3

Most of the work has been done here except I want TiKz to calculate the y values at least. I try here below but need help with the syntax to finishenter image description here

 \documentclass{article}
 \usepackage{tikz}
\usetikzlibrary{intersections,positioning,calc}

\newcommand{\pder}[2]{#1^{\prime}(#2)}

 \begin{document}
 \newcommand*{\DeltaX}{0.01}
 \newcommand*{\DrawTangent}[7][]{%
   % #1 = draw options
   % #2 = name of curve
   % #3 = ymin
   % #4 = ymax
   % #5 = x value at which tangent is to be drawn

  \path[name path=Vertical Line Left]  (#5-\DeltaX,#3) -- (#5-\DeltaX,#4);
  \path[name path=Vertical Line Right] (#5+\DeltaX,#3) -- (#5+\DeltaX,#4);

  \path [name intersections={of=Vertical Line Left and #2}];
  \coordinate (X0) at (intersection-1);
  \path [name intersections={of=Vertical Line Right and #2}];
  \coordinate (X1) at (intersection-1);

  \draw [shorten <= -1cm, shorten >= -1cm, #1] (X0) -- (X1) node[above      
   right=#6cm and #7cm] {\small $m=\pder{f}{#5}={f(#5)}$};
  }%

 \begin{minipage}{.4\textwidth}
 Theorem 1 says that for $f(x)=e^{x}$, the derivative at $x$ (the slope of 
 the tangent line) is the same as the function value at $x$. That is, on the 
 graph of $y=e^{x}$,  at the point $(0,1)$,  the slope is $m=1$;  at the 
 point $(1,e)$,  the slope is $m=e$; at the point $(2,e^{2})$, the slope is 
 $m=e^{2}$, and so on. The function $y=e^{x}$ is the only exponential 
 function for which this correlation between the function and its derivative 
 is true.\\
 In Section 3.5, we will develop a formula for the derivative of the more 
 general exponential function given by $y=a^{x}$
 \end{minipage}
 \hspace{1cm}
 \begin{minipage}{.6\textwidth}
  \begin{tikzpicture}[scale=.75, declare function={f(\x)=(2.71828)^(\x);}]
 \draw[step=1.0,gray,thin,dotted] (-3,-1) grid (7,9);
 \draw [-latex] (-3,0) -- (7.5,0) node (xaxis) [below] {$x$};
 \draw [-latex] (0,-1) -- (0,9.5) node [left] {$y$};
    \foreach \x/\xtext in {-2/-2,-1/-1,1/1,2/2,3/3,4/4,5/5,6/6}
     \draw[xshift=\x cm] (0pt,3pt) -- (0pt,0pt) 
     node[below=2pt,fill=white,font=\normalsize]
     {$\xtext$};    
   \foreach \y/\ytext in {1/1,2/2,3/3,4/4,5/5,6/6,7/7,8/8}
   \draw[yshift=\y cm] (2pt,0pt) -- (-2pt,0pt) 
   node[left,fill=white,font=\normalsize]
    {$\ytext$};
 \draw[name path=curve,domain=-3:2.2,samples=200,variable=\x,red,<->,thick] 
 plot ({\x},{(2.71828)^(\x)});
 \DrawTangent[blue,thick,-]{curve}{-1}{4}{1}{.5}{.3}
 \DrawTangent[blue,thick,-]{curve}{-1}{3}{0}{.5}{.8}
 \DrawTangent[blue,thick,-]{curve}{5}{9}{2}{.5}{.2}
 \draw[fill=red,red] (0,1) circle (3pt) node[right] {\small $y=f(0)=0$};
 \draw[fill=red,red] (1,2.71828) circle (3pt) node[right] {\small 
    $y=f(1)\approx 2.71828$};
  \draw[fill=red,red] (2,7.3890) circle (3pt) node[right] {\small 
    $y=f(2)\approx 7.3890$};
 \draw[fill=red,red] (-1,0.3679) circle (3pt) ;
 \draw[fill=red,red] (-2,0.1353) circle (3pt) node[] {$$};
 \end{tikzpicture}
 \end{minipage}
  \end{document}
3

Something like this? I only let TikZ compute the numbers in

 \draw[fill=red,red] (1,{f(1)}) circle (3pt) node[right] {\small 
     $y=f(1)\approx \pgfmathparse{f(1)}
     \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
     \pgfmathprintnumber{\pgfmathresult}$};
  \draw[fill=red,red] (2,{f(2)}) circle (3pt) node[right] {\small 
  \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
    $y=f(2)\approx \pgfmathparse{f(2)}\pgfmathprintnumber{\pgfmathresult}$};

Here, \pgfmathparse{f(1)} computes the value of f at 1 and stores it in \pgfmathresult. \pgfmathprintnumber prints it then. I set the number of digits to 2, and you may change it at will. And this is the MWE.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections,positioning,calc}

