BARUCH COLLEGEMATH 2205SPRING 2006MANUAL FOR THE UNIFORM FINAL EXAMINATIONJoseph Collison, Warren Gordon, Walter Wang, April Allen MaterowskiThe final examination for Math 2205 will consist of two parts.Part I: This part will consist of 25 questions similar to thequestions that appear in Problem Section A.No calculator will be allowed on this part.Part II: This part will consist of 10 questions. The graphing calculator is allowed on this part.At least 6 of the questions will be similar to questionsthat appear in Problem Section C (Calculator allowed).The remaining 4 questions will be similar to questionsthat appear in Sections A and C.(Note that all 10 questions might come from Section C.)There may be a few new problem types on the exam that are not similarto the problems in Sections A and C. If such problems appear, they willbe similar to problems that you have seen during the semester.GRADING: Each question will be worth 3 points. Anyone who gets 34 or 35 questions correct will be assigned a grade of 100. No points are subtracted for wrong answers.CONTENTS OF THIS MANUAL:Page showing the sample questions that correspond to each section of the current text. (When a section has been covered in class, the list indicates the problems that can be used in studying for the exam that includes that section during the semester.)TI-89 Facts for the Uniform Final Examination. (This portion indicates the minimal calculator knowledge needed.)Problem Section A. (Problems for which a calculator may not be used.)Problem Section C. (Problems for which a calculator should be used.)Answers to the problems.

Math 2205Textbook Sections Corresponding to Sample Uniform Final Exam QuestionsSpring 2006Textbook: Applied Calculus, Gordon, Wang, Materowski, Baruch College, CUNYSection Problems 1.1 (Extrema) 12, 20, 59, 82, 86, 99, 101, 129, 130, 151, 154, 155, 156, 158, C4, C5, C191.2 (1 Der. Test) 21, 22, 60, 145, 160st1.3 (Concavity) 11, 16, 17, 89, 90, 95, 100, 102, 120, 121, 122, 126, 140, 141, 144, 146, 147, 150, 153,159, C2, C8, C9, C12 1.4 (Geom. Apps.) 31, 61, 65, 103, 165, 1681.5 (Business Apps.) 23, 62, 92, 97, 104, 118, 123, 127, 137, 143, 148, 149, 152, 166, C351.6 (Linearization) 9, 25, 93, 105, 115, 116, 117, 139, 161, 162, 167, C3, C20, C21, C29, C302.1 (Inverses) 5, 38, 662.2 (Exponent. F’cns) 8, 19, 502.3 (Number e) 15, 57, 96, C7, C11, C13, C322.4 (Derivative e^x) 1, 10, 26, 27, 41, 63, 64, 107, 119, C10, C23, C25, C26, C272.5 (Logarithms) 2, 6, 24, 43, 51, C1, C142.6 (Log. Props/Der) 7, 36, 39, 40, 44, 47, 48, 53, 67, 68, 85, 106, 114, 128, 131, 142, 157, C6, C15, C16,C24, C31, C342.7 (Applications) C22, C28 3.1 (Antiderivatives) (See 3.2 Problems)3.2 (Apps. Antider.) 4, 84, 124, 125, 1383.3 (Substitution) 3, 493.4 (Approx. Area) 18, C6, C183.5 (Sigma and Area) 81, 88, C173.6 (Definite Integral) 13, 42, 45, 803.7 (Subst. Def. Int.) 37, 52, 54, 55, 58, 78, 79, 83, 87, 943.8 (Applications) 14, 46, 56, C334.1 (f(x,y)) 28, 69, 109, 1324.2 (Partial Derivs.) 29, 30, 32, 70, 71, 73, 74, 110, 112, 133, 134, 1644.3 (Extrema) 33, 34, 72, 76, 77, 108, 113, 1364.4 (Lagrange Mult.) 35, 111, 1354.5 (Econ. Apps) 75, 91, 98, 163

