MATH 2070 Test 3 (Sections , , & )

Transcription

1 Multiple Choice: Use a # pencil and completel fill in each bubble on our scantron to indicate the answer to each question. Each question has one correct answer. If ou 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 our answers on this test paper as ou will not receive our Scantron back with our test. According to market analsis D( c, p) ice cream bars will be demanded monthl b 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. 380,000 b. 393,000 c. 405,000 d. 41,000. From point C, which of the following would cause the greatest decrease in demand? a. increasing the calories b 10 b. decreasing the calories b 10 c. increasing the price b $0.10 d. decreasing the price b $ What is the correct classification of points A, B, and C? a. A: relative maximum, B: saddle point, C: not a critical point 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 minimum, B: saddle point, C: relative maximum

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 onl) for the wind chill as a function of air temperature when the wind speed is 5 miles per hour. a. C( t,5) 0.813t b. C( t,5) 0.667t 9.60 c. C( t,5) 1.497t d. C( t,5) 1.073t To estimate the wind chill when the air temperature is 30 degrees Fahrenheit and the wind speed is 17 miles per hour ou would first need to. a. use a column of data to find the cross-sectional model C(30, w ). b. use a row of data to find the cross-sectional model C(30, w ). c. use a column of data to find the cross-sectional model C( t,17). d. use a row 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 wind chill per mile per hour c. of wind chill per of air temperature d. of air temperature per of wind chill.

3 TKK Products manufactures 50-, 60-, 75-, and 100-watt electric light bulbs. Laborator tests show that the lives of these light bulbs are normall 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 probabilit that a randoml 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 probabilit densit function for the lives of these light bulbs a maximum? a. 85 hours b. 775 hours c. 750 hours d. 675 hours 10. According to the Empirical Rule, approximatel what percentage of light bulbs will last between 675 hours and 975 hours? a. 84% b % c. 8% d. 81.5% 11. From point A, when is K(s, w) increasing most rapidl? a. when s increases b. when w increases c. when s decreases d. when w decreases A

4 P( a, n) 5a 3n 48a 4n an 90 million dollars gives the profit of a one-product compan where a thousand dollars is the amount spent on advertising and n thousand units of the product are sold. Check: P(6,) 40 Use this context to answer the next four questions. 1. Complete the interpretation of P n (6.3,.1) 4. When a one-product compan spends 6.3 thousand dollars on advertising and sells.1 thousand units of their product,. a. their profit is decreasing b 4 million dollars per thousand dollars spent on advertising. b. the amount spent on advertising is decreasing b 4 thousand dollars per million dollars of profit. c. their profit is decreasing b 4 million dollars per thousand units sold. d. the number of units sold is decreasing b 4 thousand units per million dollars of profit. 13. What sstem of equations would have to be solved to determine the critical point of P( a, n )? a. 10a n 4 a 6n 48 b. 10a n 48 a 6n 4 c. 10a n 48 a 6n 4 d. 10a n 4 a 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, 1.98) b. (6, 1.476) c. (6,.691) d. (6,.387) 15. Find the slope of the tangent line at the point (4, 1.71). a..34 b c d

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

6 Free Response: RE-READ the directions at the beginning of the test. Then read each question carefull. Provide onl one clearl indicated answer to each question. If our answer is illegible, it will be graded as incorrect. Show all work. The free response portion is 53% of our test grade. When possible, set up the specific mathematical notation that is being evaluated to obtain our answer. No credit will be awarded for simpl coping 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 emploee income M ( t, v ) thousand dollars at a compan where t is the number of ears the emploee has been with the compan and v is the emploee review value assigned to the emploee. Emploee Review Value (v) t ears a. Find the quadratic cross-sectional model that could be used to model the boxed in data on the table above. Completel define our model b filling in all of the blanks below. (6 pts) M t t t (, 3) quadratic function with coefficients rounded to three decimal places thousand dollars gives the average annual emploee income output units output description when the emploee has been with the compan for t ears input description and the emploee s review value is 3, 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 likel tpo 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 emploee who has been with the compan for.5 ears and has an emploee review value of 3. Show the mathematical notation, round our answer to one decimal place and include units. (3 pts) M(.5,3) 40.6 thousand dollars 1 pt notation, 1.5 pts value, ½ pt unit; -½ pt rounding error

