Vrftjkry

You can use a bit of calculus:

The DrawTangent macro creates two vertical lines around the given x values and computes the intersection of those and the curve. Then, the tangent can be approximated by the secant between those two intersection points.

Code:

documentclass[border=2pt]{standalone}

usepackage{tikz}

usetikzlibrary{calc,intersections}





newcommand*{DeltaX}{0.01}

newcommand*{DrawTangent}[5]{%

    % #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 <= -1.00cm, shorten >= -1.00cm, #1] (X0) -- (X1);

}%



begin{document}

begin{tikzpicture}

% Axes

draw [->, name path=x] (-1,0) -- (11,0) node [right] {$x$};

draw [->] (0,-1) -- (0,6) node [above] {$y$};

% Origin

node at (0,0) [below left] {$0$};

% Points

coordinate (start) at (1,-0.8);

coordinate (c1) at (3,3);

coordinate (c2) at (5.5,1.5);

coordinate (c3) at (8,4);

coordinate (end) at (10.5,-0.8);

% show the points

foreach n in {start,c1,c2,c3,end} fill [black] (n)

    circle (2pt) node [below] {};

% join the coordinates

draw [thick,name path=curve] (start) to[out=70,in=180] (c1) to[out=0,in=180]

    (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);

% add tangets and dashed lines

foreach c in {1,2,3} {

    %draw [dashed] let p1=(cc) in (cc) -- (x1,0) node [below] {$c_c$};

    draw ($(cc)-(0.75,0)$) -- ($(cc)+(0.75,0)$) node [midway,above=4mm]{$$}; %{$f'(c_c)=0$};

}

% add a and b

path [name intersections={of={x and curve}, by={a,b}}] 

    (a) node [below left] {$a$}

    (b) node [above right] {$b$};



DrawTangent[red, thick]{curve}{-1}{4}{1.5}

DrawTangent[orange, thick]{curve}{-1}{4}{3.5}



end{tikzpicture}

end{document}

A PSTricks solution just for comparison purpose.

documentclass[pstricks]{standalone}

usepackage{pstricks-add,pst-vehicle,tikzducks}



defV{rput(1,1.5){psscalebox{-2 2}{tikz{duck[alien=red]}}}}

defF{2*cos(x)+x/3}

defFp{Derive(1,F)}



deftangent#1#2#3{psplotTangent[linecolor=blue,showpoints]{#1}{.5}{F}uput[#2](*#1 {F}){scriptsize$f'#30$}}

defnormal#1{psplotTangent[linecolor=magenta,Derive={-1/Fp},showpoints]{#1}{.2}{F}}



psset{plotpoints=200}

begin{document}

foreach i in {-0.8,-0.6,...,10.8}{

begin{pspicture}(-2.5,-2)(12.5,6)

    psaxes[ticks=none,labels=none,linecolor=gray]{->}(0,0)(-2,-1.5)(11.5,5)[$x$,0][$y$,90]   

    psplot[linecolor=red]{-1.5}{11}{F}

        normal{-1.5}

        tangent{-1}{180}{>}

        tangent{0.167}{-90}{=}

        tangent{1.334}{0}{<}

        tangent{2.974}{-90}{=}

        tangent{4.7115}{0}{>}

        tangent{6.449}{-90}{=}

        tangent{7.852}{0}{<}

        tangent{9.256}{-90}{=}

        tangent{10.5}{0}{>}

        normal{11}

    psVehicle[vehicle=SelfDefinedVehicle,ownvehicle=V,showSlope=false,frontwheel=SpokesWheelB,backwheel=SpokesWheelA,rB=1.2,rF=0.9,d=3]{.2}{i}{F}        

end{pspicture}}

end{document}

This is just for fun and not to "steal" the check mark from Peter Grill, who IMHO should get it. Conceptually it is basically the same as Jake's pioneering answer. This answer comes with two styles: tangent at and normal at, which you can pass to the curve, and utilizes decorations.markings rather than intersections. If you want to draw several tangents and/or normals, it is advantageous to use /.list, as in the MWE:

