IIT Exam Honor Code: Proofs, Algebra, Limits & Cost Function

Question Instructions and Honor Code Statement

While this will be an open book exam and you are allowed to consult books, notes and other print sources, you are not allowed to consult with one another.If you consult any (course or external) materials for a question, be sure to provide me a complete list of resources that you consulted. This information is to be provided for each question separately.No credit will be given for typed responses.In order to receive full credit for each question, be sure to provide as much detail as possible.Each response should reflect your understanding of the material. Exam will be followed by a short viva in which ask you to explain one or may be more questions of my choice.

Your total score in the exam will be determined based on the exam and viva performance.Just like homeworks, submit all your work as a single pdf file. No credit will be given for work submitted as multiple images.Be sure to submit the correct file in a single go before the deadline.Before proceeding to the answer any questions, please write and sign the following statement on the first page of your response sheet. 

1. (1 point) Write and sign the IIT Honor Code statement (given in the instructions above) on the first page of your response sheet.

2. (41/2 points) Prove by contradiction that √ 2 is an irrational number.

3. (9 points) Prove, or disprove, the following identity:

(a) (11/2 points) A − B = B − A. If A and B are two arbitrary sets, define the symmetric difference between A and B as A?B = (A − B) ∪ (B − A) One of the following identities are generally not valid. Identify which one. Verify the remaining identities.

(b) (11/2 points) A?B = B?A.

(c) (2 points) A?B = (A ∪ B) − (A ∩ B).

(d) (2 points) (A?B)?C = A?(B?C). 1 of 3 Priyanka Sharma

(e) (2 points) (A ∩ C)?B = (A?B) ∩ (C?B).

4. (6 points) If q denotes the annual demand for wheat (in tons) and p denotes its price per ton, the approximate relationship between them over a specific time period is q = 694, 500p −0.3

(a) (2 points) Using the rules for exponents, re-write the above relationship such that p appears with a positive, integer exponent.

(b) (2 points) Compute demand for wheat when p = 35, 000 and when p = 55, 000.

Proof of Irrationality of √2

(c) (2 points) Based on your answer to part (b) above, identify whether price and quantity demanded are positively or negatively related. Also, comment on whether your findings are consistent with what one can intuitively expect. 5. (13 points) The following system of equations illustrates the general form of a partial market equilibrium model, which is a model of price determination in a one-good market.

Solve the above system of equations for the equilibrium value of P expressed in terms of the parameters a, b, c and d. Denote this equilibrium value of P by P ∗ . Using the value of P ∗ to determine the equilibrium value of:

(a) (3 points) Use direct substitution method to solve for the equilibrium values of P ∗ and Q∗ 

(b) (3 points) Use elimination method to solve for the equilibrium values of P ∗ and Q∗ 

(c) (3 points) Use graphical method to solve for the equilibrium values of P ∗ and Q∗ 

(d) (4 points) Use different quotient to identify the impact on equilibrium values of P ∗ and Q∗ due to change in a.

6. (5 points) If you plan a birthday party with infinitely many guests where the person having the birthday cake takes 1 cake, the best friend is given half a cake, the next person a third, and so on, then how many cakes should you bake?

7. (5 points) Total world consumption of iron is 1971 was approximately 794 million tons. If consumption increases by 5% each year and the world’s total resources of iron are 249 billion tons, how long will these resources last?

8. (7 points) A construction firm wants to buy a building site and has the choice between three different payment schedules: 1. Pay $67, 000 in cash. 2. Pay $12, 000 per year for 8 years, where the first installment is to be paid at once. 3. Pay $22, 000 in cash and thereafter $7000 per year for 12 years, where the first installment is to be paid after 1 year. Determine which schedule is least expensive if the interest rate is 11.5% and the firm has atleast $67, 000 available to spend in cash.

9. (12 points) Compute the following limits: (a) (4 points) limx→2 x 2−2x x3−8 (b) (4 points) limx→0 √3 27+h−3 h.

(c) (4 points) limx→1 x n−1 x−1

10. (6 points) Let f(x) be defined byf (x) = x 2 − 2 for x < 0 −3x 2 + 15 for x > 2 Can you define f(x) as a linear function on [0, 2] so that f(x) is continuous for all x ∈ R? 11. (10 points) Consider f(x) = x 2−9x+20 x−4 .

(a) (11/2 points) Is it possible to apply the quotient limit theorem to find the limit of this function as x → 4? Why or why not?

(b) (21/2 points) Find limx→4 f(x)

(c) (3 points) Is this function continuous at x = 4? Why or why not?

(d) (3 points) Is this function differentiable at x = 4? Why or why not?

11. (10 points) Consider f(x) = x2−9x+20x−4
.
(a) (11/2 points) Is it possible to apply the quotient limit theorem to find the limit of this function as x → 4? Why or why not?

(b) (21/2 points) Find .

12. (9 points) Answer the following questions.

(a) (2 points) Show that (√ x + h − √ x)(√ x + h + √ x) = h.

(b) (3 points) If f(x) = √ x, calculate the difference quotient.

(c) (3 points) Use the result in part(b) to show that for x > 0, f(x) = √ x =⇒ f 0 (x) = 1 2 √ x

(d) (1 point) Show that the result could also be written as d dxx 1 2 = 1 2 x − 1 2

13. (6 points) Let C(q) = aq2 + bq + c be the cost function of a firm. (a) (3 points) Calculate the average rate of change when q is changed from q0 to q0 + h. (b) (3 points) Calculate the instantaneous rate of change when q = q0.

14. (6 points) Prove that for any positive number a, the equation x 3 = a has a unique positive solution.

15. ( 1/2 point) How many hours did you spend working on this exam?