GENTLY REMOVE THIS PAGE AND FILL OUT THE FRONT PAGE OF YOUR TEST. This page will be collected with our test, but work written here will NOT be graded. MATH 2070 Test 3 Formula Sheet When ou are asked to show work or the notation that leads to our answer, be sure to write the notation with specific parameter values and function names for that problem. No credit will be given for simpl coping a formula from this sheet or for using a function location (Y1) instead of a function name. Compensating For Change Formulas Given f( x,, ) d f x d = and x dx f The Determinant Test dx fxx fx Dab (, ) = = fxx f fx f x f f x ( ab, ) ( ab, ) If Dab (, ) > 0 and f ( ab, ) < 0, then f has a relative maximum at ( ab., ) xx If Dab (, ) > 0 and f ( ab, ) > 0, then f has a relative minimum at ( ab., ) xx If Dab (, ) < 0, f has a saddle point at ( ab., ) If Dab (, ) = 0, the test does not given an information about ( ab., ) Constrained Optimization f = g λ f x x = g λ gx (, ) = c M = cλ
Name: (PRINT) Section Instructor: DO NOT OPEN THIS EXAM UNTIL INSTRUCTED TO DO SO. Statement of Academic Integrit: An communications with an person (other than the instructor or the designated proctor) during this test in an form, including but not limited to written, signed, verbal or digital, is understood to be a violation of academic integrit. All devices, including but not limited to, computers, cell phones, and smart watches must be turned off while the student is in the testing room. All watches must be removed from our wrist and placed in our bag or in our pocket. The onl calculators to be used are TI-83, TI-84, TI-83+, TI-84+, or TI-84+ CE. No exceptions ma be made. You ma not borrow or share a calculator with another person taking this test. The test is closed notes and closed book. No outside resources ma be on our desk at an time during the testing period. As a rule, no questions are answered during the test. It is ver disruptive to other students. If ou believe a question is incorrectl written, ou ma write a note to that effect, but do tr to give the best answer ou can. Your proctor will not interpret an question for ou. B signing below, the student acknowledges that (s)he has read and understands these testing conditions. In addition, the student acknowledges that (s)he has neither given nor received an inappropriate or unauthorized information at an time before or during this test. This table is for instructor use onl Points Points Page Question Possible Earned 8 9 10 11 1a 4 1b 3 1c 5 1d 2 2 8 3a 4 3b 4 4a 2 4b 6 4c 4 4d 3 5a 2 5b 3 5c 4 5d 3 Scantron 1 Free Response (includes scantron) 58 Multiple Choice 42 Total 100 Student Signature: Page 1 of 12
GENERAL DIRECTIONS: Show work where possible. Answers without supporting work (where work is appropriate) ma receive little or no credit. Work in pencil onl. Do not round intermediate calculations. Answers in context ALWAYS require units. Round our answers to 3 decimal places UNLESS the answer needs to be rounded differentl to make sense in the context of the problem, OR the directions specif another tpe rounding, OR the complete answer has less than 3 decimal places. When ou are asked to show the mathematical notation that leads to our answer, be as specific as possbile and do not use calculator notation. i.e. Write f(3) f(2) instead of f( b) f( a) or Y1(3) Y1(2). When ou are asked to write a model, include all components of a model: an equation, a description of the output including units, a description of the input including units, and the input interval when known. When asked to write a sentence of interpretation, answer the questions When?, What?, How? and How much? using ordinar, conversational language. Do not use math words, terms, jargon or unnecessar phrases. Alwas use a ruler when estimating values off of a graph. HINTS FOR TROUBLESHOOTING YOUR CALCULATOR: If ou lose our L1, L2, etc., ou ma reinsert them using STAT 5 (set-up editor) enter. The SCATTER PLOT will not show unless PLOT 1 has been turned on and there is data in L1 and L2. Turn off/on scatter plots b using the arrows to highlight the item ou want off/on and using the enter ke to change the status. DIM MISMATCH error usuall means that the lists in L1 and L2 are not of equal length. DATA TYPE error usuall means that ou alread have something in Y1 and ou need to clear it before ou can paste a new equation. INVALID DIM error usuall means that our plot(s) are on, but that ou have no data in the lists. Refer to the second hint above. If our batteries die, raise