MAC2233 The University of Oklahoma Utility Function Measurements Test 3 please answer these simple questions correctly it is final assignment thank you for
MAC2233 The University of Oklahoma Utility Function Measurements Test 3 please answer these simple questions correctly it is final assignment thank you for your work. Name (first and last how it appears on canvas):
Section number:
MAC2233 Test 3 Summer 2020
Directions: You should write clearly and in complete sentences, show your work, and explain your logic
(where necessary) to earn full credits. Your answers should include units whenever applicable. For
example: You would not tell someone you are travelling at a speed of 7, but you may say 7 miles per
hour. 50 points = 100%.
Question 1: In economics, the utility function measures the welfare or satisfaction of a consumer as a
function of consumption of real goods such as food or clothing. Consider the following utility function
?? = ??2??3
where ?? is the utility (in utiles), ?? is the amount of food (in pounds) and ?? is the amount of water (in
gallons).
(a) (1 point) A consumer has ?? = 10 pounds of food and ?? = 2 gallons of water, what is the consumers
utility? (Hint: Use the above equation!)
(b) (4 points) Use implicit differentiation to differentiate the equation ?? = ?? 2 ?? 3 with respect to ??.
There is no need to simplify your result. (Hint: You need a product rule!)
(c) (2 points) Assume a consumer has ?? = 10 pounds of food and ?? = 2 gallons of water. Further
suppose the amount of food is decreasing at a rate of 3 pounds per day and the amount of water is
????
????
increasing at a rate of 5 gallons per day. (Hint: What does this say about ???? and ???? ?). How fast is the
utility changing? (Mathematically we are asking for
????
!)
????
Question 2: (1 point each) Provide the requested indefinite integral facts
(a) ? ?? ?? ???? =
where ?? ? ?1.
(b) ? ?? ?1 ???? =
(c) ? ?? ???? ???? =
Question 3: (3 points) Explain (in words or symbols) how to use FTOC to evaluate a definite integral of
??
the form ??? ??(??) ????.
Question 4: Consider the function ??(??) = ?? 3 on the interval [0,8].
(a) (3 points) Calculate the value of the Left End Point Riemann Sum using ?? = 4 sub-intervals. You
should show your work and explain your logic for full credits. Simply giving the correct number will not
earn any credit!
(b) (3 points) Calculate the value of the Right End Point Riemann Sum using ?? = 2 sub-intervals. You
should show your work and explain your logic for full credits. Simply giving the correct number will not
earn any credit!
Question 5: Evaluate each of the following indefinite integrals.
(a) (3 points) ?(?? ?2 + ?? ?1 + 1)????
(b) (3 points) ?(?? 5?? + 3?? ?? + 4??) ????
(c) (3 points) ?(??? ? 5?? 4 + 10?? 11 ) ????
Question 6: (3 points) Suppose the marginal cost function is given by ?? ? (??) = 8?? + 1 and the cost to
produce ?? = 5 units is ??(5) = 200 dollars. Find an expression for the cost function ??(??).
Question 7: Evaluate the following definite integrals. You should show your work for full credit. Simply
giving the correct number will not earn any points.
7
(a) (2 points) ?3 (2?? ? 4?? 3 ) ????
2
(b) (2 points) ??2(3?? 2 + 6) ????
3
2
(c) (2 points) ?3 (?? ?? )????
Question 7: (4 points) Find the area of the region(s) bounded by the graph of ??(??) = 3 ? 3?? 2 and the xaxis between ?? = ?3 and ?? = 5.
9
Question 8: (4 points) Use the graph of ??(??) evaluate the integral ?0 ??(??) ????.
Please note the points (??, ??), (??, ??), (??, ??) and (??, ?????) are on the graph and all the lines are straight!
Question 9: Suppose the supply and demand functions for a certain consumer good are given by
??(??) = 850 ? 20??,
??(??) = 50 + 6?? 2
(a) (1 points) In this case the equilibrium quantity is ???? = 10. What is the equilibrium price ???? ?
??
(b) (2 point) Recall the consumer surplus is given by the integral ?0 ?? (??(??) ? ???? ) ????. Write (but do not
evaluate) the integral expression for the consumer surplus in this case. Please gather like terms inside
the integral expression!
(c) (1 point) Evaluate the integral expression in (b).
??
(d) (2 point) Recall the producer surplus is given by the integral ?0 ?? (???? ? ??(??)) ???? . Write (but do not
evaluate) the integral expression for the producer surplus in this case. Please gather like terms inside
the integral expression
(e) (1 points) Evaluate the integral expression in (d).
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