diff --git a/.gitignore b/.gitignore index 3cb00df8c..f65fba033 100644 --- a/.gitignore +++ b/.gitignore @@ -15,6 +15,7 @@ APEX-BW.pl cli.log vshifts.ods *.log +logs/ # vs code stuff .vscode/* @@ -25,3 +26,7 @@ codechat_config.yaml .devcontainer.json .github/workflows .cache + +# validation files +*-assembled.xml +*-validation.txt diff --git a/fig_parsurf2_3D.asy b/fig_parsurf2_3D.asy deleted file mode 100644 index 1cd0db152..000000000 --- a/fig_parsurf2_3D.asy +++ /dev/null @@ -1,175 +0,0 @@ -usepackage("amsmath"); - import graph3; - bool incolor; - incolor = true; - pen apexmeshpen=rgb(0,0,.7); - pen blackmeshpen=rgb(0,0,0); - pen surfacepen=rgb(.6,.6,1)+opacity(.7); - pen surfacepen2=rgb(1,.6,.6)+opacity(1); - material simplesurfacepen=emissive(rgb(.6,.6,1)+opacity(0.7)); - material simplesurfacepen2=emissive(rgb(1,.6,.6)+opacity(0.7)); - material simplesurfacepen3=emissive(rgb(.5,.9,.5)+opacity(0.7)); - pen bluepen=blue; - pen bluemeshpen=rgb(0,0,.5); - pen bluecurvepen=rgb(.1,.1,.7); - pen dotblue=rgb(.6,.6,1); - pen redpen=red; - pen redmeshpen=rgb(.7,0,0); - pen redmeshpen2=rgb(.5,0,0); - pen redcurvepen=rgb(.9,0,0); - pen greenmeshpen=rgb(0,.5,0); - pen greencurvepen=rgb(0,.7,0); - pen curvepen=.4mm+bluepen; - pen curvepen2=.4mm+redpen; - pen darksurfacepen=rgb(.2,.2,1)+opacity(.7); - if(settings.outformat == "html") currentlight.background=opacity(0.0); -texpreamble("\newcommand{\ds}{\displaystyle} - \newcommand{\fp}{f'} - \newcommand{\fpp}{f''} - - - - \newcommand{\lz}[2]{\frac{d#1}{d#2}} - - - - \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} - - - - \newcommand{\lzo}[1]{\frac{d}{d#1}} - - - - \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} - - - - \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} - - - - \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} - - - - \newcommand{\plz}[2]{\frac{\partial#1}{\partial#2}} - - - - \newcommand{\plzoa}[3]{\left.{\frac{\partial#1}{\partial#2}}\right|_{#3}} - - - - \newcommand{\inflim}[1][n]{\lim\limits_{#1 \to \infty}} - - - - \newcommand{\infser}[1][1]{\sum_{n=#1}^\infty} - - \newcommand{\Fp}{F\primeskip'} - \newcommand{\Fpp}{F\primeskip''} - \newcommand{\yp}{y\primeskip'} - \newcommand{\gp}{g\primeskip'} - \newcommand{\dx}{\Delta x} - \newcommand{\dy}{\Delta y} - \newcommand{\ddz}{\Delta z} - \newcommand{\thet}{\theta} - \newcommand{\norm}[1]{\left\lVert#1\right\rVert} - \newcommand{\vnorm}[1]{\left\lVert\vec #1\right\rVert} - \newcommand{\snorm}[1]{\left|\left|\,#1\,\right|\right|} - \newcommand{\la}{\left\langle} - \newcommand{\ra}{\right\rangle} - \newcommand{\dotp}[2]{\vec #1 \cdot \vec #2} - \newcommand{\proj}[2]{\text{proj}_{\,\vec #2}{\,\vec #1}} - \newcommand{\crossp}[2]{\vec #1 \times \vec #2} - \newcommand{\veci}{\vec i} - \newcommand{\vecj}{\vec j} - \newcommand{\veck}{\vec k} - \newcommand{\vecu}{\vec u} - \newcommand{\vecv}{\vec v} - \newcommand{\vecw}{\vec w} - \newcommand{\vecx}{\vec x} - \newcommand{\vecy}{\vec y} - \newcommand{\vrp}{\vec r\hskip0.75pt '} - \newcommand{\vrpp}{\vec r\hskip0.75pt ''} - \newcommand{\vsp}{\vec s\hskip0.75pt '} - \newcommand{\vrt}{\vec r(t)} - \newcommand{\vst}{\vec s(t)} - \newcommand{\vvt}{\vec v(t)} - \newcommand{\vat}{\vec a(t)} - \newcommand{\px}{\partial x} - \newcommand{\py}{\partial y} - \newcommand{\pz}{\partial z} - \newcommand{\pf}{\partial f} - \newcommand{\unittangent}{\vec{{}T}} - \newcommand{\unitnormal}{\vec{N}} - \newcommand{\unittangentprime}{\vec{{}T}\hskip0.75pt '} - \newcommand{\R}{mathbb{R}} - \newcommand{\mathN}{\mathbb{N}} - \newcommand{\surfaceS}{\mathcal{S}} - \newcommand{\zerooverzero}{\ds \raisebox{8pt}{\text{``\ }}\frac{0}{0}\raisebox{8pt}{\textit{ ''}}} - \newcommand{\deriv}[2]{\myds\frac{d}{dx}\left(#1\right)=#2} - \newcommand{\myint}[2]{\myds\int #1\, dx= {\ds #2}} - \newcommand{\primeskip}{\hskip.75pt} - \newcommand{\abs}[1]{\left\lvert #1\right\rvert} - \newcommand{\sech}{\operatorname{sech}} - \newcommand{\csch}{\operatorname{csch}} - \newcommand{\curl}{\operatorname{curl}} - \newcommand{\divv}{\operatorname{div}} - \newcommand{\Hess}{\operatorname{Hess}} -\newcommand{\lt}{<} -\newcommand{\gt}{>} -\newcommand{\amp}{&} -"); - - - - - - //ASY file for figparsurf1_3D.asy in Chapter 13 - - size(200,200,IgnoreAspect); - //currentprojection=perspective(7,2,1); - currentprojection=orthographic(11.9,9.6,17.8); - defaultrender.merge=true; - - // setup and draw the axes - real[] myxchoice={-3,3}; - real[] myychoice={-3,3}; - real[] myzchoice={5,10}; - defaultpen(0.5mm); - - pair xbounds=(-3.5,3.5); - pair ybounds=(-3.5,3.5); - pair zbounds=(-1,12); - - xaxis3("",xbounds.x,xbounds.y,black,OutTicks(myxchoice),Arrow3(size=3mm)); - yaxis3("",ybounds.x,ybounds.y,black,OutTicks(myychoice),Arrow3(size=3mm)); - zaxis3("",zbounds.x,zbounds.y,black,OutTicks(myzchoice),Arrow3(size=3mm)); - - label("$x$",(xbounds.y+0.05*(xbounds.y-xbounds.x),0,0)); - label("$y$",(0,ybounds.y+0.05*(ybounds.y-ybounds.x),0)); - label("$z$",(0,0,zbounds.y+0.05*(zbounds.y-zbounds.x))); - - - //Draw the top half of the surface z^2 = x^2+2y^2 - triple f(pair t) { - return (2*t.y*cos(t.x),2*t.y*sin(t.x),(2*t.y*cos(t.x))^2+2*(2*t.y*sin(t.x))^2);// - } - surface s=surface(f,(0,0),(2pi,1),16,16,usplinetype=new splinetype[] {notaknot,notaknot,monotonic},vsplinetype=new splinetype[] {notaknot,notaknot,monotonic}); - pen p=apexmeshpen; - draw(s,surfacepen,meshpen=p); - - - triple g(real t) { - return (2*cos(t),2*sin(t),0); - } - - //triple g(real t) {return(cos(t),sin(t),t/(2*pi));} - path3 mypath=graph(g,0,2pi,operator ..); - draw(surface(mypath--cycle),curvepen+opacity(.5)); - draw(mypath,curvepen); - - - \ No newline at end of file diff --git a/project.ptx b/project.ptx index 13de612f4..c03e98bee 100644 --- a/project.ptx +++ b/project.ptx @@ -3,21 +3,11 @@ - - - - - - - - - - - - - - - + + + + + + publication="publication-color-print.ptx" xsl="apex-latex-print-color.xsl" output-filename="apex-color-print.pdf"/> + publication="publication-color-print-video.ptx" xsl="apex-latex-print-color.xsl" output-filename="apex-color-print-video.pdf"/> - + diff --git a/ptx/appendix_back_reference.ptx b/ptx/appendix_back_reference.ptx index 371bf7b97..80cea7950 100644 --- a/ptx/appendix_back_reference.ptx +++ b/ptx/appendix_back_reference.ptx @@ -1393,7 +1393,7 @@ - + Test Series

Condition(s) of Convergence

 diff --git a/ptx/chapter_limits.ptx b/ptx/chapter_limits.ptx index 5ce3a4d22..4faa54757 100644 --- a/ptx/chapter_limits.ptx +++ b/ptx/chapter_limits.ptx @@ -28,7 +28,6 @@ - Chapter Summary

In this chapter we:

    diff --git a/ptx/docinfo.ptx b/ptx/docinfo.ptx index 9473103f7..17bc991f6 100644 --- a/ptx/docinfo.ptx +++ b/ptx/docinfo.ptx @@ -1,6 +1,6 @@ - + APEX Exercise Part diff --git a/ptx/review-exercises-limits.ptx b/ptx/review-exercises-limits.ptx index 8c14ddf4f..09dd87313 100644 --- a/ptx/review-exercises-limits.ptx +++ b/ptx/review-exercises-limits.ptx @@ -272,7 +272,7 @@ For a numerical approximation, make a table:

    - + x diff --git a/ptx/sec_FTC.ptx b/ptx/sec_FTC.ptx index 189840c64..79d218056 100644 --- a/ptx/sec_FTC.ptx +++ b/ptx/sec_FTC.ptx @@ -129,7 +129,7 @@ x \geq 1 is given by A(x)=\frac12 (x)(2x)-\frac12 (1)(2)=x^2-1.