documentclass{article}

usepackage{tikz}

usetikzlibrary{decorations.markings}



begin{document}

pgfkeys{tikz/.cd,

tangent length/.store in=TangentLength,

tangent length=7mm,

normal length/.store in=NormalLength,

normal length=7mm}

tikzset{tangent/.style={red,thin},normal/.style={blue,thin},

tangent at/.style={postaction={decorate,decoration={markings,

mark=at position #1 with {draw[tangent] (-TangentLength,0) -- (TangentLength,0);

fill[tangent] (0,0) circle (2pt);}}}},

normal at/.style={postaction={decorate,decoration={markings,

mark=at position #1 with {draw[normal] (0,-NormalLength) -- (0,NormalLength);

fill[normal] (0,0) circle (2pt);}}}},

}

begin{tikzpicture}

% Axes

draw [-latex] (-1,0) -- (11,0) node [right] {$x$};

draw [-latex] (0,-1) -- (0,6) node [above] {$y$};

% Origin

node at (0,0) [below left] {$0$};

% Points

coordinate (start) at (1,-0.8);

coordinate (c1) at (3,3);

coordinate (c2) at (5.5,1.5);

coordinate (c3) at (8,4);

coordinate (end) at (10.5,-0.8);

% show the points

% foreach n in {start,c1,c2,c3,end} fill [black] (n)

%     circle (2pt) node [below] {};

% join the coordinates

draw [thick,tangent at/.list={0.15,0.3,...,0.75},

normal at/.list={0.25,0.4,...,0.7}] (start) to[out=70,in=180] (c1) to[out=0,in=180]

    (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);

end{tikzpicture}

end{document}

Just if you are wondering: yes, it is possible to draw the tangent at some specific point the curve runs through. One possibility is to use the tricks of this answer, i.e. decompose the path into segments and attach the tangents/normals to the starting or end points of those. If you want to have this spelled out, please ask a new question, this here is really just for fun.

ADDENDUM: As for your request in the comments... (and yes, I am using Jake's answer, as I mention above)....

documentclass{article}

usepackage{tikz}

usetikzlibrary{decorations.markings,intersections}



begin{document}

pgfkeys{tikz/.cd,

tangent length/.store in=TangentLength,

tangent length=3.14cm,

normal length/.store in=NormalLength,

normal length=7mm}

newcounter{tangent}

newcounter{normal}

tikzset{tangent/.style={red,thin},normal/.style={blue,thin},

tangent at/.style={postaction={decorate,decoration={markings,

mark=at position #1 with {stepcounter{tangent}

draw[tangent,name path=tangent-thetangent] (-TangentLength,0) -- (TangentLength,0);

fill[tangent] (0,0) circle (2pt);}}}},

normal at/.style={postaction={decorate,decoration={markings,

mark=at position #1 with {stepcounter{normal}

draw[normal,name path=normal-thenormal] (0,-NormalLength) -- (0,NormalLength);

fill[normal] (0,0) circle (2pt);}}}},

}

begin{tikzpicture}

% Axes

draw [-latex] (-1,0) -- (11,0) node [right] {$x$};

draw [-latex] (0,-1) -- (0,6) node [above] {$y$};

% Origin

node at (0,0) [below left] {$0$};

% Points

coordinate (start) at (1,-0.8);

coordinate (c1) at (3,3);

coordinate (c2) at (5.5,1.5);

coordinate (c3) at (8,4);

coordinate (end) at (10.5,-0.8);

% show the points

% foreach n in {start,c1,c2,c3,end} fill [black] (n)

%     circle (2pt) node [below] {};

% join the coordinates

draw [name path=curve,thick,tangent at/.list={0.15,0.3,...,0.75},

normal at/.list={0.25,0.4,...,0.7}] (start) to[out=70,in=180] (c1) to[out=0,in=180]

    (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);

fill[name intersections={of=curve and tangent-2,total=t},cyan]

(intersection-t) circle (2pt);

end{tikzpicture}

end{document}

Here is a macro inspired by @PeterGrill's answer in which I tried to automate the calculations. This macro is called secante because it allows you to draw both a tangent and a normal to the curve.

