\darg{d\mathrm{arg}} \def\Db{\mathrm{Db}} \def\dclosed{\mathrm{d-closed}} \def\deg{\mathrm{deg}} \def\dim{\mathrm{dim}} \def\divisor{\mathrm{div}} \def\dlog{d\mathrm{log}} \def\DNE{\mathrm{DNE}} \def\DR{\mathrm{DR}} \def\DST{\mathrm{DST}} \def\exp{\mathrm{exp}} \def\FLB{\mathrm{FLB}} \def\FLS{\mathrm{FLS}} \def\Gr{\mathrm{Gr}} \def\Hzar{H_{\mathrm{Zar}}} \def\Hol{\mathrm{Hol}} \def\Id{\mathrm{Id}} \def\Image{\mathrm{Im}} \def\Ka{\mathcal{K}_A} \def\Ker{\mathrm{Ker}} \def\kod{\mathrm{kod}} \def\Kx{\mathcal{K}_X} \def\Kz{\mathcal{K}_Z} \def\log{\mathrm{log}} \def\Log{\mathrm{Log}} \def\Li{\mathrm{Li}} \def\min{\mathrm{min}} \def\Mon{\mathrm{Mon}} \def\Nef{\mathrm{Nef}} \def\NS{\mathrm{NS}} \def\Oa{\mathcal{O}_A} \def\Ox{\mathcal{O}_X} \def\Oz{\mathcal{O}_Z} \def\Perp{\mathrm{Perp}} \def\Pic{\mathrm{Pic}} \def\Proj{\mathrm{Proj}} \def\rank{\mathrm{rank}} \def\Rat{\mathrm{Rat}} \def\Real{\mathrm{Re}} \def\reg{\mathrm{reg}} \def\Res{\mathrm{Res}} \def\res{\mathrm{res}} \def\Ric{\mathrm{Ric}} \def\sech{\mathrm{sech}\hspace{0.07cm}} \def\Span{\mathrm{Span}} \def\Spec{\mathrm{Spec}} \def\sing{\mathrm{sing}} \def\Singx{\mathrm{Sing}(X)} \def\sheafKer{\mathcal{\Ker}} \def\sheafIm{\mathcal{\Im}} \def\Span{\mathrm{Span}} \def\Spin{\mathrm{Spin}} \def\Str{\mathrm{Str}} \def\td{\mathrm{td}} \def\tr{\mathrm{tr}} \def\Todd{\mathrm{Todd}} \def\tor{\mathrm{tor}} \def\trdeg{\mathrm{trdeg}} \def\Zar{\mathrm{Zar}} \def\ZFLS{\mathrm{ZFLS}} \usepackage{tikz} \usepackage{tkz-graph} \usepackage{tkz-euclide} \usetikzlibrary{patterns} \usetikzlibrary{positioning} \usetikzlibrary{matrix,arrows} \usetikzlibrary{calc} \usetikzlibrary{shapes} \usetikzlibrary{through,intersections,decorations,shadows,fadings} \usepackage{pgfplots} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
1
What Is Calculus
1
Analytic Geometry
1.1
Analytic Geometry
1.1.1
The Cartesian Plane
1.1.2
Loci
1.2
Lines
1.2.1
Equations of Lines
1.2.2
Slope
1.2.3
Slope Intercept Form
1.2.4
Calcluating Equations of Lines
1.3
Conics
1.3.1
Slicing a Cone
1.3.2
Descriptions of Conics
1.4
Intersection
1.4.1
1.4.2
Intersection of Lines and Conics
1.5
Shifts
1.5.1
Shifts of Conics
1.6
Week 1 Activity
1.6.1
Equations of Lines
1.6.2
Equations of Conics
1.6.3
Intersection of Lines and Conics
1.6.4
Conceptual Review Questions
2
Functions
2.1
Functions
2.1.1
Sets, Briefly
2.1.2
Three Ideas for Functions
2.1.3
Functions on
\(\RR\)
2.2
Types of Functions
2.2.1
Constant Functions
2.2.2
Linear Functions
2.2.3
Quadratic Functions
2.2.4
Polynomial Functions
2.2.5
Rational Functions
2.2.6
Algebraic Functions
2.2.7
Trigonometric Functions
2.2.8
Inverse Trigonometri Functions
2.2.9
Exponential and Logarithmic Functions
2.2.10
Hyperbolic Functions
2.2.11
Elementary Functions
2.3
Properties of Functions
2.3.1
First Properties
2.4
Operations on Functions
2.4.1
Pointwise Operations
2.4.2
Composition
2.4.3
Inversion
2.4.4
Restriction of Domain
2.5
Models
2.5.1
The Concept of a Model
2.5.2
Regression
2.6
Week 2 Activity
2.6.1
Composing Functions
2.6.2
Recognizing Composition
2.6.3
Inverting Functions
2.6.4
Regression
2.6.5
Interpreting Parameters
2.6.6
Conceptual Review Questions
3
Week 3
3.1
Population Growth
3.1.1
A Motivating Problem
3.2
Differential Equations
3.2.1
Interpreting the Population Regression
3.2.2
The Concept of a Differential Equation
3.3
Phase Lines
3.3.1
Autonomous DEs
3.3.2
Phase Line Analysis
3.4
Week 3 Activity
3.4.1
Exponential Growth
3.4.2
Conceptual Review Questions
4
Week 4
4.1
The Concept of a Limit
4.1.1
Limit Definitions