    -
    +
    The area of the shaded region is F(x) = \int_1^x 2t\, dt diff --git a/ptx/sec_Graphical_Numerical.ptx b/ptx/sec_Graphical_Numerical.ptx index 33d7965c6..7a3759f4b 100644 --- a/ptx/sec_Graphical_Numerical.ptx +++ b/ptx/sec_Graphical_Numerical.ptx @@ -190,12 +190,11 @@

    - - - - - - + + + + +

    Adjust the initial condition in this interactive figure to answer the following.

    @@ -1073,7 +1072,7 @@ Slope field for the logistic differential equation \yp = y(1-y) from - Graph of slope field for the logistic differential equation y’=y(1-y) from the example. + Graph of slope field for the logistic differential equation y'=y(1-y) from the example.

    @@ -1161,7 +1160,7 @@ with a few representative solution curves - Graph of slope field for the logistic differential equation y’=y(1-y) with representative solution curves. + Graph of slope field for the logistic differential equation y'=y(1-y) with representative solution curves.

    @@ -2114,7 +2113,7 @@ - +

    Match each slope field below with the appropriate differential equation. diff --git a/ptx/sec_Modeling.ptx b/ptx/sec_Modeling.ptx index 29391f743..654bed141 100644 --- a/ptx/sec_Modeling.ptx +++ b/ptx/sec_Modeling.ptx @@ -42,7 +42,7 @@ \draw [firstcolor] (-1.5,0) node [text width=60pt,align=center] (a) { \centering The rate of change of the population}; \draw [firstcolor,->] (a) -- (.3,.7); - \draw [firstcolor] (2,.25) node [text width=60pt,align=center] (b) { \centering the population.}; + \draw [firstcolor] (2,0) node [text width=60pt,align=center] (b) { \centering the\\ population.}; \draw [firstcolor,->] (b) -- (1.6,.8); \draw [firstcolor] (.5,2) node [text width=32pt,align=center] (c) { \centering is}; diff --git a/ptx/sec_antider.ptx b/ptx/sec_antider.ptx index ce04b90f1..8715ddf21 100644 --- a/ptx/sec_antider.ptx +++ b/ptx/sec_antider.ptx @@ -839,8 +839,8 @@

    Fill in the blanks: - Inverse operations do the - things in the order. + Inverse operations do the + things in the order.

    @@ -882,7 +882,7 @@ -->

    - The derivative of a position function is a/an function. + The derivative of a position function is a/an function.

    @@ -904,7 +904,7 @@ -->

    - An antiderivative of an acceleration function is a/an + An antiderivative of an acceleration function is a/an function.

    diff --git a/ptx/sec_arc_length.ptx b/ptx/sec_arc_length.ptx index 995ec2f1d..d8d538f5a 100644 --- a/ptx/sec_arc_length.ptx +++ b/ptx/sec_arc_length.ptx @@ -508,7 +508,7 @@
    A table of values of y=\sqrt{1+\cos^2(x) } to evaluate a definite integral in - + x\sqrt{1+\cos^2(x) } @@ -1451,7 +1451,9 @@ \ds f(x) = \frac1{12}x^5+\frac1{5x^3} on [0.1, 1].

    - +

    + +

    diff --git a/ptx/sec_center_of_mass.ptx b/ptx/sec_center_of_mass.ptx index 54e787a83..ad050b67e 100644 --- a/ptx/sec_center_of_mass.ptx +++ b/ptx/sec_center_of_mass.ptx @@ -1358,7 +1358,7 @@

    Given a point (x,y), - the value of x is a measure of distance from the -axis. + the value of x is a measure of distance from the -axis.

    diff --git a/ptx/sec_conic_sections.ptx b/ptx/sec_conic_sections.ptx index bcbf4bbe4..798e61560 100644 --- a/ptx/sec_conic_sections.ptx +++ b/ptx/sec_conic_sections.ptx @@ -1509,7 +1509,7 @@ - +

    Horizontal Transverse Axis

    Vertical Transverse Axis     @@ -2343,7 +2343,7 @@ - + @@ -3324,7 +3324,7 @@

    - + Planet Distance fromcenter to vertex Orbiteccentricity diff --git a/ptx/sec_cross_product.ptx b/ptx/sec_cross_product.ptx index fe91400a9..8bc8d9148 100644 --- a/ptx/sec_cross_product.ptx +++ b/ptx/sec_cross_product.ptx @@ -1501,7 +1501,7 @@ -->

    - is a measure of the turning force applied to an object. + is a measure of the turning force applied to an object.

    diff --git a/ptx/sec_curvature.ptx b/ptx/sec_curvature.ptx index df08b0ba1..66da1cb70 100644 --- a/ptx/sec_curvature.ptx +++ b/ptx/sec_curvature.ptx @@ -208,7 +208,7 @@ We find it with \vec r(2) = \la 1/5, 18/5\ra.

    -
    +
    Graphing \vec r in with parameters t and s @@ -687,7 +687,7 @@ Being able to think of curvature in terms of the radius of a circle is very useful.

    -
    +
    Illustrating the osculating circles for the curve seen in @@ -757,7 +757,7 @@ .

    -
    +
    Examining the curvature of y=x^2 @@ -834,26 +834,10 @@

    -

    - While this is not a particularly nice formula, - it does explicitly tell us what the curvature is at a given t value. - To maximize \kappa(t), - we should solve \kappa'(t)=0 for t. - This is doable, but very time consuming. - Instead, consider the graph of - \kappa(t) as given in . - We see that \kappa is maximized at two t values; - using a numerical solver, we find these values are t\approx\pm 0.189. - In we graph \vrt and indicate the points where curvature is maximized. -

    - -
    - Understanding the curvature of a curve in space - -
    - The curvature of \vec{r}(t) +
    + The curvature of \vec{r}(t) in - + A plot of the curvature as a function of the parameter t.

    @@ -885,11 +869,11 @@

    -
    - A plot of the curve \vec{r}(t)=\la t, t^2, 2t^3\ra +
    + A plot of the curve \vec{r}(t)=\la t, t^2, 2t^3\ra in - + A plot of the vector-valued function in this example, with points of maximum curvature marked.

    @@ -950,9 +934,19 @@

    - -
    +

    + While this is not a particularly nice formula, + it does explicitly tell us what the curvature is at a given t value. + To maximize \kappa(t), + we should solve \kappa'(t)=0 for t. + This is doable, but very time consuming. + Instead, consider the graph of + \kappa(t) as given in . + We see that \kappa is maximized at two t values; + using a numerical solver, we find these values are t\approx\pm 0.189. + In we graph \vrt and indicate the points where curvature is maximized. +

    @@ -1071,7 +1065,7 @@ - + OperatingSpeed (mph) MinimumRadius (ft) @@ -1189,7 +1183,7 @@

    A measure of the curviness - of a curve is . + of a curve is .

    @@ -1247,7 +1241,7 @@ Rearrange the blocks to form a valid identity.

    - + \unittangentprime(s) = \kappa diff --git a/ptx/sec_def_int.ptx b/ptx/sec_def_int.ptx index 4d3f02ec3..f07a7516c 100644 --- a/ptx/sec_def_int.ptx +++ b/ptx/sec_def_int.ptx @@ -1194,7 +1194,7 @@ What is \ds \int_3^3 \sin(x) \, dx?

    - +

    diff --git a/ptx/sec_deriv_basic_rules.ptx b/ptx/sec_deriv_basic_rules.ptx index 2fa0e35e8..7a3692e39 100644 --- a/ptx/sec_deriv_basic_rules.ptx +++ b/ptx/sec_deriv_basic_rules.ptx @@ -809,7 +809,7 @@

    - +

    diff --git a/ptx/sec_deriv_implicit.ptx b/ptx/sec_deriv_implicit.ptx index 1dbba2e3c..e959b6280 100644 --- a/ptx/sec_deriv_implicit.ptx +++ b/ptx/sec_deriv_implicit.ptx @@ -1155,7 +1155,7 @@

    - +

    diff --git a/ptx/sec_deriv_interpret.ptx b/ptx/sec_deriv_interpret.ptx index 545eb19a0..52babc70e 100644 --- a/ptx/sec_deriv_interpret.ptx +++ b/ptx/sec_deriv_interpret.ptx @@ -851,7 +851,7 @@ What is the instantaneous rate of change of position called?

