MATH 2070 Test 3 (Sections , , & )

Transcription

1 Multiple Choice: Use a #2 pencil and completely fill in each bubble on your scantron to indicate the answer to each question. Each question has one correct answer. If you indicate more than one answer, or leave a blank, the question will be marked as incorrect. In this section there are 16 multiple choice questions. Each question is worth 3 points unless otherwise indicated for a total of 46 points. For future reference, circle your answers on this test paper as you will not receive your Scantron back with your test. According to market analysis D( c, p) ice cream bars will be demanded monthly by consumers when the ice cream bar has c calories and sells at a price of p dollars per bar. Use this information to answer the next three questions. 1. What is the demand at point B? a. 412,000 b. 405,000 c. 393,000 d. 380, From point C, which of the following would cause the greatest decrease in demand? a. decreasing the calories by 10 b. increasing the calories by 10 c. decreasing the price by $0.10 d. increasing the price by $ What is the correct classification of points A, B, and C? a. A: relative minimum, B: saddle point, C: relative maximum b. A: relative maximum, B: not a critical point, C: not a critical point c. A: not a critical point, B: saddle point, C: relative maximum d. A: relative maximum, B: saddle point, C: not a critical point

2 The table below gives the wind chill C(t,w) in degrees Fahrenheit, where w is the wind speed in miles per hour and t is the air temperature in degrees Fahrenheit. t w Use this context and table to answer the next four questions. 4. Find the function for the linear cross sectional model (equation only) for the wind chill as a function of air temperature when the wind speed is 25 miles per hour. a. C( t,25) 1.497t b. C( t, 25) 0.667t c. C( t, 25) 0.813t d. C( t, 25) 1.073t To estimate the wind chill when the air temperature is 30 degrees Fahrenheit and the wind speed is 17 miles per hour you would first need to. a. use a row of data to find the cross-sectional model C(30, w ). b. use a column of data to find the cross-sectional model C(30, w ). c. use a row of data to find the cross-sectional model C( t,17). d. use a column of data to find the cross-sectional model C( t,17). 6. Given the cross-sectional model C(35, w) 45.64(0.93) w, find a. 3.6 b c d What are the units on dc(35, w) dw w18 dc(35, w) dw w18? (1 pt) a. miles per hour per of wind chill b. of air temperature per of wind chill c. of wind chill per of air temperature d. of wind chill per mile per hour.

3 TKK Products manufactures 50-, 60-, 75-, and 100-watt electric light bulbs. Laboratory tests show that the lives of these light bulbs are normally distributed with a mean of 750 hr and a standard deviation of 75 hr. Use this context to answer the next three questions. 8. What is the probability that a randomly selected TKK light bulb will burn for less than 700 hours? a b c d At what lifetime was the rate of change of the probability density function for the lives of these light bulbs a maximum? a. 825 hours b. 775 hours c. 675 hours d. 750 hours 10. According to the Empirical Rule, approximately what percentage of light bulbs will last between 675 hours and 975 hours? a. 81.5% b. 82% c % d. 84% 11. From point A, when is K(s, w) increasing most rapidly? a. when s decreases b. when w decreases c. when s increases d. when w increases A

4 2 2 P( a, n) 5a 3n 48a 4n 2an 290 million dollars gives the profit of a one-product company where a thousand dollars is the amount spent on advertising and n thousand units of the product are sold. Check: P(6, 2) 402 Use this context to answer the next four questions. 12. Complete the interpretation of P n (6.3,2.1) 4. When a one-product company spends 6.3 thousand dollars on advertising and sells 2.1 thousand units of their product,. a. their profit is decreasing by 4 million dollars per thousand units sold. b. the amount spent on advertising is decreasing by 4 thousand dollars per million dollars of profit. c..their profit is decreasing by 4 million dollars per thousand dollars spent on advertising d. the number of units sold is decreasing by 4 thousand units per million dollars of profit. 13. What system of equations would have to be solved to determine the critical point of P( a, n )? a. 10a 2n 48 2a 6n 4 b. 10a 2n 4 2a 6n 48 c. 10a 2n 4 2a 6n 48 d. 10a 2n 48 2a 6n A graph of the P = 400 million dollar contour curve is shown. Which of the following is a point on the contour curve? a. (6, ) b. (6, ) c. (6, 2.387) d. (6, 2.691) 15. Find the slope of the tangent line at the point (4, 1.721). a b c d

5 16. Determine f if xy 5 f ( x, y) y ln( x) 2 y x e x. 2 a. 2 y ln(2) f x x xy f x 2 ln(2) 5e b. y xy 2 5x 1 f xy 2 ln(2) x 5e x c. y 5 1 x d. 2 y ln(2) f xy x Check your Scantron now to make sure it will successfully run. If it does, you will earn one point. When you are not working on the multiple choice portion of the test, turn your Scantron over so that it cannot be read by others in the room.

