Estimate function expression for TC and MC I have this


Labor (L)   0   1   2   3   4   5   6   7   8   9   10  11
Output      0   32  72  114 161 204 242 272 292 304 302 288


and I need to estimate function expressions for $TC$, $AVC$, and $MC$.

I have capital $K = 10$, wage $w = 30$, and rate $r = 50$.

I think that total costs are
$$
TC = wL + rK = 30 L + 50 \cdot 10 = 30 L + 500.
$$
and marginal costs are $MC = TC'(Q)$ but my $TC$ doesn't depend on $Q$. Is it $MC = TC'(L) = 30$ instead?

If I'm calculating the marginal cost ($MC_t = \frac{\Delta TC}{\Delta Output}$) in discrete quantities I get


Labor (L)   0   1   2   3   4   5   6   7   8   9     10    11
Output      0   32  72  114 161 204 242 272 292 304   302   288
MC          -   25  20  19  17  19  21  27  40  67    -400  -57


but these numbers seem very wrong.
As the OP corrected in the comments, using $K=20, w=800$, and $r=500$, so $TC = 10000 + 800L$, we get , in the $\{Q,C\}$ space


This looks pretty text-book like. The fact that there are levels of labor for which output is reduced, tells us that for these levels we have to employ more of the "fixed" factor, capital (which thus stops being fixed).
The function as is, with capital fixed at $10000$, stops having economic sense after $L=9$.