To draw the two vertical lines, I drew these vertical lines using the current bounding box, which avoids manual calculations.

This allows me to add parameters such as the length and its angle with the tangent.

  % #1 = draw options

  % #2 = name of curve

  % #3 = x value at which tangent is to be drawn

  % #4 = length of the tangent (approximation)

  % #5 = angle with tangent,0 by default, 90 for normal

The macro first determines the points where the verticals intersect with the curve. They are shown below in blue. The two points adjacent to the tangent contact point are called "left of curve" and "right of curve".

This being done, with the calc library, the contact point is projected orthogonally on the segment (left of curve)--(droite of curve)

Then, we make a translation, of these points to find the two end points of this tangent.

It is then enlarged.

Here is an example produced by the code below:

documentclass{article}

usepackage{tikz}

usetikzlibrary{calc,intersections,arrows.meta}

newcommand{secante}[5]{

tikzset{every node/.style={font=scriptsize}}

  % #1 = draw options

  % #2 = name of curve

  % #3 = x value at which tangent is to be drawn

  % #4 = length of the tangent (approximation)

  % #5 = angle avec la tangente,0 par défaut, 90 pour la normale

path[name path=verticale] (current bounding box.south west -|#3,0)--(#3,0 |-current bounding box.north east);

path[name path=gauche] (current bounding box.south west -|#3-.01,0)--(#3-.01,0 |-current bounding box.north east);

path[name path=droite] (current bounding box.south west -|#3+.01,0)--(#3+.01,0 |-current bounding box.north east);

% point de contact de la tangente

path [name intersections={of= verticale and #2,by=point of tangency}];

% points d'approximation de la sécante (parallèle)

path [name intersections={of= gauche and #2,by=gauche of #2}];

path [name intersections={of= droite and #2,by=droite of #2}];

% projection du point de contact sur la parallèle

coordinate(projection) at ($(gauche of #2)!(point of tangency)!(droite of #2)$);

% points de la sécante

coordinate(gauche of secant line) at ($(gauche of #2)+(point of tangency)-(projection)$);

coordinate(droite of secant line) at ($(droite of #2)+(point of tangency)-(projection)$);

% secant ou normale

draw[#1]($(gauche of secant line)!#4/2!#5:(droite of secant line)$)--($(droite of secant line)!#4/2!#5:(gauche of secant line)$);

}



begin{document}



begin{tikzpicture}

% Axes

draw [->, name path=x] (-1,0) -- (11,0) node [right] {$x$};

draw [->] (0,-1) -- (0,6) node [above] {$y$};

% Origin

node at (0,0) [below left] {$0$};

% Points

coordinate (start) at (1,-0.8);

coordinate (c1) at (3,3);

coordinate (c2) at (5.5,1.5);

coordinate (c3) at (8,4);

coordinate (end) at (10.5,-0.8);

% show the points

foreach n in {start,c1,c2,c3,end} fill [black] (n)

    circle (2pt) node [below] {};

%% join the coordinates

draw [semithick,name path=curve] (start) to[out=70,in=180] (c1) to[out=0,in=180]

    (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);



secante[red,thick,label=right:L]{curve}{3}{2cm}{0}

secante[red,thick]{curve}{3}{2cm}{0} % tangent to the curve

secante[red,thick]{curve}{5}{2cm}{0}

secante[red,thick]{curve}{7}{2cm}{0}

secante[red,thick]{curve}{9}{2cm}{0}

secante[blue,thick]{curve}{8}{1cm}{90}

secante[blue,thick]{curve}{10}{1cm}{90} % normal to the curve

end{tikzpicture}



end{document}

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