diff --git a/_CalculusFix.answers b/_CalculusFix.answers index 2a18305..f2e9e9d 100644 --- a/_CalculusFix.answers +++ b/_CalculusFix.answers @@ -3,5 +3,7 @@ \noindent {\Large \bf Chapter 1} \vskip \baselineskip \noindent {\bf Section 1.1} \vskip \baselineskip -\printanswers {exercises/02_07_exercises} +\printanswers {exercises/05_01_exercises} +\noindent {\bf Section 1.2} \vskip \baselineskip +\printanswers {exercises/05_05_exercises} \end {multicols}\normalsize diff --git a/_CalculusFix.pdf b/_CalculusFix.pdf index e2b66e3..a501eba 100644 Binary files a/_CalculusFix.pdf and b/_CalculusFix.pdf differ diff --git a/_CalculusFix.tex b/_CalculusFix.tex index bfacef0..b32648f 100644 --- a/_CalculusFix.tex +++ b/_CalculusFix.tex @@ -63,6 +63,10 @@ %\printinblackandwhite \printallanswers +%%\ifthenelse{\boolean{printquestions}}{\columnbreak}{} +%5\ifthenelse{\boolean{printquestions}}{\vfill\null\columnbreak}{} + + %%\input{text/front_matter_and_coverI} @@ -79,9 +83,9 @@ %% %%%%\addtocounter{chapter}{1} % -\clearpage{\pagestyle{empty}\cleardoublepage} -\chapter{Derivatives}\label{chapter:derivatives} -\thispagestyle{empty} +%\clearpage{\pagestyle{empty}\cleardoublepage} +%\chapter{Derivatives}\label{chapter:derivatives} +%\thispagestyle{empty} %% %\input{text/02_Derivative} %\input{text/02_Derivative_Meaning} @@ -89,14 +93,14 @@ \chapter{Derivatives}\label{chapter:derivatives} %\input{text/02_Product_Quotient_Rules} %\input{text/02_Chain_Rule} %\input{text/02_Implicit_Differentiation} -\input{text/02_Derivative_Inverse_Functions} +%\input{text/02_Derivative_Inverse_Functions} %%%% %%%%\addtocounter{chapter}{2} -% +%% %\clearpage{\pagestyle{empty}\cleardoublepage} %\chapter{The Graphical Behavior of Functions}\label{chapter:graphbehavior} %\thispagestyle{empty} -% + %\input{text/03_Extreme_Values} %\input{text/03_Mean_Value_Theorem} %\input{text/03_Increasing_Decreasing} @@ -133,17 +137,17 @@ \chapter{Derivatives}\label{chapter:derivatives} %%%% %%%%%\addtocounter{chapter}{4} %% -%\clearpage{\pagestyle{empty}\cleardoublepage} -%\chapter{Integration}\label{chapter:integration} -%\thispagestyle{empty} -%\addtocontents{toc}{\protect\thispagestyle{empty}} +\clearpage{\pagestyle{empty}\cleardoublepage} +\chapter{Integration}\label{chapter:integration} +\thispagestyle{empty} +\addtocontents{toc}{\protect\thispagestyle{empty}} -%\input{text/05_Antiderivatives} +\input{text/05_Antiderivatives} %\addtocontents{toc}{\protect\thispagestyle{empty}} %\input{text/05_Definite_Integral} %\input{text/05_Riemann_Sums} %\input{text/05_FTC} -%\input{text/05_Numerical_Integration} +\input{text/05_Numerical_Integration} %%% %%%%%% %%%%%%% diff --git a/_ErrorList.tex b/_ErrorList.tex index 7eb674a..617feea 100644 --- a/_ErrorList.tex +++ b/_ErrorList.tex @@ -1011,6 +1011,20 @@ % 2.5 added more chain rule problems % 2.6 added one more implicit dy/dx with graph. Lots of work!! % 2.7 actually removed a problem (old #27) to make a section have 2 problems instead of 3. If I added more, the HW page spilled into 2 pages. +% 3.1 added terms and concepts q, along with another graphed problem +% 3.3 added terms and concepts +% 3.4 took out a problem - old #11, as concavity of sin x isn't much different than cos x +% Added a problem to each of the next 3 exsets, as they are all linked and had an odd # of problems. +% 3.5 Added terms and concepts question, added problem at end (new #28) +% 4.1 added a newton's method, 5 step problem +% 4.4 added a terms and concepts question, reorganized some problems with exset +% 5.1 added terms and concepts. An exset on antiderivs has an odd # of problems, but left it as it works out ok +% 5.2 added problem about def. int. & geometry +% 5.3 added two different summations questions in two diff. exsets +% 5.4 added integration q about sin on interval of 2pi, added q about displacement given velocity function +% 5.5 added one more term/concept q + + (262) Curve sketching. Include sketches by hand? If so, need BW versions, too. diff --git a/exercises/03_01_ex_08.tex b/exercises/03_01_ex_08.tex index 07f9427..dc73eb0 100644 --- a/exercises/03_01_ex_08.tex +++ b/exercises/03_01_ex_08.tex @@ -2,7 +2,7 @@ $\ds f(x) = \frac{2}{x^2+1}$ \myincludegraphics[scale=.8]{figures/fig03_01_ex_08} -\ifthenelse{\boolean{printquestions}}{\columnbreak}{} +%\ifthenelse{\boolean{printquestions}}{\columnbreak}{} %\end{minipage} } {$\fp(0) = 0$ diff --git a/exercises/03_01_ex_25.tex b/exercises/03_01_ex_25.tex new file mode 100644 index 0000000..8f7fbf2 --- /dev/null +++ b/exercises/03_01_ex_25.tex @@ -0,0 +1,8 @@ +{%\begin{minipage}{\linewidth} +$\ds f(x) = \sqrt[3]{x^4-2x+1}$ + +\myincludegraphics[scale=.8]{figures/fig03_01_ex_25} +%\end{minipage} +} +{Both $\fp(-1)$ and $\fp(1)$ are undefined. +} diff --git a/exercises/03_01_ex_26.tex b/exercises/03_01_ex_26.tex new file mode 100644 index 0000000..4c2484b --- /dev/null +++ b/exercises/03_01_ex_26.tex @@ -0,0 +1,3 @@ +{Fill in the blanks: The critical points of a function $f$ are found where $\fp(x)$ is equal to \underline{\hskip.5in} or where $\fp(x)$ is \underline{\hskip.5in}.