Figure 1

Figure 2

Figure 3

TI-89 FACTS FOR THE UNIFORM FINAL EXAMINATIONThe following information represents the minimal calculator usage students should be familiar with whenthey take the uniform final examination. Instructors are expected to provide much more information in class.Limits can be evaluated with the TI-89 by using the limit function whosesyntax islimit(expression, variable, value).This function is found on the Home screen by using the F3 Calc key. Thecalculus menu that appears when F3 is pressed is shown in Figure 1. Thelimit function is selected by either pressing 3 or using the down arrow cursorto highlight choice 3:limit and then pressing ENTER.Example 1: Find Solution: Select the limit function as indicated above.Complete the command line as follows and press ENTER:limit((x-4)/(16-!(x)),x,16)The answer is 1 as shown in Figure 2.Example 2: Find Solution: In order to use the calculator for this problem it is best to think of !xas a single letter such as t. That is,Now enter on the command line the following:limit((1/(x+t)-1/x)/t,t,0) The result is -1/x as shown in Figure 3.2Note that example 2 could have been done without a calculator if the limit requested was recognized as thedefinition of the derivative of f(x) = 1/x for which f !(x) = -1/x .2The syntax for the solve command and for finding the first or second derivative is:solve(equation, variable)d(expression, variable) (first derivative, f !(x), where f (x) = expression)d(expression, variable, 2) (second derivative, f "(x), where f (x) = expression)

Figure 4

Figure 5

Figure 7

Figure 6

Figure 8

The solve command is obtained by pressing the F2 (Algebra menu) key in theHome Screen as shown in Figure 4 and then pressing ENTER to select choice1:solve.The differentiation operator d is choice 1 on the Home screen calculus menushown in Figure 1. It can be more easily accessed by pressing the yellow 2ndkey and then 8 (the d appears above the 8 in yellow). The purple D appearingabove the comma key cannot be used for this purpose.If an answer such as e or 15/236 appears when a decimal answer is desired, the command can be repeated3with the green diamond " key pressed before pressing ENTER.Example 3: The demand function for a product is p = 10 - ln(x), where x is the number of units ofthe product sold and p is the price in dollars. Find the value of x for which the marginal revenue is 0.Solution: The revenue function is R(x) = px = x(10 - ln(x)).The marginal revenue is the derivative, R!(x). This can be found by hand or byusing the calculator commandd(x*(10-ln(x)),x)obtained by pressing the keys2 8 ( x * ( 1 0 - 2 x x ) ) , x ) ENTERnd ndas shown in Figure 5. The result is R!(x) = 9 - ln(x).To find the value of x for which the marginal revenue is 0 the equation 9 - ln(x)= 0 must be solved.Press F2 (Algebra) ENTER as shown in Figure 4. Then complete thecommand line assolve(9-ln(x)=0,x)Pressing ENTER produces the result e . Since a decimal answer is desired,9repeat the command by first pressing the green diamond " key before pressingENTER. The final answer is 8103.08 as shown in Figure 6 and can be roundedoff to 8103.Example 4: The function has exactly one inflection point. Find it.Solution: Recall that an inflection point occurs if h"(t) = 0 and the concavitychanges sign. So first the second derivative is found with the commandd(30/(10+e^(-0.1t+5)),t,2)where e^( is obtained by pressing "x(e appears in green over the x key)xFigure 7 is the result. Recall that to see the entire result the up cursor arrowbutton must be pressed and then the right cursor arrow button must be presseduntil the rest of the result on the right can be seen as shown in Figure 8. Thus,Move the cursor down to the command line again. The easiest way to solve h"(t) = 0 is as follows. First press F2 (Algebra) and select choice 1 so that

Figure 12

Figure 13

Figure 14

Figure 9

Figure 10

Figure 11

the command line now only showssolve(Press the up cursor arrow to highlight the expression for h"(t) and pressENTER. The command line now shows the end ofsolve(above expressionComplete the command by entering=0,t)ENTERThe answer appears in Figure 9.Since only one answer appears and it is known that an inflection point exists,there is no need to check to see if the concavity changes at t = 26.9741. It onlyremains to find the value of the original function at t = 26.9741. To do so,enter 30/(10+e^(-0.1*26.9741+5))on the command line to obtain h(26.9741) = 1.5 as shown in Figure 10. Therefore, the inflection point is (26.9741, 1.5).Example 5: 0.0559 and 1.788 are the only critical numbers for f (x) = e ln(x/2). Determine5xif the critical point (0.0559, -4.73) is a relative minimum, a relative maximum or neither.Solution 1: Recall that if f "(0.0559) is positive the critical point is a relativeminimum, if it is negative the point is a relative maximum, and if it is 0 thefirst derivative test must be used.The key with the symbol # on it means “when” or “such that.”Now enter the following command:d(e^(5x)*ln(x/2),x,2)#x=0.0559Since f "(0.0559) = -304.911 is negative as shown in Figure 11, the criticalpoint is a relative maximum.Solution 2: A graph that clearly showed the point (0.0559, -4.73) wouldreveal what was true for the critical point.Press "F1 for the y= screen. Enter the functiony1=e^(5x)*ln(x/2)as shown in Figure 12.If F2 (Zoom) and choice 6:ZoomStd are selected (so that the x and y values gofrom -10 to 10), the result is Figure 13. Notice that this reveals nothing aboutthe critical point in question. So it is desirable to look at the point more closely. (If you are “expert” at using ZoomIn, this approach can be used instead of theone shown. Just make sure the new center chosen has a y value near -4.73.)A good start might be to select values of x between -1 and 1 and values of ybetween -6 and -4. So press "F2 (Window) and enter the following values.xmin=-1 xmax=1 xscl=1 ymin=-6 ymax=-4 yscl=1as shown in Figure 14.

