Problem 6
Show that the function \(V=A e^{r t}\) (with \(A, f>0\) ) and the function \(A=V e^{-r t}\) (with \(V, r>0)\) are both strictly monotonic, but in opposite directions, and that they are both strictly convex in shape (cf. Exercise \(10.2-5\) ).
Problem 7
Find the derivatives of the following by first taking the natural log of both sides: $$(a) y=\frac{3 x}{(x+2)(x+4)}$$ $$(b) y=\left(x^{2}+3\right) e^{x^{2}+1}$$
Problem 10
Show that, if the demand for money \(M_{d}\) is a function of the national income \(Y=Y(t)\) and the interest rate \(i=i(t),\) the rate of growth of \(M_{d}\) can be expressed as a weighted sum of \(r_{y}\) and \(r_{i}\) $$r_{M d}=\varepsilon_{M d} y r_{y}+\(\varepsilon_{M_{d}} i f_{i}$$$ where the weights are the elasticities of \)M_{d}\( with respect to \)Y\( and \)i$, respectively.