} +{Where $\fp(x)$ is equal to \underline{0} or where $\fp(x)$ is \underline{undefined}. +} diff --git a/exercises/03_01_exercises.tex b/exercises/03_01_exercises.tex index 9442baf..3428b73 100644 --- a/exercises/03_01_exercises.tex +++ b/exercises/03_01_exercises.tex @@ -4,8 +4,10 @@ \exinput{exercises/03_01_ex_03} \exinput{exercises/03_01_ex_04} \exinput{exercises/03_01_ex_05} +\exinput{exercises/03_01_ex_26} \printproblems \exsetinput{exercises/03_01_exset_01} +\ifthenelse{\boolean{printquestions}}{\columnbreak}{} \exsetinput{exercises/03_01_exset_02} \exsetinput{exercises/03_01_exset_03} \printreview diff --git a/exercises/03_01_exset_02.tex b/exercises/03_01_exset_02.tex index 51e115c..1fcfae7 100644 --- a/exercises/03_01_exset_02.tex +++ b/exercises/03_01_exset_02.tex @@ -4,6 +4,7 @@ \exinput{exercises/03_01_ex_09} \exinput{exercises/03_01_ex_10} \exinput{exercises/03_01_ex_11} +\exinput{exercises/03_01_ex_14} +\exinput{exercises/03_01_ex_25} \exinput{exercises/03_01_ex_12} -\exinput{exercises/03_01_ex_13} -\exinput{exercises/03_01_ex_14} \ No newline at end of file +\exinput{exercises/03_01_ex_13} \ No newline at end of file diff --git a/exercises/03_03_ex_03.tex b/exercises/03_03_ex_03.tex index d375a34..c9d1288 100644 --- a/exercises/03_03_ex_03.tex +++ b/exercises/03_03_ex_03.tex @@ -1,4 +1,4 @@ -{Sketch a graph of a function on $[0,2]$ that is increasing but not strictly increasing. +{Sketch a graph of a function on $[0,2]$ that is increasing, where it is increasing ``quickly'' near $x=0$ and increasing ``slowly'' near $x=2$. } -{Answers will vary. +{Answers will vary; graphs should be steeper near $x=0$ than near $x=2$. } diff --git a/exercises/03_03_ex_26.tex b/exercises/03_03_ex_26.tex new file mode 100644 index 0000000..43f42bc --- /dev/null +++ b/exercises/03_03_ex_26.tex @@ -0,0 +1,4 @@ +{T/F: Functions always switch from increasing to decreasing, or decreasing to increasing, at critical points. +} +{False; for instance, $y=x^3$ is always increasing though it has a critical point at $x=0$. +} diff --git a/exercises/03_03_exercises.tex b/exercises/03_03_exercises.tex index 1a17cc4..ac24e0f 100644 --- a/exercises/03_03_exercises.tex +++ b/exercises/03_03_exercises.tex @@ -3,6 +3,7 @@ \exinput{exercises/03_03_ex_02} \exinput{exercises/03_03_ex_03} \exinput{exercises/03_03_ex_04} +\exinput{exercises/03_03_ex_26} \exinput{exercises/03_03_ex_05} \printproblems \exsetinput{exercises/03_03_exset_01} diff --git a/exercises/03_04_ex_55.tex b/exercises/03_04_ex_55.tex new file mode 100644 index 0000000..525479c --- /dev/null +++ b/exercises/03_04_ex_55.tex @@ -0,0 +1,6 @@ +{$\ds f(x) = \sec x $ on $(-3\pi/2,3\pi/2)$ +} +{Possible points of inflection: $\fp'(x)$ is not defined (nor is $f$) at $x=-\pi/2,\pi/2$; +concave down on $(-3\pi/2,-\pi/2)$ and $(\pi/2,3\pi/2)$ +concave up on $(-\pi/2,\pi/2)$ +} diff --git a/exercises/03_04_ex_56.tex b/exercises/03_04_ex_56.tex new file mode 100644 index 0000000..4fa61c9 --- /dev/null +++ b/exercises/03_04_ex_56.tex @@ -0,0 +1,5 @@ +{$\ds f(x) = \sec x $ on $(-3\pi/2,3\pi/2)$ +} +{max: at $x=\pm\pi$ +min: at $x=0$ +} diff --git a/exercises/03_04_ex_57.tex b/exercises/03_04_ex_57.tex new file mode 100644 index 0000000..60a7a76 --- /dev/null +++ b/exercises/03_04_ex_57.tex @@ -0,0 +1,4 @@ +{$\ds f(x) = \sec x $ on $(-3\pi/2,3\pi/2)$ +} +{$\fp(x)$ has no relative extrema +} diff --git a/exercises/03_04_exset_01.tex b/exercises/03_04_exset_01.tex index d48d6c9..b621013 100644 --- a/exercises/03_04_exset_01.tex +++ b/exercises/03_04_exset_01.tex @@ -10,7 +10,6 @@ \exinput{exercises/03_04_ex_07} \exinput{exercises/03_04_ex_08} \exinput{exercises/03_04_ex_09} -\exinput{exercises/03_04_ex_10} \exinput{exercises/03_04_ex_11} \exinput{exercises/03_04_ex_12} \exinput{exercises/03_04_ex_13} diff --git a/exercises/03_04_exset_02.tex b/exercises/03_04_exset_02.tex index 6d0e4ae..1fd3a30 100644 --- a/exercises/03_04_exset_02.tex +++ b/exercises/03_04_exset_02.tex @@ -12,6 +12,7 @@ \exinput{exercises/03_04_ex_20} \exinput{exercises/03_04_ex_21} \exinput{exercises/03_04_ex_22} +\exinput{exercises/03_04_ex_55} \exinput{exercises/03_04_ex_23} \exinput{exercises/03_04_ex_24} \exinput{exercises/03_04_ex_25} diff --git a/exercises/03_04_exset_03.tex b/exercises/03_04_exset_03.tex index e0c56c3..81fee25 100644 --- a/exercises/03_04_exset_03.tex +++ b/exercises/03_04_exset_03.tex @@ -8,6 +8,7 @@ \exinput{exercises/03_04_ex_33} \exinput{exercises/03_04_ex_34} \exinput{exercises/03_04_ex_35} +\exinput{exercises/03_04_ex_56} \exinput{exercises/03_04_ex_36} \exinput{exercises/03_04_ex_37} \exinput{exercises/03_04_ex_38} diff --git a/exercises/03_04_exset_04.tex b/exercises/03_04_exset_04.tex index 57aa3b0..f19c7c4 100644 --- a/exercises/03_04_exset_04.tex +++ b/exercises/03_04_exset_04.tex @@ -8,6 +8,7 @@ \exinput{exercises/03_04_ex_46} \exinput{exercises/03_04_ex_47} \exinput{exercises/03_04_ex_48} +\exinput{exercises/03_04_ex_57} \exinput{exercises/03_04_ex_49} \exinput{exercises/03_04_ex_50} \exinput{exercises/03_04_ex_51} diff --git a/exercises/03_05_ex_30.tex b/exercises/03_05_ex_30.tex new file mode 100644 index 0000000..133d80e --- /dev/null +++ b/exercises/03_05_ex_30.tex @@ -0,0 +1,4 @@ +{T/F: When sketching graphs of functions, one need not plot any points at all. +} +{F +} diff --git a/exercises/03_05_ex_31.tex b/exercises/03_05_ex_31.tex new file mode 100644 index 0000000..0ef5061 --- /dev/null +++ b/exercises/03_05_ex_31.tex @@ -0,0 +1,5 @@ +{$\ds f(x) = ax^2+bx+1$ +} +{Critical point: $x=-b/(2a)$ +No points of inflection +} diff --git a/exercises/03_05_exercises.tex b/exercises/03_05_exercises.tex index 05853a2..64b13a7 100644 --- a/exercises/03_05_exercises.tex +++ b/exercises/03_05_exercises.tex @@ -4,6 +4,7 @@ \exinput{exercises/03_05_ex_03} \exinput{exercises/03_05_ex_04} \exinput{exercises/03_05_ex_05} +\exinput{exercises/03_05_ex_30} \printproblems \exsetinput{exercises/03_05_exset_01} \exsetinput{exercises/03_05_exset_02} diff --git a/exercises/03_05_exset_03.tex b/exercises/03_05_exset_03.tex index aa99e9a..d91d659 100644 --- a/exercises/03_05_exset_03.tex +++ b/exercises/03_05_exset_03.tex @@ -2,5 +2,6 @@ {, a function with the parameters $a$ and $b$ are given. Describe the critical points and possible points of inflection of $f$ in terms of $a$ and $b$. } \exinput{exercises/03_05_ex_26} +\exinput{exercises/03_05_ex_31} \exinput{exercises/03_05_ex_27} \exinput{exercises/03_05_ex_28} \ No newline at end of file diff --git a/exercises/04_01_ex_19.tex b/exercises/04_01_ex_19.tex new file mode 100644 index 0000000..22af700 --- /dev/null +++ b/exercises/04_01_ex_19.tex @@ -0,0 +1,4 @@ +{$f(x) = x^3-x^2+x-1$, $x_0=1$ +} +{$x_0=1$, $x_1=1$, $x_2=1$, $x_3=1$, $x_4=1$, $x_5=1$ +} diff --git a/exercises/04_01_exset_01.tex b/exercises/04_01_exset_01.tex index eae616f..f10c3a0 100644 --- a/exercises/04_01_exset_01.tex +++ b/exercises/04_01_exset_01.tex @@ -6,3 +6,4 @@ \exinput{exercises/04_01_ex_05} \exinput{exercises/04_01_ex_06} \exinput{exercises/04_01_ex_07} +\exinput{exercises/04_01_ex_19} \ No newline at end of file diff --git a/exercises/04_04_ex_39.tex b/exercises/04_04_ex_39.tex new file mode 100644 index 0000000..6db3de4 --- /dev/null +++ b/exercises/04_04_ex_39.tex @@ -0,0 +1,5 @@ +{T/F: In real life, differentials are used to approximate function values when the function itself is not known. +} +{T +} + diff --git a/exercises/04_04_ex_40.tex b/exercises/04_04_ex_40.tex new file mode 100644 index 0000000..4a58d35 --- /dev/null +++ b/exercises/04_04_ex_40.tex @@ -0,0 +1,5 @@ +{$f(x) = \ln\big(\sec x\big)$ +} +{$dy = \big(\tan x\big)dx$ +} + diff --git a/exercises/04_04_exercises.tex b/exercises/04_04_exercises.tex index 3382378..e0f2fd4 100644 --- a/exercises/04_04_exercises.tex +++ b/exercises/04_04_exercises.tex @@ -4,13 +4,11 @@ \exinput{exercises/04_04_ex_03} \exinput{exercises/04_04_ex_04} \exinput{exercises/04_04_ex_05} +\exinput{exercises/04_04_ex_39} \printproblems \exsetinput{exercises/04_04_exset_01} \exsetinput{exercises/04_04_exset_02} -\exinput{exercises/04_04_ex_30} -\exinput{exercises/04_04_ex_31} -\exinput{exercises/04_04_ex_32} -\exinput{exercises/04_04_ex_33} +\exsetinput{exercises/04_04_exset_04} \exsetinput{exercises/04_04_exset_03} %\printreview %\exinput{exercises/03_03_ex_24} diff --git a/exercises/04_04_exset_01.tex b/exercises/04_04_exset_01.tex index c00e615..02dae50 100644 --- a/exercises/04_04_exset_01.tex +++ b/exercises/04_04_exset_01.tex @@ -1,5 +1,7 @@ {\noindent In Exercises} -{, use differentials to approximate the given value by hand.} +{, use differentials to approximate the given value by hand.% +%Used to have \exinput{exercises/04_04_ex_15} but took it out to get an even # of problems +} \exinput{exercises/04_04_ex_06} \exinput{exercises/04_04_ex_07} \exinput{exercises/04_04_ex_08} @@ -9,6 +11,4 @@ \exinput{exercises/04_04_ex_12} \exinput{exercises/04_04_ex_13} \exinput{exercises/04_04_ex_14} -\exinput{exercises/04_04_ex_15} -\exinput{exercises/04_04_ex_16} - +\exinput{exercises/04_04_ex_16} \ No newline at end of file diff --git a/exercises/04_04_exset_02.tex b/exercises/04_04_exset_02.tex index aca5d1c..191aa83 100644 --- a/exercises/04_04_exset_02.tex +++ b/exercises/04_04_exset_02.tex @@ -12,4 +12,5 @@ \exinput{exercises/04_04_ex_26} \exinput{exercises/04_04_ex_27} \exinput{exercises/04_04_ex_28} -\exinput{exercises/04_04_ex_29} \ No newline at end of file +\exinput{exercises/04_04_ex_29} +\exinput{exercises/04_04_ex_40} \ No newline at end of file diff --git a/exercises/04_04_exset_04.tex b/exercises/04_04_exset_04.tex new file mode 100644 index 0000000..50adf0a --- /dev/null +++ b/exercises/04_04_exset_04.tex @@ -0,0 +1,7 @@ +{\noindent Exercises} +{ use differentials to approximate propagated error. +} +\exinput{exercises/04_04_ex_30} +\exinput{exercises/04_04_ex_31} +\exinput{exercises/04_04_ex_32} +\exinput{exercises/04_04_ex_33} \ No newline at end of file diff --git a/exercises/05_01_ex_42.tex b/exercises/05_01_ex_42.tex new file mode 100644 index 0000000..28acc9e --- /dev/null +++ b/exercises/05_01_ex_42.tex @@ -0,0 +1,5 @@ +{If $F(x)$ is an antiderivative of $f(x)$, and $G(x)$ is an antiderivative of $g(x)$, give an antiderivative of $f(x)+g(x)$. +} +{$F(x)+G(x)$ +} + diff --git a/exercises/05_01_exercises.tex b/exercises/05_01_exercises.tex index 207a930..f50db27 100644 --- a/exercises/05_01_exercises.tex +++ b/exercises/05_01_exercises.tex @@ -6,6 +6,7 @@ \exinput{exercises/05_01_ex_05} \exinput{exercises/05_01_ex_06} \exinput{exercises/05_01_ex_07} +\exinput{exercises/05_01_ex_42} \printproblems \exsetinput{exercises/05_01_exset_01} \exinput{exercises/05_01_ex_39} diff --git a/exercises/05_02_ex_09.tex b/exercises/05_02_ex_09.tex index 48d0f90..c5d09ae 100644 --- a/exercises/05_02_ex_09.tex +++ b/exercises/05_02_ex_09.tex @@ -13,7 +13,8 @@ \item $\ds \int_0^4 f(x)\ dx$ \item $\ds \int_0^4 5f(x)\ dx$ \end{enumerate} -\end{minipage} +\end{minipage}% +\ifthenelse{\boolean{printquestions}}{\columnbreak}{} } {\begin{enumerate} \item $\pi$ diff --git a/exercises/05_02_ex_11.tex b/exercises/05_02_ex_11.tex index 68386e7..fe6f438 