I am not certain what tangent (slope) you want. I am assuming it is from (0,0) to where the tangent line touches each parabols.
If you want a line from (0,0) to a point on each parabola, as luck would have it, that is the minimum value for the slope, since:
and by definition (here) 
, and since:
, and since: 
the lowest value of the slope will be the value that satisfies that relation. That gives:
[m_min,ix] = min(P./V);
for each parabola, and the corresponding line will be:
plot([0 V(ix)], [0 P(ix)])
That is plotted in the last plot call, producing lines that extend from the (0,0) origin to those points on the parabolas. The points differ, however the slopes are essentially all the same for each parabola.
If you want different tangents, please describe them in a bit more detail.
Try this —
W = 10000;
S = 40;
AR = 7;
cd0 = 0.005;
k = 1 / pi / AR;
clalpha = 2*pi;
Psl=25000;
figure(1);hold on; xlabel('V');ylabel('P');
hv=0:1.6484:8.242;
for k1 = 1:numel(hv)
h = hv(k1);
i=0;
for alpha = 1:0.25:15
i=i+1;
rho(i)=1.225*exp(-h/10.4);
cl(i) = clalpha * alpha * pi/180;
V(i) = sqrt(2*W/rho(i)/S/cl(i));
cd(i) = cd0 + k * cl(i) * cl(i);
D(i) = 0.5 * rho(i) * V(i) * V(i) * S * cd(i);
P(i) = D(i)*V(i);
Ph(i)=Psl*(rho(i)/1.225).^0.75;
end
Vm(k1,:) = V;
Pm(k1,:) = P;
Pmh(k1,:) = Ph;
xix = find(diff(sign(D.*V-Ph)));
if ~isempty(xix)
for k3 = 1:numel(xix)
idxrng = max(xix(k3)-1,1) : min(numel(V),xix(k3)+1);
xv(k3) = interp1(P(idxrng)-Ph(idxrng), V(idxrng), 0);
yv(k3) = interp1(V(idxrng),P(idxrng), xv(k3));
Xm{k1,k3} = xv(k3); % Intersection X-Matrix
Ym{k1,k3} = yv(k3); % Intersection Y-Matrix
end
end
figure(1)
hp{k1} = plot(V,P);
c{k1} = hp{k1}.Color;
hold on
plot(V,Ph)
xlim([0 90])
ylim([0 45000])
plot(xv, yv, 's', 'DisplayName',sprintf('Intersections %d',k1)) % Plot Intersections (Optional)
[minV,Vnidx] = min(V);
Vstall(k1,:) = minV;
Pstall(k1,:) = P(Vnidx);
[Pmin,idx] = min(P);
minP(k1,:) = Pmin;
Vm(k1,:) = V(idx);
Vminp=Vm(:,1);
% dPdV(k1,:) = gradient(P) ./ gradient(V);
[m_min,ix] = min(P./V);
% Cm = [P(ix) V(ix)]
plot([0 V(ix)], [0 P(ix)], '-', 'LineWidth',1, 'Color',c{k1})
plot(V(ix), P(ix), 'd', 'Color',c{k1})
% Q1 = size(dPdV{k1})
end
Table = cell2table(cat(2,Xm,Ym), 'VariableNames',{'V_p_max','V_p_min','Pmax','Pmin'});
Table = addvars(Table, Vstall, 'After','V_p_min');
Table = addvars(Table, Pstall, minP,Vminp, 'After','Pmin')
Vpmin=[Vstall(1:4);cell2mat(Xm(5,2));cell2mat(Xm(6,2))];
Vpmax=cell2mat(Xm(:,1));
Vfinal=[Vpmax,Vpmin];
lgdstr = [{'H=SI'} compose('H=%.4f',hv(2:end))];
figure(1);
% legend('H=Sl','','','H=1.6484','','','H=3.2968','','','H=4.9452','','','H=6.5936', '','','H=8.242');
legend([hp{:}],lgdstr, 'Location','best')
The tangent lines go from (0,0) to the appropriate points on each parabola. The tangent points on each parabola are plotted with diamond markers in the colors of the parabola.
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