Figure 15

Figure 16

Figure 17

Then press "F3 (Graph). Figure 15 is the result.Now press F3 (Trace) and then press the right cursor arrow a few times andobserve values of x and y for each point on the graph shown. Clearly the highpoint is the critical point and therefore the critical point is a relative maximum.Students should also be familiar with using F5 (Math) in the graph window todetermine the exact location of a relative maximum or relative minimumdisplayed on the graph. For the graph shown in Figure 15 press F5 (Math) toobtain the menu shown in Figure 16.Select choice 4:Maximum.In response the “Lower Bound?” request, just use the cursor movement arrowsto move the blinking cursor to the left of the maximum and press ENTER. Then, for the “Upper Bound?” request, move the blinking cursor to the right ofthe maximum and press ENTER. Figure 17 is the result and indicates that(0.05591, -4.73092) is a relative maximum.Example 6: Given f (x) = x + 14x - 24x - 126x + 1354 3 2i) Find the critical numbers.ii) Find the critical pointsiii) All of the following are graphs of f(x) in different graphical windows. Which graph most accurately portrays the function (shows its relative extrema, asymptotes, etc.)? a) b) c)

d) e)

Solution: i) The critical numbers are the solutions to0 = f !(x) = 4x + 42x - 48x - 126.3 2Enter the commandsolve(4x^3+42x^2-48x-126=0,x)into the calculator as shown in the screen on the right.The critical numbers are x = -11.3145, -1.31026 or 2.12479

ii) For the critical points the value of y is needed for each of the critical numbers found.Enter the commandx^4+14x^3-24x^2-126x+135#x=-11.3145into the calculator.Repeat this for the other critical numbers. (Pressing the right cursor arrowremoves the command line highlight and positions the cursor at the end of theline. Then the backspace key # can be used to eliminate the -11.3145. Thenenter the next critical number.)The screen shown indicates the critical points are (-11.3145, -5401.64), (-1.31026, 230.345) and(2.212479, -86.3943).iii) A graphing screen should be used that includes the critical points found. The x values shown should include all values between -12 and 3. The y valuesshown should include all values between -5402 and 231. A reasonable windowto choose would thus bexmin=-15 xmax=5 xscl=1 ymin=-5500 ymax=500 yscl=500The graph shown on the right is the result. So the answer is d.Integration and the TI-89Example 7: Evaluate

In the HOME screen, Integration is under F3(calc)->2: integrate

Figure 18 shows the entry on the TI-89. Figure 18(Note that you must insert the constant of integration on your own.)Example 8: Evaluate The procedure is the same as for indefinite integrals, but with 2 extra arguments:F3(calc)->2: integration (The last 2 arguments are the lower limit, followed by the upper limit.)Figure 19 shows the entry on the TI-89.

Figure 19

SECTION A1. Evaluatea) b) c) d) e) none of these2. Evaluate a) 1 b) 3/2 c) 2 d) 5/2 e) 93. Evaluate a) b) c) d) e) 4. The marginal cost, in dollars is given by the equation C ' (x) = 2x + 200. If the fixed overhead cost is$2000, then the total cost of producing 10 items isa) $2000 b) $2100 c) $3500 d) $4100 e) $45005. Suppose y = f(x) and y = g(x) are inverse functions with f(3) = 9 and isa) 7 b) -7 c) 1/7 d) -1/7 e) does not exist6. Evaluate a) 2 b) -2 c) 8 d) -8 e) 27/47. Evaluate a) b) c) d) e) 8. A population of bacteria is decaying continuously at a rate of 7% every hour. If there are 3.2 millionbacteria present at 9AM, which of the following gives the population of bacteria t hours after 9AM?a) b) c) d) e) 9. Given y = 3x , find the value of the differential dy if x changes from 4 to 6.2a) 48 b) 24 c) 60 d) 12 e) 3010. The equation of the tangent line to the curve defined by at the point at which x = 1 isa) y = 4x + 2 b) y = 5x + 1 c) y = 11x - 5 d) y = -11x + 17 e) y = 17x - 11