our hand and hold up our calculator. If our instructor has an extra calculator available, he/she will loan it to ou for a few minutes. An unexpected 0 value when integrating an improper integral involving an exponential function ma mean that ou have increased the bounds beond what the calculator can compute. Tr smaller bounds. NO SIGN CHANGE error usuall means ou are attempting to solve an equation that has no solution. SINGULAR MATRIX error usuall means ou are attempting to invert a matrix that is not invertible. Check our matrix entries for errors. It is recommended that ou enter all single-variable functions in our calculator in terms of x. The use of other variables ma make calculations more difficult, and will prohibit the use of the graphing capabilities of our calculator. Page 2 of 12
Multiple Choice: Use a #2 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 42 points. For future reference, circle our answers on this test paper as ou will not receive our Scantron back with our test. The total weekl revenue of a workshop associated with manufacturing and selling their rolltop 2 2 desks is given b R( x, ) = 0.2x 0.25 0.2x + 200x + 160 dollars, where x denotes the number of finished units and denotes the number of unfinished units manufactured and sold each week. Check: R (50,100) = 22,000 Use this information to answer the next 3 questions. 1. What is the total weekl revenue of the workshop associated with rolltop desks if 300 finished units and 250 unfinished units are manufactured and sold this week? a. $25,000 b. $30,000 c. $48,000 d. $51,375 2. What is the rate of change of the total weekl revenue with respect to the number of finished desks manufactured and sold each week when the workshop makes and sells 300 finished and 250 unfinished desks each week? Answer: dollars per finished desk a. 20 b. 30 c. 50 d. -25 3. Which of the following is correct notation for the rate of change ou were asked to find in the previous question? (2 pts) a. R ( 300,250) b. dx d ( 300,250) c. d dx ( 300,250) d. R x (300,250) Page 3 of 12
ANT (, ) = (178 + N) T ln( N) barrels gives the amount of apples harvested from an orchard where the soil has N percent nitrogen and T is the average temperature for the fall season. Check: A(5, 75) 1583.217 A graph of the 1800 barrel contour curve is shown here. Use this information to answer the next 4 questions. 4. If the average fall temperature is 85 and 1800 barrels of apples are harvested, find the percentage of nitrogen in the soil. (Round to 1 decimal place.) a. 47.1 % b. 18.0 % c. 17.5 % d. 15.0 % 5. Find the slope of the tangent line to the curve at the point (25,78.9). dt dn (25,78.9) = a. 0.774 b. 1.292 c. 8.843 d. 11.427 6. The units dt dn (25,78.9) are. a. barrels per percent nitrogen b. barrels per c. per percent nitrogen d. percent nitrogen per (1 pt) 7. Which of the following shows how a derivative can be used to estimate the change in nitrogen needed to compensate for a drop in the average fall temperature from 78.9 to 70, if the orchard is to maintain a harvest of 1800 barrels? A A a. N ( 8.9) c. N ( 8.9) N T (25,78.9) b. N ( ) d. N ( ) 8.9 dt dn (25,78.9) 8.9 dn dt (25,78.9) (25,78.9) Page 4 of 12
Below is the contour map of a mountainous region. The contour values reflect the altitude in meters. Note: The lines that run across the contours represent rivers or streams and ma be ignored in answering the following. 8. Classif point A (2 pts) a. Saddle point c. Relative minimum b. Relative maximum d. None of these 9. Classif point B (2 pts) a. Saddle point c. Relative minimum b. Relative maximum d. None of these 10. Classif point C (2 pts) a. Saddle point c. Relative minimum b. Relative maximum d. None of these 11. What is the relationship between point A and point E? a. Their altitudes are about the same. b. The altitude of point A is lower than the altitude of point E. c. The altitude of point A is higher than the altitude of point E. d. We do not have enough information to determine. 12. From point C, in which of the following directions is the terrain steepest? a. North c. South b. Northeast d. Southeast Page 5 of 12
A ski resort in the area is going to build a new ski run which will open next winter. Due to various cost and safet concerns, the can onl build the run along the constraint line shown on the contour map below. Constraint Line 13. Which of the following is a constrained optimal point? a. points D and E b. Point B c. point E, onl d. Point D, onl 14. What is the highest altitude the ski run could reach under the given constraint? a. 1700 meters b. 1600 meters c. 1400 meters d. 1000 meters Page 6 of 12