    - +

    @@ -911,7 +911,7 @@

    - +

    diff --git a/ptx/sec_deriv_intro.ptx b/ptx/sec_deriv_intro.ptx index b3826eac5..6eefed210 100644 --- a/ptx/sec_deriv_intro.ptx +++ b/ptx/sec_deriv_intro.ptx @@ -130,7 +130,7 @@
    Approximating the instantaneous velocity with average velocities over a small time period h - + h Average Velocity () diff --git a/ptx/sec_deriv_inverse_function.ptx b/ptx/sec_deriv_inverse_function.ptx index ebd097d2f..86bcb5d8d 100644 --- a/ptx/sec_deriv_inverse_function.ptx +++ b/ptx/sec_deriv_inverse_function.ptx @@ -180,7 +180,7 @@ <tabular> - <row bottom="medium"> + <row bottom="medium" header="yes"> <cell>Information about <m>f</m></cell> <cell>Information about <m>g=f^{-1}</m></cell> </row> @@ -585,7 +585,7 @@ <table xml:id="tab_domain_trig"> <title>Domains and ranges of the trigonometric and inverse trigonometric functions - + Function Domain Range diff --git a/ptx/sec_deriv_prodquot.ptx b/ptx/sec_deriv_prodquot.ptx index 330cab68c..b4e4ff2e1 100644 --- a/ptx/sec_deriv_prodquot.ptx +++ b/ptx/sec_deriv_prodquot.ptx @@ -915,7 +915,7 @@

    - +

    diff --git a/ptx/sec_differentials.ptx b/ptx/sec_differentials.ptx index d6f23fbd0..058c6f242 100644 --- a/ptx/sec_differentials.ptx +++ b/ptx/sec_differentials.ptx @@ -157,19 +157,19 @@

    \ell(x) closely resembles the curve near (x_0,f(x_0)).

     -

    Correct—the tangent line shares the same instantaneous rate of change, so it hugs the curve nearby.

     +

    Correctthe tangent line shares the same instantaneous rate of change, so it hugs the curve nearby.

    \ell(x) closely resembles the graph of f(x) for any x.

     -

    Not true—tangent lines only approximate well near a given point (, locally), not globally.

     +

    Not truetangent lines only approximate well near a given point (, locally), not globally.

    \ell(x) exactly matches the curve near (x_0,f(x_0)).

     -

    Not true—tangent lines only approximate well near a given point.

     +

    Not truetangent lines only approximate well near a given point.

    \ell(x) is the most accurate approximation of f(x) near (x_0,f(x_0)).

     -

    Not correct—the error grows as you move away from the tangent point.

     +

    Not correctthe error grows as you move away from the tangent point.

         @@ -326,7 +326,7 @@

    The symbol dy/dx in differential notation is literally a fraction of two quantities, dy and dx.

    -

    False — dy/dx is one symbol for the derivative, not an actual fraction. The resemblance to a fraction is helpful but can be misleading.

    +

    False dy/dx is one symbol for the derivative, not an actual fraction. The resemblance to a fraction is helpful but can be misleading.

     diff --git a/ptx/sec_directional_derivative.ptx b/ptx/sec_directional_derivative.ptx index 2bc8b80ca..b36c12b9a 100644 --- a/ptx/sec_directional_derivative.ptx +++ b/ptx/sec_directional_derivative.ptx @@ -1298,7 +1298,7 @@

    - The gradient is to level curves. + The gradient is to level curves.

    @@ -1347,7 +1347,7 @@

    It is generally more informative to view the directional derivative not as the result of a limit, - but rather as the result of a product. + but rather as the result of a product.

    diff --git a/ptx/sec_dot_product.ptx b/ptx/sec_dot_product.ptx index c2ad08285..d77ff71d2 100644 --- a/ptx/sec_dot_product.ptx +++ b/ptx/sec_dot_product.ptx @@ -417,51 +417,49 @@
    - - - - Image shows the relation between the angle between two vectors and the sign of their dot product. - - -

    - Each image has the two vectors \vec u and \vec v along with \theta shown. - The dot product along with three possible cases of <, > and equal to 0 are shown. - In the first image, the dot product has a positive value, the angle between the two vectors is acute. - In the second image, the dot product is equal to 0, the angle is 90 degrees. - In the third image, the dot product has a negative value and the angle is obtuse. -

    -     - - - \begin{tikzpicture}[scale=1.5,>=stealth] - - \begin{scope} - \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v >0$} node [right] { $\vec u$}; - \draw[->] (0,0) -- (.5,.866) node [above right] { $\vec v$}; - \draw[->] (.3,0) arc (0:60:.3); - \draw[rotate=30] (.45,0) node { $\theta$}; - \end{scope} - - \begin{scope}[shift={(3cm,0)}] - \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v =0$}node [right] { $\vec u$}; - \draw[->] (0,0) -- (0,1) node [above ] { $\vec v$}; - \draw[->] (.3,0) arc (0:90:.3); - \draw[rotate=22.5] (.7,.2) node { $\theta=\pi/2$}; - \end{scope} - - \begin{scope}[shift={(6.5cm,0)}] - \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v <0$}node [right] { $\vec u$}; - \draw[->] (0,0) -- (-.707,.707) node [above left] { $\vec v$}; - \draw[->] (.3,0) arc (0:135:.3); - \draw[rotate=67.5] (.45,0) node { $\theta$}; - \end{scope} + + + Image shows the relation between the angle between two vectors and the sign of their dot product. + + +

    + Each image has the two vectors \vec u and \vec v along with \theta shown. + The dot product along with three possible cases of <, > and equal to 0 are shown. + In the first image, the dot product has a positive value, the angle between the two vectors is acute. + In the second image, the dot product is equal to 0, the angle is 90 degrees. + In the third image, the dot product has a negative value and the angle is obtuse. +

    +     + + + \begin{tikzpicture}[scale=1.5,>=stealth] + + \begin{scope} + \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v >0$} node [right] { $\vec u$}; + \draw[->] (0,0) -- (.5,.866) node [above right] { $\vec v$}; + \draw[->] (.3,0) arc (0:60:.3); + \draw[rotate=30] (.45,0) node { $\theta$}; + \end{scope} + + \begin{scope}[shift={(3cm,0)}] + \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v =0$}node [right] { $\vec u$}; + \draw[->] (0,0) -- (0,1) node [above ] { $\vec v$}; + \draw[->] (.3,0) arc (0:90:.3); + \draw[rotate=22.5] (.7,.2) node { $\theta=\pi/2$}; + \end{scope} + + \begin{scope}[shift={(6.5cm,0)}] + \draw[->] (0,0) -- (1,0) node [pos=.5,below] { $\vec u\cdot \vec v <0$}node [right] { $\vec u$}; + \draw[->] (0,0) -- (-.707,.707) node [above left] { $\vec v$}; + \draw[->] (.3,0) arc (0:135:.3); + \draw[rotate=67.5] (.45,0) node { $\theta$}; + \end{scope} - \end{tikzpicture} + \end{tikzpicture} - - +     + -

    diff --git a/ptx/sec_double_int_polar.ptx b/ptx/sec_double_int_polar.ptx index 6d47f3cd9..6a091be6b 100644 --- a/ptx/sec_double_int_polar.ptx +++ b/ptx/sec_double_int_polar.ptx @@ -980,12 +980,16 @@ --> -

    +

    Match the correct elements on the left to the corresponding elements on the right, so that \iint_R f(x,y)\, dA is correctly converted to polar coordinates.

    +

    + When evaluating \iint_R f(x,y)\, dA using polar coordinates, + f(x,y) is replaced with and dA is replaced with . +

    - + r\cos(\theta) x diff --git a/ptx/sec_double_int_volume.ptx b/ptx/sec_double_int_volume.ptx index a2de4cef3..6281d96bc 100644 --- a/ptx/sec_double_int_volume.ptx +++ b/ptx/sec_double_int_volume.ptx @@ -1819,7 +1819,7 @@

    An integral can be interpreted as giving the signed area over an interval; - a double integral can be interpreted as giving the signed over a region. + a double integral can be interpreted as giving the signed over a region.

    diff --git a/ptx/sec_graph_extreme_values.ptx b/ptx/sec_graph_extreme_values.ptx index 93c66850a..a209cf70a 100644 --- a/ptx/sec_graph_extreme_values.ptx +++ b/ptx/sec_graph_extreme_values.ptx @@ -775,7 +775,7 @@
    - + x f(x) @@ -879,7 +879,7 @@
    - + x f(x) @@ -983,7 +983,7 @@
    - + x f(x) @@ -1094,7 +1094,7 @@
    - + x f(x) diff --git a/ptx/sec_greensthm.ptx b/ptx/sec_greensthm.ptx index 8bc7aa146..6c1c09be9 100644 --- a/ptx/sec_greensthm.ptx +++ b/ptx/sec_greensthm.ptx @@ -1423,8 +1423,8 @@

    Let \vec F be a vector field and let C be a curve. - Flow is a measure of the amount of \vec F going C; - flux is a measure of the amount of \vec F going C. + Flow is a measure of the amount of \vec F going C; + flux is a measure of the amount of \vec F going C.