4.1.2
Vertical Asymptotes
4.1.3
One-Sided Limits
4.2
Calcluating Limits at Finite Values
4.2.1
Three Steps
4.2.2
Limit Rules
4.2.3
Examples
4.3
Continuity
4.3.1
4.3.2
Intermediate Value Theorem
4.3.3
Piecewise Functions
4.3.4
Continuity of Piecewise Functions
4.4
Week 4 Activity
4.4.1
Limits at Finite Values
4.4.2
Vertical Asymptotes
4.4.3
Continuity
4.4.4
Limits and Models
4.4.5
Conceptual Review Questions
4.4.6
Extra Practice
5
Week 5
5.1
Limits at Infinity
5.1.1
Definitions
5.1.2
Horizontal Asymptotes
5.2
Asymptotic Analysis
5.2.1
Classifying Types of Limits
5.2.2
A Novel Technique
5.2.3
Asymptotic Order
5.2.4
An Asymptotic Ranking of Functions
5.2.5
Asymptotic Ranking, Sums and Product
5.2.6
Actually Calculating Limits
5.2.1
Limits at
\(-\infty\)
5.3
Extreme Values of Models
5.3.1
Models and Asymptotic Analysis
5.4
Week 5 Activity
5.4.1
Limits at Infinity
5.4.2
Horizontal Asymptotes
5.4.3
Long Term Behaviour of Models
5.4.4
Conceptual Review Questions
6
Week 6
6.1
Two Motivating Problems
6.1.1
The Velocity Problem
6.1.2
The Distance Travelled Problem
6.2
Definition of the Derivative
6.2.1
Limits of Secant Lines
6.2.2
Differential Operators
6.2.3
Differentiability
6.2.4
Interpretation
6.3
Rules for Differentiation
6.3.1
Important Derivatives
6.3.2
Power Rule
6.3.3
Linearity
6.3.4
Leibniz Rule
6.3.5
Quotient Rule
6.4
Week 6 Activity
6.4.1
Derivatives By Definition
6.4.2
Derivatives
6.4.3
Conceptual Review Questions
7
Week 7
7.1
The Chain Rule
7.1.1
Derivatives of Compositions
7.1.2
Derivatives of Inverse Functions
7.2
Higher Derivatives
7.2.1
Iterating Differentiation
7.2.2
Examples of Higher Derivatives
7.3
L’Hôpital’s Rule
7.3.1
Derivatives in Limits
7.3.2
Examples
7.4
Week 7 Activity
7.4.1
Chain Rule
7.4.2
Higher Derivatives
7.4.3
L’Hôpital’s Rule
7.4.4
Conceptual Review Questions
7.4.5
Extra Practice
8
Week 8
8.1
Sigma Notation
8.1.1
Writing Complicated Sums
8.1.2
Manipulating Sums
8.1.3
Some Important Sums
8.2
The Riemann Integral
8.2.1
Definition
8.2.2
Properties of the Definite Integral
8.3
The Fundamental Theorem
8.3.1
Integrals as Functions
8.3.2
The Fundamental Theorem
8.3.3
The Indefinite Integral
8.3.4
Calculating Definite Integrals
8.4
Week 8 Activity
8.4.1
Sigma Notation
8.4.2
Riemann Integral
8.4.3
Fundamental Theorem of Calculus
8.4.4
Conceptual Review Questions
9
Week 9
9.1
The Substitution Rule
9.1.1
Differentiating Composition
9.1.2
Substitution Examples - Indefinite Integrals
9.1.3
Substitution Examples - Definite Integrals
9.2
Solving Differential Equations
9.2.1
Solving By Direct Integration
9.2.2
Initial Value Problems
9.2.3
Separable Differential Equations
9.3
Two Population Differential Equations
9.3.1
Percentage Growth
9.3.2
Logistic Growth
9.4
Week 9 Activity
9.4.1
Substitution Rule
9.4.2
Initial Value Problems
9.4.3
Exponential Growth
9.4.4
Logistic Growth
9.4.5
Conceptual Review Questions
10
Week 10
10.1
Extrema
10.1.1
Local Minima and Maxima
10.1.2
Examples
10.2
Optimization
10.2.1
Extreme Values of Models
10.2.2
Optimized Distances
10.3
Marginal Analysis
10.3.1
An Economic Model
10.4
Week 10 Activity
10.4.1
Extrema
10.4.2
Optimization
10.4.3
Marginal Analysis
10.4.4
Conceptual Review Questions
11
Week 11
11.1
Curve Sketching
11.1.1
The Idea of Curve Sketching
11.1.2
Examples
11.2
Model Interpretation
11.2.1
Holistic Analysis of Models
11.2.2
Examples
11.3
Week 11 Activity
11.3.1
Curve Sketching
11.3.2
Model Analysis
11.3.3
Conceptual Review Questions
Backmatter
Colophon
Colophon
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