15. The function f( x, ) has a critical point at f ( 3,6) = 500 and a second partials matrix of 20 30 30 10. Use this information to classif the critical point, if possible. a. f( x, ) has a saddle point at ( 3, 6). b. f( x, ) has a relative maximum at ( 3, 6). c. f( x, ) has a relative minimum at ( 3, 6). d. It is not possible to classif the critical point of f( x, ) with the given information. 16. Given a. b. c. d. fx fx fx fx 3 2 f( x, ) 4x 2x 3 3 8 3 = x+ = 24 2 x = = +, find f x. 2 2 12x 2 = x + 2 24 2 3 Check our Scantron now to make sure it will successfull run. Refer to the last page of the test for specifics. If it does, ou will earn one point. (1 pt) 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. Page 7 of 12
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 58% 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. A local gas station carries two different brands of energ drinks. Based on past sales, the weekl demand for energ drinks is Drm (, ) cans when the Red Bowl brand costs rr dollars per can and the Munster brand costs mm dollars per can. Values of Drm (, ) are shown in the table below. m / r 2.25 2.75 3.25 3.75 2.25 200 210 170 160 2.75 188 214 165 154 150 3.25 185 197 156 137 3.75 150 145 130 125 a. There is one relative extreme point in the table above. Find and interpret it. (4 pts) The maximum demand for energ drinks is 214 cans, occurring when Red maximum or minimum Bowls cost $_2.75 per can and Munsters cost $ 2.75 per can. b. Sketch and label the D=150 contour curve. (3 pts) c. Find a quadratic cross-sectional model for D(2.75, m ). Write the full, unrounded, model below. (5 pts) 2 D(2.75, m) = 56m + 293.6m 1 67.8 can s gives the function demand for energ drinks when the Munster brand costs m dollars per can and the Red Bowl brand costs 2.75 dollars per can, when _ 2.25 m 3. 75. units input variable interval 1 pt each blank 2.5 pts curve (should fall between the correct 2 values in columns 1-3 and hit the exact value in column 1) -0.5 pt per column that it does not hit on or between the appropriate value(s) -0.5 pt if ends do not extend all the wa to the edge of the table -0.5 pt if zig-zag or excessivel wav curve instead of a smooth curve 3 pts function 0.5 pt units; 0.5 pt each input; 0.5 pt input interval - 0.5 pt if function is not in terms of m; -1 pt if coefficients are off slightl indicating data entr error No credit for functions resulting from choosing the wrong data or reversing input and output. d. Would a row or column of data be used to construct a cross-sectional model that could be used to estimate each value below? (2 pts) D (3.75, 2.50) row column 1 pt each D (3, 2.25) row column Page 8 of 12
2x 2x 2. Let f ( x, ) = 5ln( ) e + 3 7x. Find the indicated partial derivatives. Simplification of 4 our answers is not required. 2x 4 2x 2 f x = 5ln( ) 2e + 0 2 7 = 10ln( e ) 7 4 1 pt each term A/N with the following exceptions - 0.5 pt for algebra error in rewriting f(x,), but follow the work for the derivative. -0.5 pt simplification error in answer (Simplification is optional but if done, should be done correctl. -0.5 pt for poor notation (misuse of equals signs, mixing parts of rewritten f(x,) in with the derivative, etc.) 1 2x 2x 5 5e 8x 5 1 pt correct c; 2 pts multipling c b correct derivative to get P (no credit for P if wrong derivative is used); 1pt final answer (do not follow from incorrect P due to using the wrong derivative ); - 0.5 pt for calculation error if correct work is shown -2pt if answer is correct but work is missing or does not support the answer Note: A maximum of 1 pt can be earned (for c) if the wrong derivative is used. (8 pts) f = 5 e + ln(3) 3 2 x( 4 ) 7x= + ln(3) 3 + 7x 3. The dail production level of a South American factor is given b the function PLC (, ) units, when L labor hours and C thousand dollars of capital are used. Use these values to answer the questions that follow. P P P (256,16) = 3072 = 9 L C a. Estimate the dail production level of the factor when 256 labor hours and 12 thousand dollars of capital are used. Show supporting work. (4 pts) P(256,12) 2820 units Work: P P c = ( 4)(63) = 252 P(256,12) 3072 252 = 2820 c (256,16) b. If onl 12 thousand dollars of capital are used, but the compan needs to maintain their original production level of 3072 units, how should the factor owners adjust the labor hours to compensate for the loss of capital? Show