    @@ -1461,7 +1461,7 @@

    Green's Theorem states, informally, that the circulation around a closed curve that bounds a region R is equal to the sum of the - of \vec{F} across R. + of \vec{F} across R.

    @@ -1478,7 +1478,7 @@

    The Divergence Theorem states, informally, that the outward flux across a closed curve that bounds a region R - is equal to the sum of the of \vec{F} across R. + is equal to the sum of the of \vec{F} across R.

    @@ -1500,7 +1500,7 @@

    Let \vec F be a vector field and let C_1 and C_2 be any nonintersecting paths except that each starts at point A and ends at point B. - If the of \vec{F} is 0, + If the of \vec{F} is 0, then \int_{C_1} \vec F\cdot \vec T\, ds = \int_{C_2} \vec F\cdot \vec T\, ds.

    @@ -1518,7 +1518,7 @@

    Let \vec F be a vector field and let C_1 and C_2 be any nonintersecting paths except that each starts at point A and ends at point B. - If the of \vec{F} is 0, + If the of \vec{F} is 0, then \int_{C_1} \vec F\cdot \vec n\, ds = \int_{C_2} \vec F\cdot \vec n\, ds.

    diff --git a/ptx/sec_hyperbolic.ptx b/ptx/sec_hyperbolic.ptx index e6c1ef962..cfd78f0ff 100644 --- a/ptx/sec_hyperbolic.ptx +++ b/ptx/sec_hyperbolic.ptx @@ -781,7 +781,7 @@ Domains and ranges of the hyperbolic and inverse hyperbolic functions - + Function Domain Range diff --git a/ptx/sec_improper_integration.ptx b/ptx/sec_improper_integration.ptx index ce7b28413..f75acc9f3 100644 --- a/ptx/sec_improper_integration.ptx +++ b/ptx/sec_improper_integration.ptx @@ -1325,7 +1325,7 @@

    If \lim\limits_{b\to \infty} \int_0^b f(x)\, dx exists, - then the integral \ds \int_0^\infty f(x)\, dx is said to . + then the integral \ds \int_0^\infty f(x)\, dx is said to .

    @@ -1351,7 +1351,7 @@

    If \ds \int_1^\infty f(x)\, dx=10, and 0\leq g(x)\leq f(x) for all x, - then we know that \ds \int_1^\infty g(x)\, dx . + then we know that \ds \int_1^\infty g(x)\, dx .

    diff --git a/ptx/sec_int_comp_tests.ptx b/ptx/sec_int_comp_tests.ptx index 5dc0ab996..af755b346 100644 --- a/ptx/sec_int_comp_tests.ptx +++ b/ptx/sec_int_comp_tests.ptx @@ -513,6 +513,18 @@ Since the limit on the left side diverges to \infty, we can say that \lim\limits_{n \to \infty}\sum_{i=N}^n b_i also diverges to \infty.

    +

    @@ -520,18 +532,7 @@ -
    diff --git a/ptx/sec_iterated_integrals.ptx b/ptx/sec_iterated_integrals.ptx index c31635dde..9536cdcd7 100644 --- a/ptx/sec_iterated_integrals.ptx +++ b/ptx/sec_iterated_integrals.ptx @@ -922,8 +922,8 @@

    When evaluating an iterated integral, - we integrate from to , - then from to . + we integrate from to , + then from to .

    @@ -958,7 +958,7 @@

    One understanding of an iterated integral is that - \ds \int_a^b\int_{g_1(x)}^{g_2(x)} \, dy\, dx gives the of a plane region. + \ds \int_a^b\int_{g_1(x)}^{g_2(x)} \, dy\, dx gives the of a plane region.

    diff --git a/ptx/sec_lhopitals_rule.ptx b/ptx/sec_lhopitals_rule.ptx index 495c466c8..e9ac17604 100644 --- a/ptx/sec_lhopitals_rule.ptx +++ b/ptx/sec_lhopitals_rule.ptx @@ -693,8 +693,8 @@

    Fill in the blanks: The Quotient Rule is applied to - \ds \frac{f(x)}{g(x)} when taking ; - l'Hospital's Rule is applied when taking certain . + \ds \frac{f(x)}{g(x)} when taking ; + l'Hospital's Rule is applied when taking certain .

    @@ -814,7 +814,7 @@

    - \lim\limits_{x\to 1} \frac{ }{} + \lim\limits_{x\to 1} \frac{ }{}

    diff --git a/ptx/sec_limit_continuity.ptx b/ptx/sec_limit_continuity.ptx index 612bf9227..33ad59c5d 100644 --- a/ptx/sec_limit_continuity.ptx +++ b/ptx/sec_limit_continuity.ptx @@ -1425,7 +1425,7 @@

    Illustrating the relationship between the angle between vectors and the sign of their dot product Finding the extreme values of f(x)= 2x^3+3x^2-12x in Finding the extreme values of a piecewise-defined function in Finding the extrema of f(x)= \cos\mathopen{}\left(x^2\right)\mathclose{} in Finding the extrema of the half-circle in Video presentation of
    Iterations of the Bisection Method of Root Finding - + Iteration # Interval Midpoint Sign @@ -3274,7 +3274,7 @@ If f(m)\gt 0 at the midpoint m, put + for the midpoint sign. If f(m)\lt 0, put - for the midpoint sign. - + Iteration Interval Midpoint Sign @@ -3364,7 +3364,7 @@ If f(m)\gt 0 at the midpoint m, put + for the midpoint sign. If f(m)\lt 0, put - for the midpoint sign. - + Iteration Interval Midpoint Sign @@ -3454,7 +3454,7 @@ If f(m)\gt 0 at the midpoint m, put + for the midpoint sign. If f(m)\lt 0, put - for the midpoint sign. - + Iteration Interval Midpoint Sign @@ -3544,7 +3544,7 @@ If f(m)\gt 0 at the midpoint m, put + for the midpoint sign. If f(m)\lt 0, put - for the midpoint sign. - + Iteration Interval Midpoint Sign diff --git a/ptx/sec_limit_infty.ptx b/ptx/sec_limit_infty.ptx index 55dc94d27..6fb8b198b 100644 --- a/ptx/sec_limit_infty.ptx +++ b/ptx/sec_limit_infty.ptx @@ -759,7 +759,7 @@
    - + x \sin(x)/x @@ -295,7 +295,7 @@
    - + x \sin(x)/x @@ -428,7 +428,7 @@
    - + x \frac{x^2-x-6}{6x^2-19x+3} @@ -591,7 +591,7 @@
    - + x f(x) @@ -764,7 +764,7 @@
    - + x f(x) @@ -857,7 +857,7 @@
    - + x f(x) @@ -1094,7 +1094,7 @@
    - + x \sin(1/x) @@ -1434,7 +1434,7 @@
    - + h \frac{f(1+h)-f(1)}{h} @@ -1830,7 +1830,7 @@ -->

    - An expression of the form \frac{0}{0} is called . + An expression of the form \frac{0}{0} is called .

    @@ -1917,7 +1917,7 @@

    When x is near 0, - \dfrac{\sin(x)}{x} is near the value . + \dfrac{\sin(x)}{x} is near the value .

    - Two distinct lines in the plane can intersect or be . + Two distinct lines in the plane can intersect or be .

    diff --git a/ptx/sec_multi_chain.ptx b/ptx/sec_multi_chain.ptx index 2c009e2ee..a6505848e 100644 --- a/ptx/sec_multi_chain.ptx +++ b/ptx/sec_multi_chain.ptx @@ -821,7 +821,7 @@

    Fill in the blank: The single variable Chain Rule states - \ds\frac{d}{dx}\Big(f\big(g(x)\big)\Big) = \fp\big(g(x)\big)\cdot. + \ds\frac{d}{dx}\Big(f\big(g(x)\big)\Big) = \fp\big(g(x)\big)\cdot.

    @@ -921,7 +921,7 @@

    - The Multivariable Chain Rule allows us to compute implicit derivatives easily by just computing two derivatives. + The Multivariable Chain Rule allows us to compute implicit derivatives easily by just computing two derivatives.

    diff --git a/ptx/sec_multi_intro.ptx b/ptx/sec_multi_intro.ptx index 5f64ca6f9..b24ef7fc4 100644 --- a/ptx/sec_multi_intro.ptx +++ b/ptx/sec_multi_intro.ptx @@ -881,7 +881,7 @@
    - + c r @@ -980,7 +980,7 @@ -->

    - The graph of a function of two variables is a . + The graph of a function of two variables is a .

    @@ -999,7 +999,7 @@ -->

    - Most people are familiar with the concept of level curves in the context of maps. + Most people are familiar with the concept of level curves in the context of maps.