100644 --- a/exercises/05_02_ex_11.tex +++ b/exercises/05_02_ex_11.tex @@ -14,6 +14,7 @@ \item $\ds \int_0^1 f(x)\ dx$ \end{enumerate} \end{minipage} +\ifthenelse{\boolean{printquestions}}{\vskip1in\ \columnbreak}{} } {\begin{enumerate} \item $4/\pi$ diff --git a/exercises/05_02_ex_12.tex b/exercises/05_02_ex_12.tex index bb9bd9f..d788bce 100644 --- a/exercises/05_02_ex_12.tex +++ b/exercises/05_02_ex_12.tex @@ -14,7 +14,6 @@ \item $\ds \int_0^1 f(x)\ dx$ \end{enumerate} \end{minipage} -\ifthenelse{\boolean{printquestions}}{\vskip1in\ \columnbreak}{} } {\begin{enumerate} \item $4$ diff --git a/exercises/05_02_ex_30.tex b/exercises/05_02_ex_30.tex new file mode 100644 index 0000000..a427fbe --- /dev/null +++ b/exercises/05_02_ex_30.tex @@ -0,0 +1,27 @@ +{\noindent +\begin{minipage}{\linewidth} +\myincludegraphics[scale=.8]{figures/fig05_02_ex_30} +\end{minipage} +\noindent\begin{minipage}[t]{.5\linewidth} +\begin{enumerate} +\item $\ds \int_0^5 f(x)\ dx$ +\item $\ds \int_3^7 f(x)\ dx$ +\end{enumerate} +\end{minipage} +\begin{minipage}[t]{.5\linewidth} +\begin{enumerate}\addtocounter{enumii}{2} +\item $\ds \int_0^0 f(x)\ dx$ +\item $\ds \int_a^b f(x)\ dx$, where + +$0\leq a\leq b\leq 10$ +\end{enumerate} +\end{minipage} +} +{\begin{enumerate} +\item $15$ +\item $12$ +\item $0$ +\item $3(b-a)$ +\end{enumerate} +} + diff --git a/exercises/05_02_exercises.tex b/exercises/05_02_exercises.tex index 1088e97..f1556c1 100644 --- a/exercises/05_02_exercises.tex +++ b/exercises/05_02_exercises.tex @@ -5,11 +5,11 @@ \exinput{exercises/05_02_ex_04} \printproblems \exsetinput{exercises/05_02_exset_01} -\ifthenelse{\boolean{printquestions}}{\columnbreak}{} +%\ifthenelse{\boolean{printquestions}}{\columnbreak}{} \exsetinput{exercises/05_02_exset_02} \exsetinput{exercises/05_02_exset_03} \exinput{exercises/05_02_ex_16} -\ifthenelse{\boolean{printquestions}}{\columnbreak}{} +%\ifthenelse{\boolean{printquestions}}{\columnbreak}{} \exinput{exercises/05_02_ex_17} \exsetinput{exercises/05_02_exset_04} \ifthenelse{\boolean{printquestions}}{\columnbreak}{} diff --git a/exercises/05_02_exset_01.tex b/exercises/05_02_exset_01.tex index 9af6a6c..fe536d0 100644 --- a/exercises/05_02_exset_01.tex +++ b/exercises/05_02_exset_01.tex @@ -4,4 +4,5 @@ \exinput{exercises/05_02_ex_06} \exinput{exercises/05_02_ex_07} \exinput{exercises/05_02_ex_08} -\exinput{exercises/05_02_ex_09} \ No newline at end of file +\exinput{exercises/05_02_ex_09} +\exinput{exercises/05_02_ex_30} \ No newline at end of file diff --git a/exercises/05_03_ex_02.tex b/exercises/05_03_ex_02.tex index 0a67596..24cdd94 100644 --- a/exercises/05_03_ex_02.tex +++ b/exercises/05_03_ex_02.tex @@ -1,4 +1,6 @@ -{What is the upper bound in the summation $\ds \sum_{i=7}^{14} (48i-201)$? +{What is the upper bound in the summation + +$\ds \sum_{i=7}^{14} (48i-201)$? } {14 } diff --git a/exercises/05_03_ex_45.tex b/exercises/05_03_ex_45.tex new file mode 100644 index 0000000..ffcf39b --- /dev/null +++ b/exercises/05_03_ex_45.tex @@ -0,0 +1,5 @@ +{$\ds \sum_{i=1}^{10} 5$ +} +{$5+5+\ldots+5=50$ +} + diff --git a/exercises/05_03_ex_46.tex b/exercises/05_03_ex_46.tex new file mode 100644 index 0000000..7d7c61b --- /dev/null +++ b/exercises/05_03_ex_46.tex @@ -0,0 +1,5 @@ +{$\ds \sum_{i=1}^{10} 5$ +} +{$5\cdot 10=50$ +} + diff --git a/exercises/05_03_exset_01.tex b/exercises/05_03_exset_01.tex index 1b4dbe9..49b14fa 100644 --- a/exercises/05_03_exset_01.tex +++ b/exercises/05_03_exset_01.tex @@ -3,6 +3,7 @@ \exinput{exercises/05_03_ex_05} \exinput{exercises/05_03_ex_06} \exinput{exercises/05_03_ex_07} +\exinput{exercises/05_03_ex_45} \exinput{exercises/05_03_ex_08} \exinput{exercises/05_03_ex_09} \exinput{exercises/05_03_ex_10} diff --git a/exercises/05_03_exset_03.tex b/exercises/05_03_exset_03.tex index 12594f1..c076ab9 100644 --- a/exercises/05_03_exset_03.tex +++ b/exercises/05_03_exset_03.tex @@ -1,5 +1,6 @@ {\noindent In Exercises} {, evaluate the summation using Theorem \ref{thm:summation}.} +\exinput{exercises/05_03_ex_46} \exinput{exercises/05_03_ex_16} \exinput{exercises/05_03_ex_17} \exinput{exercises/05_03_ex_18} diff --git a/exercises/05_04_ex_59.tex b/exercises/05_04_ex_59.tex new file mode 100644 index 0000000..27d8669 --- /dev/null +++ b/exercises/05_04_ex_59.tex @@ -0,0 +1,5 @@ +{Explain why $\ds \int_{a}^{a+2\pi} \sin t\ dt = 0$ for all values of $a$. +} +{$\int_{a}^{a+2\pi} \sin t\ dt = \cos(a+2\pi)-\cos(a)$. Since cosine is periodic with period $2\pi$, $\cos(a+2\pi) = \cos(a)$, and hence the integral is 0. +} + diff --git a/exercises/05_04_ex_60.tex b/exercises/05_04_ex_60.tex new file mode 100644 index 0000000..55fb56d --- /dev/null +++ b/exercises/05_04_ex_60.tex @@ -0,0 +1,5 @@ +{$v(t) = 10$ft/s on $[0,3]$. +} +{$30$ft +} + diff --git a/exercises/05_04_exercises.tex b/exercises/05_04_exercises.tex index 9826e7a..e419ce4 100644 --- a/exercises/05_04_exercises.tex +++ b/exercises/05_04_exercises.tex @@ -6,6 +6,8 @@ \printproblems \exsetinput{exercises/05_04_exset_01} \exinput{exercises/05_04_ex_28} +\exinput{exercises/05_04_ex_59} +\ifthenelse{\boolean{printquestions}}{\columnbreak}{} \exsetinput{exercises/05_04_exset_02} \exsetinput{exercises/05_04_exset_03} \exsetinput{exercises/05_04_exset_04} diff --git a/exercises/05_04_exset_04.tex b/exercises/05_04_exset_04.tex index 1415789..0794dda 100644 --- a/exercises/05_04_exset_04.tex +++ b/exercises/05_04_exset_04.tex @@ -3,6 +3,7 @@ } \exinput{exercises/05_04_ex_40} \exinput{exercises/05_04_ex_41} +\exinput{exercises/05_04_ex_60} \exinput{exercises/05_04_ex_42} \exinput{exercises/05_04_ex_43} \exinput{exercises/05_04_ex_44} \ No newline at end of file diff --git a/exercises/05_05_ex_26.tex b/exercises/05_05_ex_26.tex new file mode 100644 index 0000000..7811875 --- /dev/null +++ b/exercises/05_05_ex_26.tex @@ -0,0 +1,4 @@ +{Simpson's Rule is based on approximating portions of a function with what type of function? +} +{A quadratic function (i.e., parabola) +} diff --git a/exercises/05_05_exercises.tex b/exercises/05_05_exercises.tex index 3680a32..d786003 100644 --- a/exercises/05_05_exercises.tex +++ b/exercises/05_05_exercises.tex @@ -2,6 +2,7 @@ \exinput{exercises/05_05_ex_23} \exinput{exercises/05_05_ex_24} \exinput{exercises/05_05_ex_25} +\exinput{exercises/05_05_ex_26} \printproblems \exsetinput{exercises/05_05_exset_02} \exsetinput{exercises/05_05_exset_03} diff --git a/figures/Calculus_figures.pdf b/figures/Calculus_figures.pdf index 4943b65..3f8f1ac 100644 Binary files a/figures/Calculus_figures.pdf and b/figures/Calculus_figures.pdf differ diff --git a/figures/Calculus_figures.tex b/figures/Calculus_figures.tex index e3c93e1..9236cab 100644 --- a/figures/Calculus_figures.tex +++ b/figures/Calculus_figures.tex @@ -145,52 +145,13 @@ \begin{document} -\mysettikzname{fig02_06_ex_38} -\input{fig_02_06_ex_38} +\mysettikzname{fig05_02_ex_30} +\input{fig_05_02_ex_30} -Image File Name: \texttt{\detokenize{figlimit1}}\par -TeX File Name: \texttt{\detokenize{fig_limit1}}\par +Image File Name: \texttt{\detokenize{fig05_02_ex_30}}\par +TeX File Name: \texttt{\detokenize{fig_05_02_ex_30}}\par %%%% -% -% -%Image File Name: \texttt{\detokenize{figsketch3_blank}}\par -%TeX File Name: \texttt{\detokenize{fig_sketch3_blank}}\par - - -%\mysettikzname{figsketch3_blank} -%\input{fig_sketch3_blank} -% -%Image File Name: \texttt{\detokenize{figsketch3_blank}}\par -%TeX File Name: \texttt{\detokenize{fig_sketch3_blank}}\par - -%\mysettikzname{figsimply_connected_c} -%\input{fig_simply_connected_c} -% -%Image File Name: \texttt{\detokenize{figsimply_connected_a}}\par -%TeX File Name: \texttt{\detokenize{fig_simply_connected_a}}\par -% -% -%\mysettikzname{figparsurfareaC} -%\input{fig_parsurfareaC} - -%Image File Name: \texttt{\detokenize{figparsurfareaC}}\par -%eX File Name: \texttt{\detokenize{fig_parsurf3a}}\par - -% -%\mysettikzname{fig13_02_ex_18} -%\input{fig_13_02_ex_18} -% -%Image File Name: \texttt{\detokenize{fig13_02_ex_18}}\par -%TeX File Name: \texttt{\detokenize{fig_13_02_ex_18}}\par - -%\mysettikzname{website_header_12} -%\input{website_header_12} -% -%Image File Name: \texttt{\detokenize{figfwithderiv}}\par -%TeX File Name: \texttt{\detokenize{figfwithderiv}}\par - - @@ -203,6 +164,19 @@ {% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\mysettikzname{fig03_01_ex_25} +%\input{fig_03_01_ex_25} +% +%Image File Name: \texttt{\detokenize{fig03_01_ex_25}}\par +%TeX File Name: \texttt{\detokenize{fig_03_01_ex_25}}\par + +%\mysettikzname{fig02_06_ex_38} +%\input{fig_02_06_ex_38} +% +%Image File Name: \texttt{\detokenize{figlimit1}}\par +%TeX File Name: \texttt{\detokenize{fig_limit1}}\par + + %\mysettikzname{figsimply_connected_a} %\input{fig_simply_connected_a} % diff --git a/figures/fig03_01_ex_25.pdf b/figures/fig03_01_ex_25.pdf new file mode 100644 index 0000000..066a251 Binary files /dev/null and b/figures/fig03_01_ex_25.pdf differ diff --git a/figures/fig03_01_ex_25BW.pdf b/figures/fig03_01_ex_25BW.pdf new file mode 100644 index 0000000..7ee2288 Binary files /dev/null and b/figures/fig03_01_ex_25BW.pdf differ diff --git a/figures/fig05_02_ex_30.pdf b/figures/fig05_02_ex_30.pdf new file mode 100644 index 0000000..600d000 Binary files /dev/null and b/figures/fig05_02_ex_30.pdf differ diff --git a/figures/fig05_02_ex_30BW.pdf b/figures/fig05_02_ex_30BW.pdf new file mode 100644 index 0000000..35cdaf2 Binary files /dev/null and b/figures/fig05_02_ex_30BW.pdf differ diff --git a/figures/fig_03_01_ex_25.tex b/figures/fig_03_01_ex_25.tex new file mode 100644 index 0000000..421340c --- /dev/null +++ b/figures/fig_03_01_ex_25.tex @@ -0,0 +1,23 @@ +\begin{tikzpicture}[baseline=10pt] +\begin{axis}[width=\marginparwidth+25pt,tick label style={font=\scriptsize},axis y line=middle,axis x line=middle,name=myplot,% + xtick={-2,-1,1,2}, +% ytick={-1,1,2,3}, +% minor y tick num=1, +% extra x ticks={-6.28,-3.14,3.14,6.28},% +% extra x tick labels={$-2\pi$, $-\pi$, $\pi$, $2\pi$}, + ymin=-.5,ymax=3.5,% + xmin=-2.5,xmax=2.5,% +% grid=major +] + +\addplot [thick,{\colorone},domain=-2:-1,samples=25,smooth] {(x^4-2*x^2+1)^(1/3)}; +\addplot [thick,{\colorone},domain=-1:1,samples=50,smooth] {(x^4-2*x^2+1)^(1/3)}; +\addplot [thick,{\colorone},domain=1:2,samples=25,smooth] {(x^4-2*x^2+1)^(1/3)}; + +\draw [{\colorone},fill={\colorone}] (axis cs: 1,0) node [above right,black] {\scriptsize $(1,0)$} circle (1.5pt); +\draw [{\colorone},fill={\colorone}] (axis cs: -1,0) node [above left,black] {\scriptsize $(-1,0)$} circle (1.5pt); + +\end{axis} +\node [right] at (myplot.right of origin) {\scriptsize $x$}; +\node [above] at (myplot.above origin) {\scriptsize $y$}; +\end{tikzpicture} diff --git a/figures/fig_05_02_ex_30.tex b/figures/fig_05_02_ex_30.tex new file mode 100644 index 0000000..3fbb3ba --- /dev/null +++ b/figures/fig_05_02_ex_30.tex @@ -0,0 +1,32 @@ +\begin{tikzpicture} +\begin{axis}[width=\marginparwidth+25pt,% +tick label style={font=\scriptsize},axis y line=middle,axis x line=middle,name=myplot,axis on top,% + %x=.37\marginparwidth, + %y=.37\marginparwidth, + %xtick={1,2,3,4},% +% extra x ticks={.33}, +% extra x tick labels={$1/3$}, + ytick={1,2,3}, + %minor y tick num=1,%extra y ticks={-5,-3,...,7},% + ymin=-.5,ymax=3.5,% + xmin=-1,xmax=10.9% +] + +%\addplot [{\coloronefill},fill={\coloronefill},domain=0:10,area style] {5} |- (axis cs:0,0) --cycle; +\addplot [thick, {\colorone},smooth,samples=2,domain=0:10] ({x}, {3}); + +\draw (axis cs:5,3.4) node {\scriptsize $f(x) = 3$}; + + +\end{axis} + +\node [right] at (myplot.right of origin) {\scriptsize $x$}; +\node [above] at (myplot.above origin) {\scriptsize $y$}; +\end{tikzpicture} + + + + + + + diff --git a/text/03_Extreme_Values.tex b/text/03_Extreme_Values.tex index 322af92..fc9585a 100644 --- a/text/03_Extreme_Values.tex +++ b/text/03_Extreme_Values.tex @@ -120,11 +120,11 @@ \section{Extreme Values}\label{sec:extreme_values} {Let a function $f$ have a relative extrema at the point $(c,f(c))$. Then $c$ is a critical number of $f$. } -Be careful to understand that this theorem states ``All relative extrema occur at critical points.'' It does not say ``All critical numbers produce relative extrema.'' For instance, consider $f(x) = x^3$. Since $\fp(x) = 3x^2$, it is straightforward to determine that $x=0$ is a critical number of $f$. However, $f$ has no relative extrema, as illustrated in Figure \ref{fig:extreme4}. \\ - +Be careful to understand that this theorem states ``All relative extrema occur at critical points.'' It does not say ``All critical numbers produce relative extrema.'' For instance, consider $f(x) = x^3$. Since $\fp(x) = 3x^2$, it is straightforward to determine that $x=0$ is a critical number of $f$. However, $f$ has no relative extrema, as illustrated in Figure \ref{fig:extreme4}. \mfigure{.8}{A graph of $f(x)=x^3$ which has a critical value of $x=0$, but no relative extrema.}{fig:extreme4}{figures/figextreme4} -Theorem \ref{thm:extreme_val} states that a continuous function on a closed interval will have absolute extrema, that is, both an absolute maximum and an absolute minimum. These extrema occur either at the endpoints or at critical values in the interval. We combine these concepts to offer a strategy for finding extrema. +Theorem \ref{thm:extreme_val} states that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum. Common sense tells us ``extrema occur either at the endpoints or somewhere in between.'' It is easy to check for extrema at endpoints, but there are infinite points to check that are ``in between.'' Our theory tells us we need only check at the critical points that are in between the endpoints. %These extrema occur either at the endpoints or at critical values in the interval. +We combine these concepts to offer a strategy for finding extrema. \keyidea{idea:extrema}{Finding Extrema on a Closed Interval}% {Let $f$ be a continuous function defined on a closed interval $[a,b]$. To find the maximum and minimum values of $f$ on $[a,b]$:\index{extrema!finding} @@ -139,70 +139,93 @@ \section{Extreme Values}\label{sec:extreme_values} We practice these ideas in the next examples.\\ \example{ex_extval4}{Finding extreme values}{ -Find the extreme values of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$, graphed in Figure \ref{fig:extval4}.} +Find the extreme values of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$, graphed in Figure \ref{table:ext4}(a).} {We follow the steps outlined in Key Idea \ref{idea:extrema}. We first evaluate $f$ at the endpoints: $$f(0) = 0 \quad \text{and}\quad f(3) =45.$$ -\mfigure{.35}{A graph of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$ as in Example \ref{ex_extval4}.}{fig:extval4}{figures/figextval4} +%\mfigure{.35}{A graph of $f(x) = 2x^3+3x^2-12x$ on $[0,3]$ as in Example \ref{ex_extval4}.}{fig:extval4}{figures/figextval4} Next, we find the critical values of $f$ on $[0,3]$. $\fp(x) = 6x^2+6x-12 = 6(x+2)(x-1)$; therefore the critical values of $f$ are $x=-2$ and $x=1$. Since $x=-2$ does not lie in the interval $[0,3]$, we ignore it. Evaluating $f$ at the only critical number in our interval gives: $f(1) = -7$. -The table in Figure \ref{table:ext4} gives $f$ evaluated at the ``important'' $x$ values in $[0,3]$. We can easily see the maximum and minimum values of $f$: the maximum value is $45$ and the minimum value is $-7$. +The table in Figure \ref{table:ext4}(b) gives $f$ evaluated at the ``important'' $x$ values in $[0,3]$. We can easily see the maximum and minimum values of $f$: the maximum value is $45$ and the minimum value is $-7$. -\mtable{.85}{Finding the extreme values of $f$ in Example \ref{ex_extval4}.}{table:ext4}{\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} 0 & 0 \\ 1 & $-7$\\3 & 45 \end{tabular}} +\mtable{.75}{Finding the extreme values of $f(x)= 2x^3+3x^2-12x$ in Example \ref{ex_extval4}.