Midterm Exam - Math for Economists!

Tonight I had a Midterm Exam in my Math for Economists class! I did ok but I probably did make a few stupid mistakes. Here are three of the questions from the exam:

#2) Use Matrix Algebra to solve the following model:
Y= C + I + G
C=a + b(Y-T
I=c +dY +eR
where Y=income, C=Consumption, I=Investment, G=Government Spending, T=Taxes, and R=the Interest Rate. Also a>0, 1>b>0, and 1>d>0. Treat Y, C, and I as endogenous and T,G, and R as exogenous variables. a,b,c,d, and e are simply exogenous coefficients. Use the implicit function theorem to determine the impact of a change in R on each of the endogenous variables.

#6) The value of a tree is given by the following expression V(t) = 2 ^ sqrt(t) where ti s time and r is the discount rate. The present value is V(t) = 2 ^ sqrt(t) * e ^ -rt. What is the optimal time to cut down the tree (the point at which the present value is maximized?

#7) Write Total Costs function as TC=TC(Q) and Average Costs as AC(Q)=TC(Q)/Q, and use calculus to demonstrate that the Marginal Cost Curve intersects the Average Cost Curve at the minimum point of the Average Cost Curve.