mjhjmtu

Hi everyone I am looking for a smoother program using the intersection library to calculate where the tangent line intersects the vertical line of the x-coordinate of the second coordinate. I have so far:

documentclassarticle
usepackagetikz
usepackagegeometry
usetikzlibrarydecorations.pathreplacing
usetikzlibraryintersections

begindocument

newcommand*DeltaX0.01
newcommand*DrawTangentLabel[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 <= -3cm, shorten >= -3cm, #1] (X0) -- (X1) node $$;
%

begincenter
begintikzpicture[scale=1.75,cap=round]
 tikzsetaxes/.style=
%draw[style=help lines,step=1cm, dotted] (-5.25,-5.25) grid (5.25,5.25);
% The graphic
beginscope[style=axes]
draw[->] (-.5,0) -- (4.5,0) node[below] $x$;
draw[->] (0,-.5)-- (0,3) node[left] $y$;
foreach x/xtext in 2.25/x
 draw[xshift=x cm] (0pt,2pt) -- (0pt,-2pt) 
 node[below,fill=white,font=normalsize]
 $xtext$; 
 %%%
 draw[name path=curve, domain=.5:3.25,smooth,variable=x,black,<->,thick] 
 plot (x,.5*(x-1.5)*(x-1.5)+1);
 DrawTangentLabel[red,thick,<->]curve-132.25
 draw[name path=curve, domain=.5:3.25,smooth,variable=x,black,<->,thick] plot (x,.5*(x-1.5)*(x-1.5)+1);
 %%%
 filldraw[black] (2.25,1.28125) circle (1pt) node $$;
 filldraw[black] (3,1.28125) circle (1pt) node $$;
 filldraw[black] (3,2.125) circle (1pt) node $$;
 filldraw[black] (3,1.775) circle (1pt) node $$;%%Found by slope formula then trial and error
 %%%
 draw[dashed] (2.25,1.28125)--(3,1.28125); 
 draw[dashed] (3,2.125)--(3,1.28125);
 draw[dashed] (2.9,1.28125)--(2.9,1.38125)--(3,1.38125);
 %%%
 draw[decoration=brace,raise=5pt,decorate,thick]
 (4,2.125) -- node[right=6pt] textcolorblue$Delta y$ (4,1.28125);
 draw[dashed] (4,2.125)--(3,2.125);
 draw[dashed] (4,1.28125)--(3,1.28125); 
 draw[decoration=brace,mirror,raise=5pt,decorate,thick]
 (2.25,1.28125) -- node[below=6pt] textcolorblue$Delta x$ 
 (3,1.28125);
 draw[dashed] (2.25,1.28125)--(2.25,0);
 node at (.75,1.75) $y=f(x)$;
%%%
 filldraw[black] (3,2.125) circle (1pt) node[left] ;
 endscope
 endtikzpicture
 endcenter

enddocument

This outputs:

I would like tikz to calculate the point rather than an estimate.

If you instead of shorten use the syntax of the calc library to draw the tangent line, you can use the intersections library to find the intersection.

documentclassarticle
usepackagetikz
usetikzlibrarydecorations.pathreplacing
usetikzlibrarycalc % <-- added
usetikzlibraryintersections

begindocument

newcommand*DeltaX0.01
newcommand*DrawTangentLabel[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 [#1] ($(X0)!-2cm!(X1)$) -- ($(X1)!-2cm!(X0)$); % <-- modified
%

begincenter
begintikzpicture[
 scale=1.75,
 cap=round,
 axes/.style=->,
 declare function=f(x)=.5*(x-1.5)*(x-1.5)+1; % <-- added
]
%draw[style=help lines,step=1cm, dotted] (-5.25,-5.25) grid (5.25,5.25);
% The graphic

draw[axes] (-.5,0) -- (4.5,0) node[below] $x$;
draw[axes] (0,-.5)-- (0,3) node[left] $y$;

foreach x/xtext in 2.25/x
 draw (x,2pt) -- (x,-2pt) node[below,fill=white,font=normalsize] $xtext$; 


draw[name path=curve, domain=.5:3.25,smooth,<->,thick] plot (x,f(x));

DrawTangentLabel[red,thick,<->, name path=tangent]curve-132.25

foreach [count=i] x in 2.25,3
 filldraw (x,f(x)) circle[radius=1pt] coordinate(ni);

draw [dashed,name path=dash] (n1) -| coordinate (n3) (n2);
filldraw (n3) circle[radius=1pt];

fill[name intersections=of=dash and tangent] (intersection-1) circle[radius=1pt];


draw[decoration=brace,raise=5pt,decorate,thick] (n2 -| 4,0) -- node[right=6pt,blue] $Delta y$ (n3 -| 4,0);
draw[decoration=brace,mirror,raise=5pt,decorate,thick] (n1) -- node[below=6pt,blue] $Delta x$ (n3);

draw[dashed] (n1) -- (n1 |- 0,0)
 (n2) -- (n2 -| 4,0)
 (n3) -- (n3 -| 4,0);

node [above]at (.5,f(.5)) $y=f(x)$;
%%%

endtikzpicture
endcenter

enddocument