    @@ -1037,7 +1037,7 @@ -->

    - The analogue of a level curve for functions of three variables is a level . + The analogue of a level curve for functions of three variables is a level .

    diff --git a/ptx/sec_newton.ptx b/ptx/sec_newton.ptx index b94333e79..4fd44fe46 100644 --- a/ptx/sec_newton.ptx +++ b/ptx/sec_newton.ptx @@ -530,19 +530,19 @@

    Approximating the solution to \cos(x)=x.

     -

    Correct—this equation cannot be solved algebraically but Newton's Method can approximate the solution.

     +

    Correctthis equation cannot be solved algebraically but Newton's Method can approximate the solution.

    Finding the exact solution to x^2=9.

     -

    Not needed—this equation has an exact solution, so Newton's Method is unnecessary.

     +

    Not neededthis equation has an exact solution, so Newton's Method is unnecessary.

    Programming a calculator or computer to approximate roots efficiently.

     -

    Correct—Newton's Method is fast and widely used in computational root finding.

     +

    CorrectNewton's Method is fast and widely used in computational root finding.

    Estimating the slope of a tangent line at a point.

     -

    No—Newton's Method uses tangent slopes, but its goal is to find roots, not slopes.

     +

    NoNewton's Method uses tangent slopes, but its goal is to find roots, not slopes.

         diff --git a/ptx/sec_numerical_integration.ptx b/ptx/sec_numerical_integration.ptx index f79155539..509a1280a 100644 --- a/ptx/sec_numerical_integration.ptx +++ b/ptx/sec_numerical_integration.ptx @@ -377,7 +377,7 @@
    - + x_i Exact Approx. @@ -710,7 +710,7 @@
    - + x_i e^{-x_i^2} @@ -905,7 +905,7 @@
    - + x_i e^{-x_i^2} @@ -972,7 +972,7 @@
    - + \overline{x_i} Exact Approx. @@ -1313,7 +1313,7 @@
    - + x_i \sin(x_i^3) @@ -1971,7 +1971,7 @@
    - + TimeSpeed @@ -2191,7 +2191,7 @@ Simpson's Rule is based on approximating portions of a function with what type of function?

    - +

    diff --git a/ptx/sec_par_calc.ptx b/ptx/sec_par_calc.ptx index 5f91f4226..98fa296f5 100644 --- a/ptx/sec_par_calc.ptx +++ b/ptx/sec_par_calc.ptx @@ -371,7 +371,7 @@ any line that passes through the center of a circle intersects the circle at right angles.

    -
    +
    @@ -661,7 +661,7 @@

    -
    +
    @@ -789,38 +789,12 @@ .

    -

    - The points of inflection are found by setting \frac{d^2y}{dx^2}=0. - This is not trivial, - as equations that mix polynomials and trigonometric functions generally do not have nice solutions. -

    - -

    - In we see a plot of the second derivative. - It shows that it has zeros at approximately t=0.5,\,3.5,\,6.5,\,9.5,\,12.5 and 16. - These approximations are not very good, - made only by looking at the graph. - Newton's Method provides more accurate approximations. - Accurate to 2 decimal places, we have: - - t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74 - . -

    - -

    - The corresponding points have been plotted on the graph of the parametric equations in . - Note how most occur near the x-axis, - but not exactly on the axis. -

    - -
    -
    - -
    -
    + + - - Graph of the second derivative is a sinusoid with increasing amplitude. + Graph of the second derivative is a sinusoid with increasing amplitude.

    The image shows the graph y = 2\cos(t)-4t\sin(t), which is a graph of \frac{d^2y}{dx^2}. @@ -848,15 +822,40 @@ \end{tikzpicture} - + -

    -
    + + - - Graph of the parametric curve in this example, with points of inflection marked. + Graph of the parametric curve in this example, with points of inflection marked.

    The graph shows a curve that appears to be sinusoidal, but with a frequency that increases with x. @@ -892,9 +891,6 @@ - - - @@ -1281,7 +1277,7 @@ Find the surface area if this shape is rotated about the x-axis, as shown in .

    -
    +
    diff --git a/ptx/sec_param_eqs.ptx b/ptx/sec_param_eqs.ptx index 7e221ffa2..fb1b71318 100644 --- a/ptx/sec_param_eqs.ptx +++ b/ptx/sec_param_eqs.ptx @@ -187,8 +187,8 @@
    @@ -1538,7 +1538,7 @@ illustrating the cusp at (1,4).

    -
    +
    diff --git a/ptx/sec_partial_fraction.ptx b/ptx/sec_partial_fraction.ptx index bb35a2594..c30515b11 100644 --- a/ptx/sec_partial_fraction.ptx +++ b/ptx/sec_partial_fraction.ptx @@ -683,7 +683,7 @@ -->

    - Partial Fraction Decomposition is a method of rewriting functions. + Partial Fraction Decomposition is a method of rewriting functions.

    diff --git a/ptx/sec_polar.ptx b/ptx/sec_polar.ptx index 9a61b34a8..3bc2c1380 100644 --- a/ptx/sec_polar.ptx +++ b/ptx/sec_polar.ptx @@ -48,7 +48,7 @@ coordinatespolar

    -
    +
    Illustration of polar coordinates relative to a pole and initial ray. @@ -706,7 +706,7 @@ This graph is also plotted in .

    -
    +
    @@ -811,7 +811,7 @@
    - Pt.\theta\cos(2\theta) + Pt.\theta\cos(2\theta) 101 2\pi/60.5 3\pi/40 diff --git a/ptx/sec_polarcalc.ptx b/ptx/sec_polarcalc.ptx index 3403e0cf5..f0175ac7d 100644 --- a/ptx/sec_polarcalc.ptx +++ b/ptx/sec_polarcalc.ptx @@ -126,7 +126,7 @@ .

    -
    +
    @@ -940,7 +940,7 @@ Find the area bounded between the polar curves r=1 and r=2\cos(2\theta), as shown in .

    -
    +
    A zoomed in view of a region bounded by a circle, a rose curve, and the x axis. @@ -995,7 +995,7 @@ .

    -
    +
    A zoomed in view of a polar region, showing it divided into two parts. @@ -1157,7 +1157,7 @@ .

    -
    +
    diff --git a/ptx/sec_power_series.ptx b/ptx/sec_power_series.ptx index 69b5f6f5a..b284f7502 100644 --- a/ptx/sec_power_series.ptx +++ b/ptx/sec_power_series.ptx @@ -246,43 +246,41 @@ Determining the Radius and Interval of Convergence - -

    - Given the power series \ds \infser[0] a_n(x-c)^n, - apply the ratio test to the series \ds \infser[0]\abs{a_n (x-c)^n}. - The result will be L\abs{x-c}, where \ds L=\lim_{n\to\infty}\frac{\abs{a_{n+1}}}{\abs{a_n}}. -

    - -

    -

      -
    1. -

      - If L=0, then the power series converges for every x - by the ratio test, since L\abs{x-c}=0\lt 1. -

      -
      2. +

      + Given the power series \ds \infser[0] a_n(x-c)^n, + apply the ratio test to the series \ds \infser[0]\abs{a_n (x-c)^n}. + The result will be L\abs{x-c}, where \ds L=\lim_{n\to\infty}\frac{\abs{a_{n+1}}}{\abs{a_n}}. +

      + +

      +

        +
      1. +

        + If L=0, then the power series converges for every x + by the ratio test, since L\abs{x-c}=0\lt 1. +

        +
        2. -
      2. -

        - If L=\infty, then power series converges only when x=c. -

        -
        3. +
      3. +

        + If L=\infty, then power series converges only when x=c. +

        +
        4. -
      4. -

        - If 0\lt L\lt \infty, then R=1/L is the radius of convergence: - by the ratio test, the series converges when \abs{x-c}\lt R. -

        +
      5. +

        + If 0\lt L\lt \infty, then R=1/L is the radius of convergence: + by the ratio test, the series converges when \abs{x-c}\lt R. +

        -

        - To determine the interval of convergence, plug the endpoints (x=c-R and x=c+R) - into the power series, and test the resulting series for convergence. - If the series converges, we include the endpoint. If it diverges, we exclude the endpoint. -

        -
        6. -
      -

      - +

      + To determine the interval of convergence, plug the endpoints (x=c-R and x=c+R) + into the power series, and test the resulting series for convergence. + If the series converges, we include the endpoint. If it diverges, we exclude the endpoint. +

      + +
    +

    @@ -924,7 +922,7 @@ -->

    - We adopt the convention that x^0=, + We adopt the convention that x^0=, regardless of the value of x.

    @@ -995,7 +993,7 @@ \ds\infser[0] a_nx^n is 5, then the radius of convergence of \ds\infser[0] (-1)^na_nx^n is - . + .