supporting work. (4 pts) To compensate for the 4 thousand dollar loss in capital while maintaining a production level of 3072, the factor owners should increase the labor hours from 256 hours to 284 hours. dl Work: L c = ( 4)( 7) = 28 dc (256,16) = 63 (256,16) (256,16) (256,16) increase or decrease = 7 2 pts multipling c from above b correct derivative to get L (no credit for L if wrong derivative is used; follow c from above); 0.5 pt 1 st blank; 0.5 pt 2 nd blank (follow from sign shown in work); 1 pt 3 rd blank (follow from work for miscalculations; do not follow if incorrect derivative was used) - 0.5 pt for calculation error if correct work is shown Page 9 of 12-2pt if answer is correct but work is missing or does not support the answer Note: A maximum of 1 pt can be earned (for 1 st two blanks) if the wrong derivative is used. dl dc
4. A Kentuck bourbon distiller sells two different tpes of bourbon, which are referred to as blended and single-barrel. 2 2 The annual profit is given b P( x, ) = 12.5x+ 36.5 0.05x 0.15x 0.3 thousand dollars when x hundred bottles of blended bourbon and hundred bottles of single-barrel bourbon are sold. Check: P (2,3) = 130.9 The first partial derivatives of Px (, ) are given as follows: P = 12.5 0.05 0.3x and P = 36.5 0.05x 0.6 x a. Write a sstem of equations that can be used to find the critical point of Px (, ). (2 pts) 12.5 0.05 0.3x= 0 1 pt per equation (ALL OR NOTHING); Equivalent equations receive full credit, 36.5 0.05x 0.6 = 0 including matrix equations. An augmented matrix is not acceptable here. -½ pt for an notational errors such as equating the partials to a number other than 0, misuse of =, etc. Example: P x=-0.3x-0.05=-12.5 b. Solve the sstem to find the critical point of Px (, ). Show all work (algebraic process or matrices and matrix operations). Report our answers as requested below. (6 pts) 12.5 0.05 0.3x= 0 0.3x 0.05 = 12.5 0.3 0.05 x 12.5 = 36.5 0.05x 0.6 = 0 0.05x 0.6 = 36.5 0.05 0.6 36.5 x 0.3 0.05 12.5 31.972 = = 0.05 0.6 36.5 58.169 1 x = 31.97 hundred bottles 2 decimal places with units = 58.17 hundred bottles 2 decimal places with units P = 1261.408 thousand dollars 3 decimal places with units 3 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 (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; 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 1.5 pts for correct x and pair (must match ke AND follow the sstem of matrices that was solved; do not follow work); 0.5 pt x & units 0.5 pt for P value that follows from student s x & ; 0.5 pt P units -½ pt if x & are found and labeled correctl in the work but copied into the wrong blanks; -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 matrices are used but a matrix equation such as [A][X]=[B] OR [A] -1 [B]=[X] or a matrix operation such as [X]=rref[A B] is never shown. -½ pt TOTAL for rounding errors Page 10 of 12
c. Find the second partial derivatives matrix for Px (, ). Then calculate the determinant of the second partials matrix at the critical point. (4 pts) D (31.97, 58.17) = ( 0.3)( 0.6) ( 0.05)( 0.05) 2 nd Partials Matrix: 0.3 0.05 0.05 0.6 Determinant: 0.1775 4 decimal places 2 pts matrix (0.5 pt each entr, A/N); 2 pts for the determinant of their matrix (follow work) -1 pt for calculation error if supporting work is shown. d. Classif the critical point. Justif our answer b completing the sentence below. (3 pts) The critical point identified in part b is a relative maximum relative minimum or relative maximum or saddle point because Determinant > 0 or 0.1775 > 0 or D(31.97,58.17) = 0.1775 > 0 and Reason 1: Show the specific value and comparison made (if necessar) P (31.97,58.17) = 0.3 < 0. xx Reason 2: Show the specific value and comparison made If the student s determinant is positive, 1 pt correct classification as rel max or rel min based on their P xx value from part c 1 pt first reason (½ pt for specific value; ½ pt for comparison) Deduct the full point if profit value is used instead of the determinant. Notation ma be accepted in place of the value ONLY if it is determinant notation (D instead of P or f) and is clearl labeled in part c. Example: Stating D>0 is oka here if student clearl labeled their answer in part c as D 1 pt second reason (½ pt for specific value, ½ pt for comparison) Deduct ½ pt if fxx is referenced instead of P xx. Notation ma be accepted in place of the value ONLY if it is correct notation (P xx instead of f xx) and the value associated with it is clearl labeled somewhere in the work. If the student s determinant in part c is negative, 1 pt classification as saddle point 1 pt first reason (½ pt for specific value; ½ pt for comparison) Same guidelines as above. Automatic deduction of 1 pt for making the problem simpler than the intended problem Deduct all 3 pts if the determinant is never found in part c. Page 11 of 12