6 Free Response: RE-READ the directions at the beginning of the test. Then read each question carefully. Provide only one clearly indicated answer to each question. If your answer is illegible, it will be graded as incorrect. Show all work. The free response portion is 53% of your test grade. When possible, set up the specific mathematical notation that is being evaluated to obtain your answer. No credit will be awarded for simply copying generic formulas from the formula sheet. Little or no credit will be awarded for answers without the corresponding notation. 1. The following data shows the average annual employee income M ( t, v ) thousand dollars at a company where t is the number of years the employee has been with the company and v is the employee review value assigned to the employee. Employee Review Value (v) t years a. Find the quadratic cross-sectional model that could be used to model the boxed in data on the table above. Completely define your model by filling in all of the blanks below. (6 pts) M t t t 2 (, 4) quadratic function with coefficients rounded to three decimal places thousand dollars gives the average annual employee income output units output description when the employee has been with the company for t years input description and the employee s review value is 4, 1 t 6. input interval / domain 1 pt for function name (½ each input); -½ pt if the order is reversed 3 pts for the function -1 pt for a quadratic function where the coefficients are close but incorrect due to a likely typo ½ pt for the output units ½ pt for the output description ½ pt for the input description (-½ pt if unclear or includes v) ½ pt for the domain -½ pt for inconsistent variable name (notation, function, input description, and domain) -½ pt rounding error or incorrect sign on a coefficient b. Find the average annual income for an employee who has been with the company for 3.5 years and has an employee review value of 4. Show the mathematical notation, round your answer to one decimal place and include units. (3 pts) M (3.5, 4) 49.9 thousand dollars 1 pt notation, 1.5 pts value, ½ pt unit; -½ pt rounding error

7 2. Determine if f ( x ) is a valid probability density function. Show specific work (graph, values, etc.) to justify your conclusion. (4 pts) Is f ( x ) a valid p.d.f.? No Justification: Yes or No 1 pt conclusion (must follow reasonable supporting work) 1 pt stating the trapped area 1 (or > 1 or < 1 depending on version) 2 pts calculating the trapped area -1 pt for an error in the area calculation and then follow work -1 pt for incorrect reasons such as f(x) < 0 3. The amount of snowfall in feet is a remote region of Alaska in the month of January has the 2 2x 2x if 0 x 3 probability density function f ( x) 3 9. Check: f (1.5) otherwise a. Find the probability that this region receives more than 1.75 feet of snow in January. (4 pts) Show the specific probability and mathematical notation. Round your answer to four decimal places. 3 P( x 1.75) f ( x) dx ½ pt probability notation; 1.5 pts specific integral notation; 2 pts answer -½ pt rounding error, missing dx notation b. Find the average amount of snow this region receives in January. (4 pts) Show the specific mathematical notation and include units with the answer. 3 x f ( x) dx 1.5 feet pts specific integral notation; 2 pts answer; ½ pt units -½ pt missing dx notation c. Find the variance in the amount of snow this region receives in January. (4 pts) Show the specific mathematical notation and give the answer. 3 2 The total trapped area is not equal to 1. A (1 2) (0.5)(0.6) x 1.5 f ( x) dx pts specific integral notation (1 pt bounds, 1 pt mu plugged in); 2 pts answer (follow work for different mu values); -½ pt missing dx notation; -1 pt for not showing the specific value of μ in the formula unless μ is explicitly defined in part b; -2 pts for only finding σ

8 4. R(s,n) million dollars describes the annual revenue for a fast food company, where s in the number of locations the company operates in South Carolina (SC) and n is the number of locations the company operates in North Carolina (NC). The partial derivatives for the annual revenue function are: Rs = 6s + 2n 100 million dollars per SC restaurant and Rn = 2s + 4n 80 million dollars per NC restaurant. The company currently operates 30 locations in SC and 20 locations in NC which generates an annual revenue of million dollars. a. Write the specific formula for dn. Your final answer should be in terms of n and s, not R. ds (4 pts) dn Rs 6s 2n 100 6s 2n 100 OR ds R 2s 4n 80 2s 4n 80 n b. What are the units on dn? Circle one. ds 1 pt negative (parentheses must be shown if the negative sign is written in the numerator) 3 pts partial flipped 1 pt partial earned for R s/r n notation -½ pt for notation errors such as dn/ds = - ds/dn -2 pts for giving a numerical answer instead of a formula (1 pt) NC restaurants SC restaurant SC restaurants NC restaurant million dollars NC restaurant million dollars SC restaurant c. Suppose the company is planning to close two locations in SC. In order to maintain the same revenue, how many new restaurants would they need to open in NC to compensate for the lost revenue in SC? Show all necessary work to solve this problem. (6 pts) Work: dn ds (30,20) 6(30) 2(20) 100 = - 2 2(30) 4(20) 80 n dn s ds dn n s 2 ( 2) 4 ds 1.5 pts for correctly finding dn/ds at (30,20). 1 pt for correct s, including the negative (all or nothing). 2 pts finding n (follow work when possible) ½ pt for each blank below (follow their work for the first two blanks provided answers are reasonable; negatives and non-integers received no credit) pts for correct answers below without any work. -3 for solving for s instead of n -½ pt for notational errors such as misuse of equal signs. Conclusion: To maintain their annual revenue of million dollars, the company would have to open 4 new locations in NC bringing the total number of restaurants to _24 in NC and 28 in SC.