}{table:ext4}{\begin{tabular}{c}%outer table +\myincludegraphics{figures/figextval4}\\[5pt] +(a)\\[10pt] +\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} 0 & 0 \\ 1 & $-7$\\3 & 45 \\ \end{tabular}\\[20pt] +(b) +\end{tabular}} \vskip-\baselineskip }\\ -Note that all this was done without the aid of a graph; this work followed an analytic algorithm and did not depend on any visualization. Figure \ref{fig:extval4} shows $f$ and we can confirm our answer, but it is important to understand that these answers can be found without graphical assistance. - - +Note that all this was done without the aid of a graph; this work followed an analytic algorithm and did not depend on any visualization. Figure \ref{table:ext4} shows $f$ and we can confirm our answer, but it is important to understand that these answers can be found without graphical assistance. We practice again.\\ \example{ex_extval5}{Finding extreme values}{ -Find the maximum and minimum values of $f$ on $[-4,2]$, where $$f(x) = \left\{\begin{array}{cc} (x-1)^2 & x\leq 0 \\ x+1 & x>0 \end{array}\right. .$$ +Find the maximum and minimum values of $f$ on $[-4,2]$, where $$f(x) = \left\{\begin{array}{cc} (x-1)^2 & x\leq 0 \\ x+1 & x>0 \end{array}\right. ,$$ graphed in Figure \ref{table:ext5}(a). } {Here $f$ is piecewise--defined, but we can still apply Key Idea \ref{idea:extrema} as it is continuous on $[-4,2]$ (one should check to verify that $\ds\lim_{x\to0}f(x) =f(0)$). Evaluating $f$ at the endpoints gives: $$ f(-4) = 25 \quad \text{and} \quad f(2) = 3.$$ We now find the critical numbers of $f$. We have to define $\fp$ in a piecewise manner; it is $$\fp(x) =\left\{\begin{array}{cc} 2(x-1) & x < 0 \\ 1 & x>0 \end{array}\right. .$$ Note that while $f$ is defined for all of $[-4,2]$, $\fp$ is not, as the derivative of $f$ does not exist when $x=0$. (From the left, the derivative approaches $-2$; from the right the derivative is 1.) Thus one critical number of $f$ is $x=0$. -We now set $\fp(x) = 0$. When $x >0$, $\fp(x)$ is never 0. When $x<0$, $\fp(x)$ is also never 0. (We may be tempted to say that $\fp(x) = 0 $ when $x=1$. However, this is nonsensical, for we only consider $\fp(x) = 2(x-1)$ when $x<0$, so we will ignore a solution that says $x=1$.) +We now set $\fp(x) = 0$. When $x >0$, $\fp(x)$ is never 0. When $x<0$, $\fp(x)$ is also never 0, so we find no critical values from setting $\fp(x)=0$. %(We may be tempted to say that $\fp(x) = 0 $ when $x=1$. However, this is nonsensical, for we only consider $\fp(x) = 2(x-1)$ when $x<0$, so we will ignore a solution that says $x=1$.) -So we have three important $x$ values to consider: $x= -4, 2$ and $0$. Evaluating $f$ at each gives, respectively, $25$, $3$ and $1$, shown in Figure \ref{table:ext5}. Thus the absolute minimum of $f$ is 1; the absolute maximum of $f$ is $25$. Our answer is confirmed by the graph of $f$ in Figure \ref{fig:extval5}. -\mtable{.6}{Finding the extreme values of $f$ in Example \ref{ex_extval5}.}{table:ext5}{\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-4$ & 25 \\ 0 & 1 \\ 2 & 3\end{tabular}} -\mfigure{.4}{A graph of $f(x)$ on $[-4,2]$ as in Example \ref{ex_extval5}.}{fig:extval5}{figures/figextval5} +So we have three important $x$ values to consider: $x= -4, 2$ and $0$. Evaluating $f$ at each gives, respectively, $25$, $3$ and $1$, shown in Figure \ref{table:ext5}(b). Thus the absolute minimum of $f$ is 1, the absolute maximum of $f$ is $25$, confirmed by the graph of $f$. +\mtable{.4}{Finding the extreme values of a piecewise--defined function in Example \ref{ex_extval5}.}{table:ext5}{\begin{tabular}{c}% outer table +\myincludegraphics{figures/figextval5}\\[5pt] +(a)\\[10pt] +\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-4$ & 25 \\ 0 & 1 \\ 2 & 3\end{tabular}\\[20pt]% ends inner table +(b) +\end{tabular}} +%\mfigure{.4}{A graph of $f(x)$ on $[-4,2]$ as in Example \ref{ex_extval5}.}{fig:extval5}{figures/figextval5} }\\ -\clearpage + +%\clearpage \example{ex_extval6}{Finding extreme values}{ -Find the extrema of $f(x) = \cos (x^2)$ on $[-2,2]$.} +Find the extrema of $f(x) = \cos (x^2)$ on $[-2,2]$, graphed in Figure \ref{table:ext6}(a).} {We again use Key Idea \ref{idea:extrema}. Evaluating $f$ at the endpoints of the interval gives: $f(-2) = f(2) = \cos (4) \approx -0.6536.$ We now find the critical values of $f$. Applying the Chain Rule, we find $\fp(x) = -2x\sin (x^2)$. Set $\fp(x) = 0$ and solve for $x$ to find the critical values of $f$. We have $\fp(x) = 0$ when $x = 0$ and when $\sin (x^2) = 0$. In general, $\sin t = 0$ when $t = \ldots -2\pi, -\pi, 0, \pi, \ldots$ Thus $\sin (x^2) = 0$ when $x^2 = 0, \pi, 2\pi, \ldots$ ($x^2$ is always positive so we ignore $-\pi$, etc.) So $\sin (x^2)=0$ when $x= 0, \pm \sqrt{\pi}, \pm\sqrt{2\pi},$ etc. The only values to fall in the given interval of $[-2,2]$ are $0$ and $\pm\sqrt{\pi}$, where $\sqrt{\pi} \approx 1.77$. -We again construct a table of important values in Figure \ref{table:ext6}. In this example we have 5 values to consider: $x= 0, \pm 2, \pm\sqrt{\pi}$. +We again construct a table of important values in Figure \ref{table:ext6}(b). In this example we have 5 values to consider: $x= 0, \pm 2, \pm\sqrt{\pi}$. -\mtable{.85}{Finding the extrema of $f(x)= \cos (x^2)$ in Example \ref{ex_extval6}.}{table:ext6}{\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-2$ & $-0.65$ \\ $-\sqrt{\pi}$ & $-1$ \\ 0 & 1\\ $\sqrt{\pi}$ & $-1$ \\ 2 & $-0.65$ \end{tabular}} +\mtable{.75}{Finding the extrema of $f(x)= \cos (x^2)$ in Example \ref{ex_extval6}.