5. Refer back the profit model for the Kentuck bourbon distiller from the previous page. The function and its partial derivatives are repeated here for our convenience: 2 2 P( x, ) = 12.5x+ 36.5 0.05x 0.15x 0.3 P = 12.5 0.05 0.3x and P = 36.5 0.05x 0.6 x The distiller has decided that the will onl use 52 barrels of their 12-ear stock for production this ear. Based on the number of bottles that can be filled from each barrel and the amount of 12-ear stock goes into each tpe of bourbon, the constraint equation is given b: gx (, ) = 0.25x+ = 52 barrels a. Find the first partial derivatives of gx. (, ) g = 0.25 and g = 1 x 1 pt each, A/N (2 pts) b. Write a sstem of equations (not matrices) in terms of x,, and λ that can be used to find the optimal point of Px (, ) subject to the constraint gx. (, ) DO NOT SOLVE. The solution is provided below. (3 pts) 1 pt per equation (follow from part a); Equivalent equations receive full credit. If correct equation is shown, but then manipulated incorrectl, ½ pt partial 12.5 0.05 0.3x= 0.25λ credit can be awarded. Otherwise no partial credit. 36.5 0.05x 0.6 = 1λ -½ pt for notational errors such calling the equation P x or P after it has been 0.25x+ = 52 manipulated, misuse of equals, etc. (onl deduct once) Solution: x= 27.44, = 45.14, λ 8.04 and P(27.44, 45.14) 1204.449 c. Use the close point test to classif the optimal point. (4 pts) x Px (, ) Close point 1 32 44 1201.2 Optimal Point 27.44 45.14 1204.449 Close point 2 24 46 1202.6 Circle our classification here: constrained maximum / constrained minimum / relative maximum / relative minimum 1 pt each x value, A/N; ½ pt each P value (do not follow from incorrect x) 1 pt classification that follows from P values (no credit if P is not shown) d. Use lambda to estimate the optimal profit earned b the distiller if the use 55 barrels of 12-ear stock instead of 52. Show our work. Include units. (3 pts) Work: New Optimal Profit _$1228.569 thousand_ M c λ = (3)(8.04) = 24.12 1204.449 + 24.12 = 1228.569 1.5 pts M calculation 1 pt new profit = old profit + M; 0.5 pt units -1/2 pt for calculation error if correct work is shown -1 pt if c is incorrect (sign error, etc) Page 12 of 12-2 pts if c is never considered (i.e. if student just adds lambda to the profit) -0.5 pt for notational errors
FILL OUT YOUR SCANTRON At the top of the scantron fill out: Name Your last name and first name Course: Math 2070 Section: see table below Instructor: see table below Test No. (see top, right page header) Test Version: (see top, right page header) Write AND Bubble in our CUID. Bubble in a 0 underneath the C. Bubble in the Test Version: (see top, right page header) The Scantron will not successfull run if: The XID is not filled in correctl: Check that ou have the correct number and that ou have bubbled in the number below where ou write it in. The initial C is bubbled in as a zero. The Test Version is bubbled in incorrectl. Check to make sure that the Test Version ou bubbled in (below the XID) agrees with the Test Version in the header of this page. Your responses are not bubbled in boldl enough. If our responses are onl lightl shaded, the machine will not pick up our answers. Check to make sure our marks are heav enough. You also will not earn the point if: Your name, our instructor s name and our section number are missing. Check to make sure all of these are clearl and legibl indicated. A list of section numbers, class times and instructors are included below for our convenience. Sec Time Instructor Sec Da/Time Instructor 1 TTh 8:00-9:15am Haodong Li 11 TTh 11:00am-12:15pm Jennifer Newton 2 TTh 9:30-10:45am Haodong Li 12 TTh 8:00-9:15am Travis Baumbaugh 3 TTh 11:00am-12:15pm Thanh To 13 TTh 8:00-9:15am Thanh To 4 TTh 12:30-1:45pm Todd Morra 14 TTh 3:30-4:45pm Anna Bachstein 5 TTh 2:00-3:15pm Todd Morra 15 TTh 2:00-3:15pm Anna Bachstein 6 TTh 3:30-4:45pm Madd St. Ville 16 TTh 9:30-10:45am Travis Baumbaugh 7 MW 2:30-3:45pm Joe Liu 17 TTh 11:00am-12:15pm Madd St. Ville 8 MW 4:00-5:15pm Joe Liu 18 TTh 12:30-1:45pm Jennifer Newton 9 MW 2:30-3:45pm Moll Honecker 19 TTh 12:30-1:45pm Thanh To 10 MW 4:00-5:15pm Moll Honecker 20 TTh 3:30-4:45pm Todd Morra Page 13 of 12