Spline Derivatives and Root Finding

A reactor is thermally stratified as in the following table:

Depth, m= [0 0.5 1 1.5 2 2.5 3]

Temperature, ◦C= [70 70 55 22 13 10 10]

Based on these temperatures, the tank can be idealized as two zones separated by a strong temperature gradient or thermocline. The depth of the thermocline can be defined as the inflection point of the temperature-depth curve — at is, the point at which d^2T/dz^2=0.

At this depth, the heat flux from the surface to the bottom layer can be computed with Fourier’s law:q=k*(dT/dz)

Use a clamped cubic spline fit with zero end derivatives to determine the thermocline depth. If k =0.01 cal/(s · cm · ◦C) compute the flux across this interface.

So I know I need a spline function and I understand the typical format is yy= spline(x,Y,xx) but I am very confused about labeling. If someone could help me break this down or point me in the right direction thatd be great

[Ttill]

this is what i have thus far

%hw8_1 d= 0:3; y= [70 70 55 22 13 10 10]; dd= 0:.5:3; k=0.01 cs= spline(dd,[0 y 0]);

M= diag(0:.5:3,1);

%first derivative cs1= cs; cs1.coefs=cs1.coefs.*M;

%second derivative cs2= cs1; cs2.coefs=cs2.coefs.*M;

x= linspace(dd(1),dd(end),1001);

q= k.*cs1 q