Problem 1
Find the zeros of the following functions graphically: (a) \(f(x)=x^{2}-8 x+15\) (b) \(g(x)=2 x^{2}-4 x-16\)
Problem 2
Let the national-income model be:
\\[
\begin{array}{ll}
Y=C+l_{0}+C \\
C=a-b\left(Y-T_{0}\right) & (a>0, \quad 0
Problem 3
The demand and supply functions of a two-commodity market model are as follows: $$\begin{array}{ll}Q_{d 1}=18-3 P_{1}+P_{2} & Q_{d 2}=12+P_{1}-2 P_{2} \\\Q_{s 1}=-2+4 P_{1} & Q_{52}=-2 \quad+3 P_{2}\end{array}$$ Find \(P_{i}^{*}\) and \(Q_{i}^{*}(i=1,2) .\) (Use fractions rather than decimals.)
Problem 3
(a) Find a cubic equation with roots \(6,-1,\) and 3 (b) Find a quartic equation with roots \(1,2,3,\) and 5
Problem 3
Find \(Y=\) and \(C^{*}\) from the following: \\[ \begin{array}{l} Y=C+1_{0}+C_{0} \\ C=25+6 Y^{1 / 2} \\ l_{0}=16 \\ C_{0}=14 \end{array} \\]
Problem 4
For each of the following polynomial equations, determine if \(x=1\) is a root. (a) \(x^{3}-2 x^{2}-3 x-2=0\) (c) \(3 x^{4}-x^{2}+2 x-4=0\) (b) \(2 x^{3}-\frac{1}{2} x^{2}+x-2=0\)
Problem 5
Find the rational roots, if any, of the following: (a) \(x^{3}-4 x^{2}-x+6=0\) (c) \(x^{3}+\frac{3}{4} x^{2}-\frac{3}{8} x-\frac{1}{8}=0\) (b) \(8 x^{3}+6 x^{2}-3 x-1=0\) (d) \(x^{4}-6 x^{3}+7 \frac{3}{4} x^{2}-\frac{3}{2} x-2=0\)
Problem 6
Find the equilibrium solution for each of the following models: (a) \(Q_{d}=Q_{\text {s }}\) \(Q_{d}=3-p^{2}\) \(Q_{\mathrm{s}}=6 P-4\) (b) \(Q_{d}=Q_{s}\) \(Q_{d}=8-p^{2}\) \(Q_{3}=p^{2}-2\)