    - A fundamental calculus technique is to use + A fundamental calculus technique is to use to refine approximations to get an exact answer.

    diff --git a/ptx/sec_sequences.ptx b/ptx/sec_sequences.ptx index fdd2bb488..806f90308 100644 --- a/ptx/sec_sequences.ptx +++ b/ptx/sec_sequences.ptx @@ -733,7 +733,7 @@

    -
    + A scatter plot showing a representative sample of points from the third sequence in this example. @@ -1514,7 +1514,7 @@

      -
    1. +

    2. a_{n+1}-a_n \amp = \frac{n+2}{n+1} - \frac{n+1}{n} @@ -1523,84 +1523,13 @@ \amp \lt 0 \text{ for all \(n\). } Since a_{n+1}-a_n\lt 0 for all n, - we conclude that the sequence is decreasing. -

      -
      3. - -
    3. -

      - - a_{n+1}-a_n \amp = \frac{(n+1)^2+1}{n+2} - \frac{n^2+1}{n+1} - \amp = \frac{\big((n+1)^2+1\big)(n+1)- (n^2+1)(n+2)}{(n+1)(n+2)} - \amp = \frac{n^2+3n}{(n+1)(n+2)} - \amp \gt 0 \text{ for all \(n\). } - - Since a_{n+1}-a_n\gt 0 for all n, - we conclude the sequence is increasing. -

      -
      4. - -
    4. -

      - We can clearly see in , - where the sequence is plotted, that it is not monotonic. - However, it does seem that after the first 4 terms it is decreasing. - To understand why, perform the same analysis as done before: - - - a_{n+1}-a_n \amp = \frac{(n+1)^2-9}{(n+1)^2-10(n+1)+26} - \frac{n^2-9}{n^2-10n+26} - \amp = \frac{n^2+2n-8}{n^2-8n+17}-\frac{n^2-9}{n^2-10n+26} - \amp = \frac{(n^2+2n-8)(n^2-10n+26)-(n^2-9)(n^2-8n+17)}{(n^2-8n+17)(n^2-10n+26)} - \amp = \frac{-10n^2+60n-55}{(n^2-8n+17)(n^2-10n+26)} - . + we conclude that the sequence is decreasing, illustrated in .

      -

      - We want to know when this is greater than, or less than, 0. - The denominator is always positive, - therefore we are only concerned with the numerator. - For small values of n, - the numerator is positive. - As n grows large, - the numerator is dominated by -10n^2, - meaning the entire fraction will be negative; - , for large enough n, a_{n+1}-a_n \lt 0. - Using the quadratic formula we can determine that the numerator is negative for n\geq 5. - In short, the sequence is simply not monotonic, - though it is useful to note that for n\geq 5, - the sequence is monotonically decreasing. -

      -
      5. - -
    5. - -

      - Again, the plot in - shows that the sequence is not monotonic, - but it suggests that it is monotonically decreasing after the first term. - We perform the usual analysis to confirm this. - - a_{n+1}-a_n \amp = \frac{(n+1)^2}{(n+1)!} - \frac{n^2}{n!} - \amp = \frac{(n+1)^2-n^2(n+1)}{(n+1)!} - \amp = \frac{-n^3+2n+1}{(n+1)!} - - When n=1, the above expression is \gt 0; - for n\geq 2, the above expression is \lt 0. - Thus this sequence is not monotonic, - but it is monotonically decreasing after the first term. -

      -
      6. -
    -

    - -
    -
    - - -
    -
    - + Plot of the first sequence in this example. It is decreasing and bounded below.

    @@ -1631,11 +1560,24 @@ + + +

  • +

    + + a_{n+1}-a_n \amp = \frac{(n+1)^2+1}{n+2} - \frac{n^2+1}{n+1} + \amp = \frac{\big((n+1)^2+1\big)(n+1)- (n^2+1)(n+2)}{(n+1)(n+2)} + \amp = \frac{n^2+3n}{(n+1)(n+2)} + \amp \gt 0 \text{ for all \(n\). } + + Since a_{n+1}-a_n\gt 0 for all n, + we conclude the sequence is increasing, illustrated in . +

    -
    -
    • - + Scatter plot for the second sequence in this example. It is increasing but not bounded.

    @@ -1667,13 +1609,43 @@ - + + +

  • +

    + We can clearly see in , + where the sequence is plotted, that it is not monotonic. + However, it does seem that after the first 4 terms it is decreasing. + To understand why, perform the same analysis as done before: + + + a_{n+1}-a_n \amp = \frac{(n+1)^2-9}{(n+1)^2-10(n+1)+26} - \frac{n^2-9}{n^2-10n+26} + \amp = \frac{n^2+2n-8}{n^2-8n+17}-\frac{n^2-9}{n^2-10n+26} + \amp = \frac{(n^2+2n-8)(n^2-10n+26)-(n^2-9)(n^2-8n+17)}{(n^2-8n+17)(n^2-10n+26)} + \amp = \frac{-10n^2+60n-55}{(n^2-8n+17)(n^2-10n+26)} + . +

    - -
    -
    • - + Scatter plot for the third sequence in this example. It is not monotonic.

    @@ -1710,12 +1682,32 @@ -

    -
    - + Scatter plot for the last sequence in this example. It is not monotonic.

    @@ -1754,9 +1746,17 @@ - - - + + + +

    + + + + + + + Video solution @@ -1864,7 +1864,7 @@ -->

    - The domain of a sequence is the numbers. + The domain of a sequence is the numbers.

    diff --git a/ptx/sec_series.ptx b/ptx/sec_series.ptx index 68de229a4..7fcb7c391 100644 --- a/ptx/sec_series.ptx +++ b/ptx/sec_series.ptx @@ -983,7 +983,7 @@ Partial sums of the series are plotted in .

    -
    +
    @@ -1087,7 +1087,7 @@

      -
    1. +

    2. We can decompose the fraction 2/(n^2+2n) as @@ -1121,47 +1121,11 @@ so \infser \frac1{n^2+2n} = \frac32. This is illustrated in .

      -
      3. - -
    3. -

      - We begin by writing the first few partial sums of the series: - - S_1 \amp = \ln\left(2\right) - S_2 \amp = \ln\left(2\right)+\ln\left(\frac32\right) - S_3 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right) - S_4 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right)+\ln\left(\frac54\right) - - At first, this does not seem helpful, - but recall the logarithmic identity: - \ln(x) +\ln(y) = \ln(xy). - Applying this to S_4 gives: - - S_4 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right)+\ln\left(\frac54\right) - \amp = \ln\left(\frac21\cdot\frac32\cdot\frac43\cdot\frac54\right) = \ln\left(5\right) - . - We can conclude that \{S_n\} = \big\{\ln(n+1)\big\}. - This sequence does not converge, - as \lim\limits_{n\to\infty}S_n=\infty. - Therefore \ds\infser \ln\left(\frac{n+1}{n}\right)=\infty; - the series diverges. - Note in how the sequence of partial sums grows slowly; - after 100 terms, it is not yet over 5. - Graphically we may be fooled into thinking the series converges, - but our analysis above shows that it does not. -

      -
      4. -
    -

    - -
    -
    - -
    -
    + + Scatter plots of the sequence, and corresponding partial sums, for the first part of this example.

    @@ -1204,13 +1168,42 @@ - - + + + -

    -
    + + Scatter plots of the sequence, and corresponding partial sums, for the second part of this example.

    @@ -1255,8 +1248,10 @@ - - + + +

    + Video solution @@ -1388,7 +1383,7 @@

      -
    1. +

    2. We start by using algebra to break the series apart: @@ -1398,31 +1393,10 @@ . This is illustrated in .

      -
      3. - -
    3. -

      - This looks very similar to the series that involves e in . - Note, however, - that the series given in this example starts with n=1 and not n=0. - The first term of the series in the Key Idea is 1/0! = 1, - so we will subtract this from our result below: - - \infser \frac{1000}{n!} \amp = 1000\cdot\infser \frac{1}{n!} - \amp = 1000\cdot (e-1) \approx 1718.28 - . - This is illustrated in . - The graph shows how this particular series converges very rapidly. -

      - -
      -
    - - -
    -
    + + Scatter plots of the sequence, and corresponding partial sums, for the first part of this example.

    @@ -1468,13 +1442,28 @@ - + + -

    -
    + + Scatter plots of the sequence, and corresponding partial sums, for the first part of this example.

    @@ -1514,13 +1503,11 @@ - - - - + + -

  • +

  • The denominators in each term are perfect squares; we are adding \ds \sum_{n=4}^\infty \frac{1}{n^2} diff --git a/ptx/sec_shell_method.ptx b/ptx/sec_shell_method.ptx index 2185229f5..d7df5a394 100644 --- a/ptx/sec_shell_method.ptx +++ b/ptx/sec_shell_method.ptx @@ -1447,7 +1447,7 @@

    - + Washer Method diff --git a/ptx/sec_space_coord.ptx b/ptx/sec_space_coord.ptx index 967e484ca..2ccd8fb14 100644 --- a/ptx/sec_space_coord.ptx +++ b/ptx/sec_space_coord.ptx @@ -2186,7 +2186,7 @@ - + Plane Trace @@ -2361,7 +2361,7 @@ - + Plane Trace @@ -2626,7 +2626,7 @@ - + Plane Trace @@ -2818,7 +2818,7 @@ - + Plane Trace @@ -3027,7 +3027,7 @@ - + Plane Trace @@ -3235,7 +3235,7 @@ - + Plane Trace @@ -4196,8 +4196,8 @@

    In the plane, - the equation x=2 defines a ; - in space, x=2 defines a . + the equation x=2 defines a ; + in space, x=2 defines a .

    diff --git a/ptx/sec_substitution.ptx b/ptx/sec_substitution.ptx index e740624be..245aaf373 100644 --- a/ptx/sec_substitution.ptx +++ b/ptx/sec_substitution.ptx @@ -1485,7 +1485,7 @@

    - +

    diff --git a/ptx/sec_surface_area.ptx b/ptx/sec_surface_area.ptx index 0262eacac..66d4bdd8d 100644 --- a/ptx/sec_surface_area.ptx +++ b/ptx/sec_surface_area.ptx @@ -811,7 +811,7 @@

    To approximate the area of a small portion of a surface, - we computed the area of its plane. + we computed the area of its plane.

