1. Find the derivative with respect to x for the following functions.
a. f (x)= ax3+ bx2+x;
b. f(x)=ax ln x;

2. What is the price elasticity of the demand curve Q (p) = a - bln (p)?

3. Suppose that the demand for water balloons fit the function D(p,ps)= .2 p-0.5ps-0.2; where p is the price of water balloons and ps is the price of super soakers (an advanced squirt gun technology).
Based upon this demand curve, are super soakers a compliment or a substitute for water balloons? Demonstrate why.

4. Suppose that the current price of oil is $60 per barrel and the quantity sold is 90 million barrels per day. Assume that the supply and demand curves for oil are linear. The current estimates of the price elasticity of supply and demand in the U.S. are η = 1 and ε = - .2 respectively. Assume that the supply and demand curves for oil are linear. In other words, supply can be expressed as S(p) = a + bp, and demand as D(p) = e - fp.
a. Derive a and b, the horizontal (quantity) intercept and slope of the supply curve .
b. Derive the components of the demand curve, e and f .
Now assume that the U.S. government has decided to purchase an additional 2 million barrels of oil per day and put it into the strategic petroleum reserve. In other words, this additional oil is pumped into the ground and not circulated in the existing U.S. oil market.
c. Draw (using supply and demand curves) the impact of this new government policy on the U.S. oil market.
d. Calculate the new equilibrium price resulting from the government's new policy.