\newcommand{\pder}[2]{#1^{\prime}(#2)}

\begin{document}
\newcommand*{\DeltaX}{0.01}
\newcommand*{\DrawTangent}[7][]{%
   % #1 = draw options
   % #2 = name of curve
   % #3 = ymin
   % #4 = ymax
   % #5 = x value at which tangent is to be drawn

  \path[name path=Vertical Line Left]  (#5-\DeltaX,#3) -- (#5-\DeltaX,#4);
  \path[name path=Vertical Line Right] (#5+\DeltaX,#3) -- (#5+\DeltaX,#4);

  \path [name intersections={of=Vertical Line Left and #2}];
  \coordinate (X0) at (intersection-1);
  \path [name intersections={of=Vertical Line Right and #2}];
  \coordinate (X1) at (intersection-1);

  \draw [shorten <= -1cm, shorten >= -1cm, #1] (X0) -- (X1) node[above      
   right=#6cm and #7cm] {\small $m=\pder{f}{#5}={f(#5)}$};
  }%

 \begin{minipage}{.4\textwidth}
 Theorem 1 says that for $f(x)=e^{x}$, the derivative at $x$ (the slope of 
 the tangent line) is the same as the function value at $x$. That is, on the 
 graph of $y=e^{x}$,  at the point $(0,1)$,  the slope is $m=1$;  at the 
 point $(1,e)$,  the slope is $m=e$; at the point $(2,e^{2})$, the slope is 
 $m=e^{2}$, and so on. The function $y=e^{x}$ is the only exponential 
 function for which this correlation between the function and its derivative 
 is true.\\
 In Section 3.5, we will develop a formula for the derivative of the more 
 general exponential function given by $y=a^{x}$
 \end{minipage}
 \hspace{1cm}
 \begin{minipage}{.6\textwidth}
  \begin{tikzpicture}[scale=.75, declare function={f(\x)=(2.71828)^(\x);}]
 \draw[step=1.0,gray,thin,dotted] (-3,-1) grid (7,9);
 \draw [-latex] (-3,0) -- (7.5,0) node (xaxis) [below] {$x$};
 \draw [-latex] (0,-1) -- (0,9.5) node [left] {$y$};
    \foreach \x/\xtext in {-2/-2,-1/-1,1/1,2/2,3/3,4/4,5/5,6/6}
     \draw[xshift=\x cm] (0pt,3pt) -- (0pt,0pt) 
     node[below=2pt,fill=white,font=\normalsize]
     {$\xtext$};    
   \foreach \y/\ytext in {1/1,2/2,3/3,4/4,5/5,6/6,7/7,8/8}
   \draw[yshift=\y cm] (2pt,0pt) -- (-2pt,0pt) 
   node[left,fill=white,font=\normalsize]
    {$\ytext$};
 \draw[name path=curve,domain=-3:2.2,samples=200,variable=\x,red,<->,thick] 
 plot ({\x},{(2.71828)^(\x)});
 \DrawTangent[blue,thick,-]{curve}{-1}{4}{1}{.5}{.3}
 \DrawTangent[blue,thick,-]{curve}{-1}{3}{0}{.5}{.8}
 \DrawTangent[blue,thick,-]{curve}{5}{9}{2}{.5}{.2}
 \draw[fill=red,red] (0,{f(0)}) circle (3pt) node[right] {\small $y=f(0)=0$};
 \draw[fill=red,red] (1,{f(1)}) circle (3pt) node[right] {\small 
     $y=f(1)\approx \pgfmathparse{f(1)}
     \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
     \pgfmathprintnumber{\pgfmathresult}$};
  \draw[fill=red,red] (2,{f(2)}) circle (3pt) node[right] {\small 
  \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
    $y=f(2)\approx \pgfmathparse{f(2)}\pgfmathprintnumber{\pgfmathresult}$};
 \draw[fill=red,red] (-1,0.3679) circle (3pt) ;
 \draw[fill=red,red] (-2,0.1353) circle (3pt) node[] {$$};
 \end{tikzpicture}
 \end{minipage}
\end{document}

enter image description here

  • yes I got everything to work. Is there a way to control the digit where the rounding takes place in the command \approx \pgfmathparse{f(2)}\pgfmathprint{\pgfmathresult} for example? Out of curiosity. – MathScholar Feb 2 at 16:04
  • 2
    @MathScholar Yes. I added it to the MWE. (One has to use \pgfmathprintnumber and not \pgfmathprint as I incorrectly wrote before.) – marmot Feb 2 at 16:10
  • 1
    @MathScholar Of course it does. Just try \pfmathparse{pi}\pgfmathprintnumber{\pgfmathresult} or something like this. – marmot Feb 2 at 16:48
  • 2
    @MathScholar \pfmathparse{(exp(1)}\pgfmathprintnumber{\pgfmathresult} – marmot Feb 2 at 17:08
  • 1
    @MathScholar Sure: declare function={f(\x)=exp(\x);}. – marmot Feb 2 at 17:10

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