    @@ -839,16 +839,14 @@ - - surface area + +

    + Your answer should be plural. +

    +     - -

    - Your answer should be plural. -

    -     diff --git a/ptx/sec_surface_integral.ptx b/ptx/sec_surface_integral.ptx index 72cf716ad..1418633de 100644 --- a/ptx/sec_surface_integral.ptx +++ b/ptx/sec_surface_integral.ptx @@ -790,8 +790,8 @@

    In the plane, - flux is a measurement of how much of the vector field passes across a ; - in space, flux is a measurement of how much of the vector field passes across a . + flux is a measurement of how much of the vector field passes across a ; + in space, flux is a measurement of how much of the vector field passes across a .

    @@ -830,7 +830,7 @@

    When \surfaceS is a closed surface, - we choose the normal vector so that it points to the of the surface. + we choose the normal vector so that it points to the of the surface.

    @@ -847,7 +847,7 @@

    If \surfaceS is a plane, and \vec F is always parallel to \surfaceS, - then the flux of \vec F across \surfaceS will be . + then the flux of \vec F across \surfaceS will be .

    diff --git a/ptx/sec_tan_norm.ptx b/ptx/sec_tan_norm.ptx index 098580294..1ae2e6201 100644 --- a/ptx/sec_tan_norm.ptx +++ b/ptx/sec_tan_norm.ptx @@ -68,7 +68,7 @@ since they are only length 1.)

    -
    +
    • @@ -382,7 +382,7 @@ These are sketched in .

    -
    +
    @@ -447,17 +447,7 @@ - +

    The previous example was once again @@ -517,7 +507,7 @@ we compute the unit tangent and normal vectors for t=-1,0 and 1 and sketch them in .

    -
    +
    @@ -607,6 +597,18 @@

    + +

    + A brief consideration of this theorem may make one wonder: what if the graph of \vec r does + not have a concave side? What if \vec r is a line? +

    +

    + This exposes a shortcoming in our definition of \unitnormal(t), where + we require that \unittangent(t) be smooth, i.e., that \unittangentprime(t) \neq \vec 0, + a requirement that lines do not fulfill. One may still want to compute a normal vector for a given line, though. + For straight lines in the x,y plane, it is most common to orient the normal vector 90^\circ + counterclockwise from the tangent vector. For lines in three dimensions, there is no preferred choice of normal vector. +

    @@ -802,7 +804,7 @@ gives a graph of the path for reference.

    -
    +
    @@ -875,7 +877,7 @@ which we plot in .

    -
    +
    @@ -966,7 +968,7 @@
    - + t a_\text{T} a_\text{N} @@ -1044,10 +1046,7 @@

    If \unittangent(t) is a unit tangent vector, what is \norm{\unittangent(t)}? -

    - -

    - +

    @@ -1073,10 +1072,7 @@

    If \unitnormal(t) is a unit normal vector, what is \unitnormal(t)\cdot \vrp(t)? -

    - -

    - +

    @@ -1119,7 +1115,7 @@ -->

    - a_\text{T} measures how much the acceleration is affecting the of an object. + a_\text{T} measures how much the acceleration is affecting the of an object.

    diff --git a/ptx/sec_taylor_poly.ptx b/ptx/sec_taylor_poly.ptx index d9fa63ddf..254dbb084 100644 --- a/ptx/sec_taylor_poly.ptx +++ b/ptx/sec_taylor_poly.ptx @@ -75,15 +75,13 @@
    - -

    - - f(0) \amp = 2 \amp \fp''(0) \amp =-1 - \fp(0) \amp = 1 \amp f^{(4)}(0) \amp = -12 - \fpp(0) \amp = 2 \amp f^{(5)}(0) \amp = -19 - -

    -     +

    + + f(0) \amp = 2 \amp \fp''(0) \amp =-1 + \fp(0) \amp = 1 \amp f^{(4)}(0) \amp = -12 + \fpp(0) \amp = 2 \amp f^{(5)}(0) \amp = -19 + +

    @@ -1923,7 +1909,7 @@ Note how similar they are near x=0.

    -
    +
    diff --git a/ptx/sec_taylor_series.ptx b/ptx/sec_taylor_series.ptx index b1c4b89e7..c6afcefdf 100644 --- a/ptx/sec_taylor_series.ptx +++ b/ptx/sec_taylor_series.ptx @@ -95,22 +95,20 @@
    - -

    - - f(x) \amp = \cos(x) \amp f(0) \amp = 1 - f'(x) \amp = -\sin(x) \amp \fp(0) \amp = 0 - \fp'(x) \amp = -\cos(x) \amp \fp'(0) \amp = -1 - \fp''(x) \amp = \sin(x) \amp \fp''(0) \amp = 0 - f^{(4)}(x) \amp = \cos(x) \amp f^{(4)}(0) \amp =1 - f^{(5)}(x) \amp = -\sin(x) \amp f^{(5)}(0) \amp =0 - f^{(6)}(x) \amp = -\cos(x) \amp f^{(6)}(0) \amp =-1 - f^{(7)}(x) \amp = \sin(x) \amp f^{(7)}(0) \amp =0 - f^{(8)}(x) \amp = \cos(x) \amp f^{(8)}(0) \amp =1 - f^{(9)}(x) \amp = -\sin(x) \amp f^{(9)}(0) \amp =0 - -

    -     +

    + + f(x) \amp = \cos(x) \amp f(0) \amp = 1 + f'(x) \amp = -\sin(x) \amp \fp(0) \amp = 0 + \fp'(x) \amp = -\cos(x) \amp \fp'(0) \amp = -1 + \fp''(x) \amp = \sin(x) \amp \fp''(0) \amp = 0 + f^{(4)}(x) \amp = \cos(x) \amp f^{(4)}(0) \amp =1 + f^{(5)}(x) \amp = -\sin(x) \amp f^{(5)}(0) \amp =0 + f^{(6)}(x) \amp = -\cos(x) \amp f^{(6)}(0) \amp =-1 + f^{(7)}(x) \amp = \sin(x) \amp f^{(7)}(0) \amp =0 + f^{(8)}(x) \amp = \cos(x) \amp f^{(8)}(0) \amp =1 + f^{(9)}(x) \amp = -\sin(x) \amp f^{(9)}(0) \amp =0 + +

    @@ -750,7 +750,7 @@ The displacement of \vec r(t) on [-1,1] is thus \vec d = \la 0,1\ra - \la 0,-1\ra = \la 0,2\ra.

    -
    +
    @@ -924,7 +924,7 @@

    When sketching vector-valued functions, - technically one isn't graphing points, but rather . + technically one isn't graphing points, but rather .

    @@ -943,7 +943,7 @@ -->

    - It can be useful to think of as a vector that points from a starting position to an ending position. + It can be useful to think of as a vector that points from a starting position to an ending position.

    @@ -960,7 +960,7 @@

    In the context of vector-valued functions, - average rate of change is divided by time. + average rate of change is divided by time.

    diff --git a/ptx/sec_vvf_calc.ptx b/ptx/sec_vvf_calc.ptx index e63c127f2..ac5d300b4 100644 --- a/ptx/sec_vvf_calc.ptx +++ b/ptx/sec_vvf_calc.ptx @@ -19,7 +19,7 @@ The theorem following the definition shows that in practice, taking limits of vector-valued functions is no more difficult than taking limits of real-valued functions.

    -     @@ -1070,7 +1071,7 @@

    -
    +
    @@ -1454,7 +1455,7 @@ -->

    - Limits, derivatives and integrals of vector-valued functions are all evaluated -wise. + Limits, derivatives and integrals of vector-valued functions are all evaluated -wise.

    @@ -1473,7 +1474,7 @@ -->

    - The definite integral of a velocity function gives . + The definite integral of a velocity function gives .

    @@ -2312,7 +2313,7 @@

    - \displaystyle\int_0^{\pi} \la -\sin(t) ,\cos(t) \ra\,dt=. + \displaystyle\int_0^{\pi} \la -\sin(t) ,\cos(t) \ra\,dt=.

    diff --git a/ptx/sec_vvf_motion.ptx b/ptx/sec_vvf_motion.ptx index 783d9a15b..b1f342e2b 100644 --- a/ptx/sec_vvf_motion.ptx +++ b/ptx/sec_vvf_motion.ptx @@ -863,6 +863,8 @@ .

