Math 205 Final Exam 6:20p.m. 8:10p.m., Wednesday, Dec. 14, 2011

Transcription

1 Math 205 Final Exam 6:20p.m. 8:10p.m., Wednesday, Dec. 14, 2011 Instructor: Class Time: Name: No books or notes are allowed. Please read the problems carefully and do all you are asked to do. You must show your work! You can use the back page as a scratch paper. Do the problems at the space provided. Total/200 #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 1 (total points: 12). A $20, 000 car depreciates exponentially down to $4, 000 in 8 years, i.e. its value after t years can be described by an exponentially decaying function. (a)(6 points) Find the formula for the value P of the car after t years for 0 t 8. (b)(6 points) After how many years is the value of the car the half of its original price? 2 (total points: 15). Find the derivative of the following functions: (a)(5 points) y = e 1+3x2. (b)(5 points) t = s 2 ln(s 1). (c)(5 points) w = 5 2u u 2. 1

2 3 (total points: 14). The demand and supply curves for a certain product are given in terms of price, p (in dollars), by D(p) = 100 2p and S(p) = 3p 50. (a)(7 points) Find the equilibrium price and quantity. (b)(7 points) If a sales tax of 6% is imposed on consumers, find the new equilibrium price and quantity. 4 (total points: 14). The demand for yams is given by q = p 2, where q is in pounds of yams and p is the price of a pound of yams. (a)(6 points) Compute the elasticity of demand as a function of the prize. (a)(4 points) Is the demand at $2 per pound elastic or inelastic? Explain what this means in terms of necessary and luxury items. (b)(4 points) Find the prize p which maximizes the total revenue. 2

3 5 (total points: 10). The following information about a function f(x, y) is given: f(2, 3) = 7.5, f x (2, 3) = 0.4, and f y (2, 3) = 1.1. Estimate f(2.2, 3.4). 6 (total points: 15). (a)(5 points) Find the partial derivative f x of f(x,y) = 3x4 y 2 x 2 y + 7. (b)(5 points) Find the partial derivative g y of g(x,y) = x y 2. (c)(5 points) Find the partial derivative h st of h(s,t) = (s + t) 3. 3

4 7 (total points: 15). Revenue is given by R(q) = 450q and cost is given by C(q) = 10, q 2. (a)(3 points) What is the fixed cost? (b)(3 points) Determine the profit function, π(q). (c)(9 points) What is the maximum total profit? 8 (total points: 14). (a)(7 points) Find all the critical points (a,b) of the function f(x,y) = x 2 2y 2 + xy 7x (b)(7 points) Use the D-test to decide for each of the critical points (a,b) found in part (a) if it is a local minimum, a local maximum, or neither. (D = f xx (a,b)f yy (a,b) f 2 xy(a,b).) 4

5 9 (total points: 12). A marginal cost function C (q) is given in the figure below If the fixed costs are $10, 000, estimate: (a)(6 points) The total cost to produce 30 units. (b)(6 points) The additional cost if the company increases production from 30 to 40 units. 10 (total points: 12). At a fast food restaurant, four cheeseburgers and two chocolate shakes cost a total of $7.90. Two shakes cost 15 cent more than one cheeseburger. What is the cost of a cheeseburger and of a chocolate shake? 5

6 11 (total points: 12). Find all values for x, y and z such that the following matrix equation is satisfied: [ ] [ ] z x + y 1 1/z 2x + 3y =. 7 x y 7 x 12 (total points: 15). Given are the matrices A = [ ] [ 4 0 1, B = Compute the following matrix expressions or indicate that this is impossible: (a)(5 points) 7B 3(A B) = ] and C = (b)(5 points) C( A) = (c)(5 points) B( A) = 6

7 13 (total points: 12). Solve the following system of equations by the inverse matrix method (no credit for other solutions!): x + 2y + 3z = 18 2x + 3y + 4z = 4 x + 2y + z = /2 2 1/2 The inverse matrix of A = is A 1 = /2 0 1/2 14 (total points: 12). For each of the following augmented matrices, check the appropriate box to indicate if the corresponding system of linear equations has a unique solution, has no solution, or has infinitely many solutions. (a)(4 points) (b)(4 points) (c)(4 points) has a unique solution..., has no solution..., has infinitely many solutions. has a unique solution..., has no solution..., has infinitely many solutions. has a unique solution..., has no solution..., has infinitely many solutions. 7

8 15 (total points: 16). The following table describes the interrelationship between the yearly production of the fishing industry F (in tons) and the boat industry B (in number of boats) on a small island in the pacific ocean. The external demand of their products from neighboring islands is also listed. Demand of F Demand of B External Demand Total Output Production of F Production of B After some of the neighbor islands have been submerged by a tsunami, the prediction for the external demand has been revised to 10 tons of fish and 50 boats. What should be the total output of both industries next year? 8