9 The annual revenue generated at a college is given by R x, y 1.5 x 2y 3xy 6x 18y million dollars when x thousand in-state students and y thousand out-of-state students are enrolled. Check: R(4,5) = 52 The first partial derivatives of R( x, y ) are R 3x 3y 6 and R 4y 3x 18. x a. Set up the system of equations that is used to find the critical point of R( x, y ). The equations should be in terms of x and y, not R. (2 pts) y 3x 3y 6 = 0 4y 3x 18= 0 OR 3x 3 y = 6 4y 3x = 18 OR 3 3 x y 18 1 pt per equation (ALL OR NOTHING); Equivalent equations receive full credit. -½ pt for notational errors such as equating R x or R y to a number other than zero. b. Solve the system of equations to find the critical point of R( x, y ). Show all work (algebraic process or matrices). (5 pts) 3x 3 y = x 6 x x 4y= y 18 y pts valid work. If matrices are used; award 2 pts for [A] and 1 pt for [B] - No partial credit for [A] if the order of coefficients is not consistent. - No partial credit for [B] if order is inconsistent with the rows of [A]. - No partial credit for [B] if signs are incorrect due to NOT isolating the constants on one side of the equations. - If errors in [A] or [B] are a result of careless mistakes (copy error, dropping a sign) partial credit should be awarded (½ pt for each correct entry) If elimination is used, award 1 pt for each step: 1) multiplying one or both equations by the proper constant; 2) adding equations; 3) substituting 1 st value value to find the 2 nd If substitution is used, award 1 pt for each correct step: 1) isolating 1 st variable; 2) substituting expression into 2 nd equation and solving; 3) substituting 1 st value to find the 2 nd 1 pt each for correct x and y value (must match key AND follow the system of matrices that was solved; do not follow work). -½ pt for any notational errors such as equating the partials to a number other than 0, misuse of = in the matrix equations, equating [A]=[B], equating [A]=rref[A], etc. -½ pt if a matrix equation such as [A][X]=[B] OR [A] -1 [B]=[X] is never shown. 1 The critical point occurs when 10 thousand in-state students and 12 thousand out-of-state students are enrolled at the college.

10 c. Find the second partial derivatives matrix and the value of the determinant at the critical point of R( x, y ). (7 pts) Matrix: Determinant: D(10,12) ( 3)( 4) (3)(3) 3 1 pt for each 2 nd partial derivative; ½ pt partial credit can be earned if the 2 nd partials are explicitly defined outside of the matrix but placed in the incorrect order in the matrix. 3 pts for the determinant of their matrix (follow work) -1 pt for simplification error if supporting work is shown. d. Classify the critical point by completing the statement below. (3 pts) The critical point identified in part b is a relative maximum relative minimum or relative maximum or saddle point because D(10,12) = 3 > 0 OR D(10,12) > 0 OR D = 3 > 0 OR 3 > 0 and Reason 1: Show the specific value and comparison made (if necessary) Rxx(10,12) = -3 < 0 OR Rxx(10,12) < 0 OR Rxx = -3 < 0 OR -3 < 0. Reason 2: Show the specific value and comparison made If the student s determinant is positive, 1 pt correct classification 1 pt first reason (½ pt for specific value or notation, ½ pt for comparison) Deduct the full point if R(9,12) or the output value of the function is used 1 pt second reason (½ pt for specific value or notation, ½ pt for comparison) Deduct ½ pt if fxx is referenced instead of R xx If the student s determinant is negative, 1 pt correct classification 1 pt first reason (½ pt for specific value or notation, ½ pt for comparison) -1 pt for making the problem simpler than the intended problem Deduct 3 pts if the determinant is never found in part c.