}{table:ext6}{% +\begin{tabular}{c}% +\myincludegraphics{figures/figextval6}\\[5pt] +(a)\\[10pt] +\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-2$ & $-0.65$ \\ $-\sqrt{\pi}$ & $-1$ \\ 0 & 1\\ $\sqrt{\pi}$ & $-1$ \\ 2 & $-0.65$ \end{tabular}\\[20pt] +(b) +\end{tabular}% +} -From the table it is clear that the maximum value of $f$ on $[-2,2]$ is 1; the minimum value is $-1$. The graph in Figure \ref{fig:extval6} confirms our results. -\mfigure{.67}{A graph of $f(x)=\cos(x^2)$ on $[-2,2]$ as in Example \ref{ex_extval6}.}{fig:extval6}{figures/figextval6} +From the table it is clear that the maximum value of $f$ on $[-2,2]$ is 1; the minimum value is $-1$. The graph of $f$ confirms our results. +%\mfigure{.67}{A graph of $f(x)=\cos(x^2)$ on $[-2,2]$ as in Example \ref{ex_extval6}.}{fig:extval6}{figures/figextval6} }\\ We consider one more example.\\ \example{ex_extval7}{Finding extreme values}{ -Find the extreme values of $f(x) = \sqrt{1-x^2}$.} +Find the extreme values of $f(x) = \sqrt{1-x^2}$, graphed in Figure \ref{table:ext7}(a).} {A closed interval is not given, so we find the extreme values of $f$ on its domain. $f$ is defined whenever $1-x^2\geq 0$; thus the domain of $f$ is $[-1,1]$. Evaluating $f$ at either endpoint returns 0. -\mfigure{.35}{A graph of $f(x)=\sqrt{1-x^2}$ on $[-1,1]$ as in Example \ref{ex_extval7}.}{fig:extval7}{figures/figextval7} -\mtable{.5}{Finding the extrema of the half--circle in Example \ref{ex_extval7}.}{table:ext7}{\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-1$ & 0 \\ 0 & 1 \\ 1 & 0 \end{tabular}} +%\mfigure{.35}{A graph of $f(x)=\sqrt{1-x^2}$ on $[-1,1]$ as in Example \ref{ex_extval7}.}{fig:extval7}{figures/figextval7} +\mtable{.4}{Finding the extrema of the half--circle in Example \ref{ex_extval7}.}{table:ext7}{% +\begin{tabular}{c}% +\myincludegraphics{figures/figextval7}\\[5pt] +(a)\\[10pt] +\begin{tabular}{cc} $x$ & $f(x)$ \\ \hline \rule{0pt}{10pt} $-1$ & 0 \\ 0 & 1 \\ 1 & 0 \end{tabular}\\[20pt] +(b) +\end{tabular}% +} -Using the Chain Rule, we find $\ds \fp(x) = \frac{-x}{\sqrt{1-x^2}}$. The critical points of $f$ are found when $\fp(x) = 0$ or when $\fp$ is undefined. It is straightforward to find that $\fp(x) = 0$ when $x=0$, and $\fp$ is undefined when $x=\pm 1$, the endpoints of the interval. The table of important values is given in Figure \ref{table:ext7}. The maximum value is 1, and the minimum value is 0. +Using the Chain Rule, we find $\ds \fp(x) = \frac{-x}{\sqrt{1-x^2}}$. The critical points of $f$ are found when $\fp(x) = 0$ or when $\fp$ is undefined. It is straightforward to find that $\fp(x) = 0$ when $x=0$, and $\fp$ is undefined when $x=\pm 1$, the endpoints of the interval. The table of important values is given in Figure \ref{table:ext7}(b). The maximum value is 1, and the minimum value is 0. }\\ \mnote{.18}{\textbf{Note:} We implicitly found the derivative of $x^2+y^2=1$, the unit circle, in Example \ref{ex_implicit7} as $\frac{dy}{dx} = -x/y$. In Example \ref{ex_extval7}, half of the unit circle is given as $y=f(x) = \sqrt{1-x^2}$. We found $\fp(x) = \frac{-x}{\sqrt{1-x^2}}$. Recognize that the denominator of this fraction is $y$; that is, we again found $\fp(x) = \frac{dy}{dx} = -x/y.$} - +\enlargethispage{2\baselineskip} We have seen that continuous functions on closed intervals always have a maximum and minimum value, and we have also developed a technique to find these values. In the next section, we further our study of the information we can glean from ``nice'' functions with the Mean Value Theorem. On a closed interval, we can find the \textit{average rate of change} of a function (as we did at the beginning of Chapter 2). We will see that differentiable functions always have a point at which their \textit{instantaneous} rate of change is same as the \textit{average} rate of change. This is surprisingly useful, as we'll see. diff --git a/text/05_Numerical_Integration.tex b/text/05_Numerical_Integration.tex index 77f5c94..0eb2388 100644 --- a/text/05_Numerical_Integration.tex +++ b/text/05_Numerical_Integration.tex @@ -1,6 +1,6 @@ \section{Numerical Integration}\label{sec:numerical_integration} -The Fundamental Theorem of Calculus gives a concrete technique for finding the exact value of a definite integral. That technique is based on computing antiderivatives. Despite the power of this theorem, there are still situations where we must \textit{approximate} the value of the definite integral instead of finding its exact value. The first situation we explore is where we \textit{cannot} compute the antiderivative of the integrand. The second case is when we actually do not know the integrand, but only its value when evaluated at certain points.\index{integration!numerical}\index{numerical integration}\\ %While we handle both situations in the same way, we address them separately here.\\ +The Fundamental Theorem of Calculus gives a concrete technique for finding the exact value of a definite integral. That technique is based on computing antiderivatives. Despite the power of this theorem, there are still situations where we must \textit{approximate} the value of the definite integral instead of finding its exact value. The first situation we explore is where we \textit{cannot} compute the antiderivative of the integrand. The second case is when we actually do not know the function in the integrand, but only its value when evaluated at certain points.\index{integration!numerical}\index{numerical integration}\\ %While we handle both situations in the same way, we address them separately here.\\ %\noindent\textbf{\large An Antiderivative Cannot Be Computed} %\vskip\baselineskip