Assignment:
Q1. Consider the function f(x) = x2/1+x
a) Determine the critical points. (Apply the quotient rule carefully to find derivative.)
b) For what intervals in the domain of f is the function increasing?
c) For what intervals in the domain of f is the function decreasing?
Q2. Find all maximum and minimum values of the function f(x) = 5x3 -10x2 .
Q3. The daily cost to manufacture generic trinkets for gullible tourists is given by the cost function:
c(x) = -0.001x2 + 0.3x + 500dollars
a) Determine the marginal cost function.
Q4.Determine the derivatives of the following functions:
a) f(x) = x2
b) f(x) = 3x2
c) f(x) =(x + 3)2
d) f(x) = (x2 - 3)3
Q5. The cost of controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is one possible model:
c(q) = 4000+ 100q
Where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm in dollars of this reduction. Government clean-air subsidies amount ot $500 per pound of pollutanat removed. How many pounds of pollutant should the firm remove each day in order to minimize the net cost (cost – subsidy)?
Q6. Suppose that during a prolonged recession, property values depreciated 2% every six months. If a house originally cost $180000, determine its value at the end of five years.
Q7. Suppose you want to be earning an annual salary of $80000 in 10 years. You have been offered a job with a guaranteed 5% increase in salary per year. The initial salary is negotiable. What initial salary should you request to meet your goal of $80k in 10 years?
Provide complete and step by step solution for the question and show calculations and use formulas.