Let $s(t), 0 \leq t \leq \infty$, denote a spot rate curve; that is, the present value of a dollar to be received at time $t$ is $e^{-s(t) t}$. For $t_{1}

(a) Find an expression for $f\left(t_{1}, t_{2}\right)$.

(b) Let $r(t)=\lim _{t_{2} \rightarrow t} f\left(t, t_{2}\right)$. We can call $r(t)$ the instantaneous interest rate at time $t$. Show that $r(t)=s(t)+s^{\prime}(t) t$.

(c) Suppose an amount $x_{0}$ is invested in a bank account at $t=0$ which pays the instantaneous rate of interest $r(t)$ at all $t$ (compounded). Then the bank balance $x(t)$ will satisfy $\mathrm{d} x(t) / \mathrm{d} t=r(t) x(t)$. Find an expression for $x(t)$. [Hint: Recall in general that $y \mathrm{~d} z+z \mathrm{~d} y=\mathrm{d}(y z)$.