mjhjmtu

I am trying to replicate a diagram and have a minimal example. I don't know how to add additional tangents between the specified points and preserve smoothness.

documentclassarticle
usepackagetikz
usetikzlibrarycalc,intersections

begindocument
begintikzpicture
% 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] $b$;
endtikzpicture
enddocument

Outputs:

My Goal is create the following:

Feel free to make any changes that make the minimal example approach the goal diagram

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
usepackagetikz
usetikzlibrarycalc,intersections


newcommand*DeltaX0.01
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);
%

begindocument
begintikzpicture
% 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-141.5
DrawTangent[orange, thick]curve-143.5

endtikzpicture
enddocument

A PSTricks solution just for comparison purpose.

documentclass[pstricks]standalone
usepackagepstricks-add,pst-vehicle,tikzducks

defVrput(1,1.5)psscalebox-2 2tikzduck[alien=red]
defF2*cos(x)+x/3
defFpDerive(1,F)

deftangent#1#2#3psplotTangent[linecolor=blue,showpoints]#1.5Fuput[#2](*#1 F)scriptsize$f'#30$
defnormal#1psplotTangent[linecolor=magenta,Derive=-1/Fp,showpoints]#1.2F

pssetplotpoints=200
begindocument
foreach i in -0.8,-0.6,...,10.8
beginpspicture(-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]-1.511F
 normal-1.5
 tangent-1180>
 tangent0.167-90=
 tangent1.3340<
 tangent2.974-90=
 tangent4.71150>
 tangent6.449-90=
 tangent7.8520<
 tangent9.256-90=
 tangent10.50>
 normal11
 psVehicle[vehicle=SelfDefinedVehicle,ownvehicle=V,showSlope=false,frontwheel=SpokesWheelB,backwheel=SpokesWheelA,rB=1.2,rF=0.9,d=3].2iF 
endpspicture
enddocument

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:

documentclassarticle
usepackagetikz
usetikzlibrarydecorations.markings

begindocument
pgfkeystikz/.cd,
tangent length/.store in=TangentLength,
tangent length=7mm,
normal length/.store in=NormalLength,
normal length=7mm
tikzsettangent/.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);,

begintikzpicture
% 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);
endtikzpicture
enddocument

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)....

documentclassarticle
usepackagetikz
usetikzlibrarydecorations.markings,intersections

begindocument
pgfkeystikz/.cd,
tangent length/.store in=TangentLength,
tangent length=3.14cm,
normal length/.store in=NormalLength,
normal length=7mm
newcountertangent
newcounternormal
tikzsettangent/.style=red,thin,normal/.style=blue,thin,
tangent at/.style=postaction=decorate,decoration=markings,
mark=at position #1 with stepcountertangent
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 stepcounternormal
draw[normal,name path=normal-thenormal] (0,-NormalLength) -- (0,NormalLength);
fill[normal] (0,0) circle (2pt);,

begintikzpicture
% 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);
endtikzpicture
enddocument