    +

    + Projectile Motion

    diff --git a/ptx/sec_work.ptx b/ptx/sec_work.ptx index e92b03bfe..80667c831 100644 --- a/ptx/sec_work.ptx +++ b/ptx/sec_work.ptx @@ -454,7 +454,7 @@

    - + x f(x) @@ -2442,7 +2442,7 @@
    1. - + x f(x) @@ -2465,7 +2465,7 @@
    2. - + x f(x) @@ -2562,7 +2562,7 @@
      1. - + x f(x) @@ -2585,7 +2585,7 @@
      2. - + x f(x) @@ -2682,7 +2682,7 @@
        1. - + x f(x) @@ -2705,7 +2705,7 @@
        2. - + x f(x) @@ -2802,7 +2802,7 @@
          1. - + x f(x) @@ -2825,7 +2825,7 @@
          2. - + x f(x) diff --git a/ptx/sec_limit_intro.ptx b/ptx/sec_limit_intro.ptx index 2a49b0a3d..75dc711bb 100644 --- a/ptx/sec_limit_intro.ptx +++ b/ptx/sec_limit_intro.ptx @@ -245,7 +245,7 @@
    Values of \sin(x)/x with x near 1 Values of \sin(x)/x with x near 0 Numerically approximating a limit in Numerically approximating a limit in Values of f(x) near x=1 in Values of f(x) near x=1 in Observing that f(x)=\sin(1/x) has no limit as x\to0 in The difference quotient evaluated at values of h near 0 A table of c values and the corresponding radius r of the spheres of constant value in Values used to approximate \int_{-\frac{\pi}4}^{\frac{\pi}2}\sin(x^3)\, dx in A table of values of e^{-x^2} A table of values of e^{-x^2} Values used to approximate \int_{-\frac{\pi}4}^{\frac{\pi}2}\sin(x^3)\, dx in - + x_i e^{-x_i^2} @@ -1439,7 +1439,7 @@
    Values used to approximate \int_{-\frac{\pi}4}^{\frac{\pi}2}\sin(x^3)\, dx in Speed data collected at 30 second intervals for Illustrating how a circle's normal lines pass through its center Graphing the parametric equations in to demonstrate concavity In (a), a graph of \frac{d^2y}{dx^2}, showing where it is approximately 0. In (b), graph of the parametric equations in along with the points of inflection +
    +
    Graphing \frac{d^2y}{dx^2} in , showing where it is approximately 0 +

    + The points of inflection are found by setting \frac{d^2y}{dx^2}=0. + This is not trivial, + as equations that mix polynomials and trigonometric functions generally do not have nice solutions. +

    + +

    + In we see a plot of the second derivative. + It shows that it has zeros at approximately t=0.5,\,3.5,\,6.5,\,9.5,\,12.5 and 16. + These approximations are not very good, + made only by looking at the graph. + Newton's Method provides more accurate approximations. + Accurate to 2 decimal places, we have: + + t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74 + . +

    + +

    + The corresponding points have been plotted on the graph of the parametric equations in . + Note how most occur near the x-axis, + but not exactly on the axis. +

    + +
    +
    A graph of the parametric equations in along with the points of inflection Rotating a teardrop shape about the x-axis in A table of values of the parametric equations in along with a sketch of their graph - - + + t x y @@ -306,8 +306,8 @@ A table of values of the parametric equations in along with a sketch of their graph - - + + t x y @@ -1072,7 +1072,7 @@

    -
    +
    Graphing the parametric equations x=4\cos(t) +3, y=2\sin(t) +1 in Graphing the curve in ; note it is not smooth at (1,4) Illustrating polar coordinates Plotting standard polar plots - + \thetar=1+\cos(\theta) @@ -987,7 +987,7 @@
    Table of points for plotting a polar curve in The limaçon in with its tangent line at \theta=\pi/4 and points of vertical and horizontal tangency The region bounded by the functions in Breaking the region bounded by the functions in into its component parts The limaçon in whose arc length is measured Scatter plot for the sequence in Scatter plot for the sequence in of Plots of sequences in +
    +
    Plot of the sequence in of +
    +
    Plot of the sequence in of +

    + We want to know when this is greater than, or less than, 0. + The denominator is always positive, + therefore we are only concerned with the numerator. + For small values of n, + the numerator is positive. + As n grows large, + the numerator is dominated by -10n^2, + meaning the entire fraction will be negative; + , for large enough n, a_{n+1}-a_n \lt 0. + Using the quadratic formula we can determine that the numerator is negative for n\geq 5. + In short, the sequence is simply not monotonic, + though it is useful to note that for n\geq 5, + the sequence is monotonically decreasing. +

    + +
    +
    Plot of the sequence in of + + +
  • + +

    + Again, the plot in of + shows that the sequence is not monotonic, + but it suggests that it is monotonically decreasing after the first term. + We perform the usual analysis to confirm this. + + a_{n+1}-a_n \amp = \frac{(n+1)^2}{(n+1)!} - \frac{n^2}{n!} + \amp = \frac{(n+1)^2-n^2(n+1)}{(n+1)!} + \amp = \frac{-n^3+2n+1}{(n+1)!} + + When n=1, the above expression is \gt 0; + for n\geq 2, the above expression is \lt 0. + Thus this sequence is not monotonic, + but it is monotonically decreasing after the first term. +

    + +
    +
    • Plot of the sequence in of Scatter plots relating to the series of Scatter plots relating to the series in - - +
    +
    Scatter plots of the sequence and corresponding partial sums in of - - +
  • +

    + We begin by writing the first few partial sums of the series: + + S_1 \amp = \ln\left(2\right) + S_2 \amp = \ln\left(2\right)+\ln\left(\frac32\right) + S_3 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right) + S_4 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right)+\ln\left(\frac54\right) + + At first, this does not seem helpful, + but recall the logarithmic identity: + \ln(x) +\ln(y) = \ln(xy). + Applying this to S_4 gives: + + S_4 \amp = \ln\left(2\right)+\ln\left(\frac32\right)+\ln\left(\frac43\right)+\ln\left(\frac54\right) + \amp = \ln\left(\frac21\cdot\frac32\cdot\frac43\cdot\frac54\right) = \ln\left(5\right) + . + We can conclude that \{S_n\} = \big\{\ln(n+1)\big\}. + This sequence does not converge, + as \lim\limits_{n\to\infty}S_n=\infty. + Therefore \ds\infser \ln\left(\frac{n+1}{n}\right)=\infty; + the series diverges. + Note in how the sequence of partial sums grows slowly; + after 100 terms, it is not yet over 5. + Graphically we may be fooled into thinking the series converges, + but our analysis above shows that it does not. +

    + +
    +
    • Scatter plots of the sequence and corresponding partial sums in of Scatter plots relating to the series in - - +
    +
    Scatter plots of the sequence and corresponding partial sums in of - - +
  • +

    + This looks very similar to the series that involves e in . + Note, however, + that the series given in this example starts with n=1 and not n=0. + The first term of the series in the Key Idea is 1/0! = 1, + so we will subtract this from our result below: + + \infser \frac{1000}{n!} \amp = 1000\cdot\infser \frac{1}{n!} + \amp = 1000\cdot (e-1) \approx 1718.28 + . + This is illustrated in . + The graph shows how this particular series converges very rapidly. +

    +
    +
    • Scatter plots of the sequence and corresponding partial sums in of Plotting unit tangent vectors in Plotting unit tangent and normal vectors in Plotting unit tangent and normal vectors in Graphing \vec r(t) in Plotting the position of a thrown ball, with 1s increments shown A table of values of a_T and a_N in Derivatives of f evaluated at 0 A graph of y=-1/(x-1) and y=p_3(x) from Derivatives of f(x)=\cos (x) evaluated at x=0 Graphing the displacement of a position function in Graphing \vec r(t) and \vec u(t) in Graphing some of the derivatives of \vec u(t) in
    Weight and Mass densities - + Fluid lb/ft^3 kg/m^3 @@ -1062,8 +1062,8 @@

    - Some integrals in this section are set up by multiplying a variable by a constant distance; - others are set up by multiplying a constant force by a variable . + Some integrals in this section are set up by multiplying a variable by a constant distance; + others are set up by multiplying a constant force by a variable .

    diff --git a/publication/publication-accelerated-print.ptx b/publication/publication-accelerated-print.ptx index fcb911414..eb88e1c5e 100644 --- a/publication/publication-accelerated-print.ptx +++ b/publication/publication-accelerated-print.ptx @@ -4,7 +4,7 @@ - + @@ -14,6 +14,7 @@ + diff --git a/publication/publication-accelerated-web.ptx b/publication/publication-accelerated-web.ptx index 111b3f6fb..ca45e79db 100644 --- a/publication/publication-accelerated-web.ptx +++ b/publication/publication-accelerated-web.ptx @@ -4,7 +4,7 @@ - + @@ -14,9 +14,11 @@ + +