In tutorial 4 question 2 ii (attached below) we were told to estimate the force of morality and initial rate of mortality using the UDD assumption. So I calculate \(\hat{u_x}\) which is an estimate for \(u_{x+0.5}\). Now to get an estimate for \(q_x\) I can either use the relationship which we prove in question 1 OR I assume that \(u_{x+t}\) is linear between ages x and x+1 and solve using $$ q_x = 1 - exp(-{\int {u_{x+t}dt}}) = 1 - exp(-0.5*(u_{x}+u_{x+1})) = 1 - exp(-u_{x+0.5}) = 1 - exp(- \hat{u_x} )$$

which is what I did in the tutorial test and was marked correctly, but the relationship proven in question 1 contradicts the way I did it. If you could shed some light on what I am missing I would really appreciate it.