Spring 2007
May 7th, 2007
ECON 1078 Math Tool for Economists I
Final Examination
Name: ______
Your instructor (Please check):
______ECON 1078-001: Yiting An
______ECON 1078-003: Mariya Burdina
______ECON 1078-004: Watcharapong Ratisukpimol
Directions:There are 12 questions and 100 points total. Give the best answer for the following questions. Show all your work instead of just putting down the answer. If you are unable to complete a problem, you might get partial credit for explaining where you are lost.
No graphing calculators are allowed. Please find a blank page for scratch work attached. If you need more scratch paper, please raise your hand during the exam period.
Please make your final answer clear by writing legibly and placing a box around it.
The questions begin from the back of this page. Don’t turn this page until you are told to do so.
You have 150 minutes to complete the exam. Good Luck.
On my honor, as a University of Colorado at Boulder student, I have neither given nor received unauthorized assistance on this work.
Signature: ______Date: 05/07/07
1. (8 pts) Define all the changes you go through to transform the graph of to the graph of? *The changes refer to the horizontal/vertical shift of the graph, or reflect the graph about x/y axis.
Answer: To transform from the graph of to the graph of , four changes are needed.
1) : reflect the graph about the y axis;
2) : shift the graph to the left (horizontally) by 2 units;
3) : reflect the graph about the x axis;
4) : shift the graph up (vertically) by 3 units;
2. (8 pts) Graph of the function is a circle. Try to write center’s coordinate and the radius of this circle.
Answer:
The distance between point (x,y) and (-3,0) is a constant number 4. So the circle centers at point (-3,0) with radius 4.
3. (10 pts) Compute the following limits:
a)
Answer: Since the function is continuous when x=-1, the limit value is equal to the function value.
b)
Answer: Since the function is NOT continuous when x=1, we cannot plug in x=1 into the function to calculate its limit value. Simplify the function first.
4. (8 pts) Consider the function f defined by the formula . Find its inverse function (use x as the free variable).
Answer:
Use x as the free variable, then the inverse function is:
5. Given a function . Calculate:
a) (2 pts) =
Answer:
b) (3 pts) =
Answer:
c) (3 pts) Interpret your answer from b):
Answer:
is actually the derivative of the function at the point x=2. It gives the information about the slope of the tangent line to the function f(x) at the point x=2.
6. (8 pts) Marginal Analysis
Little Anne is opening a lemonade stand for the summer. Her cost of producing x glasses of lemonade is given by . Find the marginal cost function and then the marginal cost of producing the 10th glass of lemonade. If Anne can sell the 10th glass of lemonade at a price of $1, should she do so and why?
Answer:
à
If the price charged is higher than the marginal cost of producing the extra lemonade, Anne gains positive profit. Price = $1 is higher than the marginal cost = $.5. Anne will do so.
Cumulative part:
1. (8 pts) Evaluate:
a.
Answer:
b.
Answer:
2. (8 pts) Define.
Evaluate
Answer:
3. (8 pts) Let U be the universal set which is U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
A = {x: xU and x is an odd number}
B = {x: xU and x is an even number}
C = {x: xU and x is a solution of the equation}
Answer:
A = {1, 3, 5, 7, 9}
B = {0, 2, 4, 6, 8, 10}
C = {2, 3} (-2 is not a member in C since it is not in U)
List all the elements in the following sets:
a.
Answer:
b.
Answer: {1,2,3,5,7,9}
c.
Answer: {3}
d. Which of the following set is equal to set {2, 3}? (Please circle.)
No correct answer
Answer:
4. (8 pts) 3 sandwiches and 2 beers cost $21, whereas 3 beers and 2 sandwiches cost $19. What is the price of each sandwich and each beer?
Answer:
Set the price of sandwiches as S, and price of beers as b:
Solve that S= $5 and b=$3.
5. (8 pts) Natural Logarithmic Function: Let and, express and simplify in terms of A and B.
Answer:
6. (10 pts) The total cost C of producing x units of calculators is a linear function. The company knows that it will cost $5,300 to produce 100 calculators and $12,800 to produce 250 calculators.
- Write an equation for the cost function C(x) and the average cost function AC(x).
- Graph the cost function C(x) and specify its domain and range in order to assure positive cost and quantity.
- If a company decides to produce 1,200 calculators, what is the lowest price charged in order to obtain a positive profit?
Answer:
a. Find the slope of the cost function.
Since it is a linear cost function, . Next, find the intercept;
Therefore, the cost function C(x) is
The average cost function AC(x) is
b. Domain is , Range is
c.
4