7 . Determine if f ( x ) is a valid probabilit densit function. Show specific work (graph, values, etc.) to justif our conclusion. (4 pts) Is f ( x ) a valid p.d.f.? No Justification: Yes or No 3. The amount of snowfall in feet is a remote region of Alaska in the month of Januar has the x x if 0 x 3 probabilit densit function f ( x) 3 9. Check: f (1.5) otherwise a. Find the probabilit that this region receives more than feet of snow in Januar. (4 pts) Show the specific probabilit and mathematical notation. Round our answer to four decimal places. 3 P( x ) f ( x) dx ½ pt probabilit notation; 1.5 pts specific integral notation; pts answer -½ pt rounding error, missing dx notation b. Find the average amount of snow this region receives in Januar. (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; pts answer; ½ pt units -½ pt missing dx notation c. Find the variance in the amount of snow this region receives in Januar. (4 pts) Show the specific mathematical notation and give the answer. 3 x f x dx ( ) 0.45 The total trapped area is not equal to 1. A (1 ) (0.5)(0.4) pt conclusion (must follow reasonable supporting work) 1 pt stating the trapped area 1 (or > 1 or < 1 depending on version) 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 pts specific integral notation (1 pt bounds, 1 pt mu plugged in); 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 explicitl defined in part b; - pts for onl finding σ

8 4. R(s,n) million dollars describes the annual revenue for a fast food compan, where s in the number of locations the compan operates in South Carolina (SC) and n is the number of locations the compan operates in North Carolina (NC). The partial derivatives for the annual revenue function are: Rs = 6s + n 0 million dollars per SC restaurant and Rn = s + 4n 60 million dollars per NC restaurant. The compan currentl operates 30 locations in SC and 0 locations in NC which generates an annual revenue of 5.4 million dollars. a. Write the specific formula for dn. Your final answer should be in terms of n and s, not R. ds 1 pt negative (parentheses must be shown if (4 the pts) negative dn Rs 6s n 0 6s n 0 sign is written in the numerator) OR 3 pts partial flipped ds Rn s 4n 60 s 4n 60 1 pt partial earned for R s/r n notation -½ pt for notation errors such as dn/ds = - ds/dn - pts for giving a numerical answer instead of a formula b. What are the units on dn? Circle one. ds (1 pt) NC restaurants SC restaurant SC restaurants NC restaurant million dollars NC restaurant million dollars SC restaurant c. Suppose the compan is planning to close two locations in SC. In order to maintain the same revenue, how man new restaurants would the need to open in NC to compensate for the lost revenue in SC? Show all necessar work to solve this problem. (6 pts) Work: dn ds (30,0) 6(30) (0) 0 = -.5 (30) 4(0) 60 n dn s ds dn n s.5 ( ) 5 ds 1.5 pts for correctl finding dn/ds at (30,0). 1 pt for correct s, including the negative (all or nothing). 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 an 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 5.4 million dollars, the compan would have to open 5 new locations in NC bringing the total number of restaurants to 5 in NC and 8 in SC.

9 5. The annual revenue generated at a college is given b R x, 1.5 x 3x 9x 1 million dollars when x thousand in-state students and thousand out-of-state students are enrolled. Check: R(4,5) = 55 The first partial derivatives of R( x, ) are R 3x 3 9 and R 4 3x 1. x a. Set up the sstem of equations that is used to find the critical point of R( x, ). The equations should be in terms of x and, not R. ( pts) 3x 3 9 = 0 4 3x 1= 0 OR 3x 3 = 9 4 3x = 1 OR 3 3 x pt per equation (ALL OR NOTHING); Equivalent equations receive full credit. -½ pt for notational errors such as equating R x or R to a number other than zero. b. Solve the sstem of equations to find the critical point of R( x, ). Show all work (algebraic process or matrices). (5 pts) 3x 3 = x 3x 4= x pts valid work. If matrices are used; award 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 (cop error, dropping a sign) partial credit should be awarded (½ pt for each correct entr) If elimination is used, award 1 pt for each step: 1) multipling one or both equations b the proper constant; ) adding equations; 3) substituting 1 st value value to find the nd If substitution is used, award 1 pt for each correct step: 1) isolating 1 st variable; ) substituting expression into nd equation and solving; 3) substituting 1 st value to find the nd 1 pt each for correct x and value (must match ke AND follow the sstem of matrices that was solved; do not follow work). -½ pt for an 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. The critical point occurs when 9 thousand in-state students and 1 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, ). (7 pts) Matrix: Determinant: D(9,1) ( 3)( 4) (3)(3) 3 1 pt for each nd partial derivative; ½ pt partial credit can be earned if the nd partials are explicitl 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. Classif the critical point b 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(9,1) = 3 > 0 OR D(9,1) > 0 OR D = 3 > 0 OR 3 > 0 and Reason 1: Show the specific value and comparison made (if necessar) Rxx(9,1) = -3 < 0 OR Rxx(9,1) < 0 OR Rxx = -3 < 0 OR -3 < 0. Reason : 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(9,1) 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.