11. The position equation for a moving body is for t $ 0.Find the time when the velocity is 0.a) 1 b) 3/2 c) 0 d) 5/6 e) 2/312. Find all of the points at which the graph of f (x) = x - 4x + 5 has horizontal tangent lines.4a) (2,0) only b) (1,0) only c) (1,2) and (-1,10) d) (1,0) and (2,13) e) (1,2) only13. Evaluate .a) 28 b) 32 c) 4 d) 16 e) 1014. The consumer’s surplus for the demand and supply functions x + y = 4 and -x + y = 2 isa) ! b) 3/2 c) 5/2 d) 7/2 e) 415. Which of the following best represents the graph of ? (All x- and y- scales are 1.)a) b) c)

d) e)

16. (0, 5) (i.e. x = 0 and y = 5) is the only critical point of f (x) = 2x + 3x + 5. You do not have to verify6 4this. Determine what is true of (0, 5).a) (0, 5) is a relative maximum. b) (0, 5) is a relative minimum c) (0, 5) is a saddle pointd) (0, 5) is not a relative extremum e) no conclusion is possible17. Solve f "(x) = 0 for the function f (x) = 4x - 9x + 1.3a) x = 4 b) x = 12 c) x = -9 d) x = 1 e) x = 018. The area of the region bounded by f(x) = 3 , the lines x = 1 and x =2 and the x-axis is to be approximatedxby 10 rectangles using the right endpoint of each rectangle. The sum isa) b) c) d) e) none of these

19. When writing the formula for an exponential function passing through the points and (2, 36)using the form , what would be the base, b?a) 4/9 b) 81 c) 1/81 d) 3 e) 1/320. Find the critical numbers for g(x) = x - 2x .4 2a) 0, ±!2 b) 0, ±1 c) ±!(3)/3 d) only 0, -1 e) only 021. If the derivative of g(x) is g!(x) = (x - 1) (x - 2) then the function is2a) decreasing on (-%, %) b) decreasing on (-%, 1) and increasing on (1, %)c) decreasing on (-%, 2) and increasing on (2, %) d) increasing on (-%, 1) and decreasing on (1, %)e) decreasing on (-%, 1), increasing on (1, 2) and decreasing on (2, %)22. Find the absolute maximum of f (x) = 12x - x on the closed interval [-4, 1].3a) -4 b) -2 c) 16 d) 11 e) 023. Find the number of units to be produced in order to maximize revenue if the demand function is given byp = 600x - 0.02x .2a) 15,000 b) 20,000 c) 25,000 d) 30,000 e) 35,00024. Evaluate a) 7/5 b) c) -5/7 d) e) none of these25. Let the profit function for a particular item be P(x) = -10x + 160x - 100. Use differentials to2approximate the increase in profit when production is increased from 5 to 6.a) 30 b) 40 c) 55 d) 60 e) 7026. If , then f !(x) =a) b) e c) e (2x - 3) d) (2x - 3) e) e (x - 3x)2x - 3 2x - 3 2x - 3 227. Find dy/dx if xe + 1 = xyy

a) 0 b) c) d) e) 28. For f (x, y) = 3x - y find f (3, -4).2 3a) 37 b) 91 c) 18 d) 6x e) 6x - 3y2

x29. For f (x, y) = 3x - y find f2 3a) 6x b) 6x - 3x c) -6y d) 0 e) cannot be determined2 2

x30. For f (x, y) = 4x y + e find f .2 3 xya) 24xy + e b) 8xy + e c) 12x y + xe d) 8xy + ye e) 12x y + ye2 xy 3 xy 2 2 xy 3 xy 2 2 xy31. The sum of a number and twice another number is 10. Find the largest possible product. (Notice thatyour answer is not one or both of the numbers, it is the product.)a) 12.5 b) 8 c) 12 d) 12.6 e) 13

y32. For f (x, y) = 4x y + e find f .2 3 xya) 24xy + e b) 8xy + e c) 12x y + xe d) 8xy + ye e) 12x y + ye2 xy 3 xy 2 2 xy 3 xy 2 2 xy33. For f (x, y) = x + y + 4x - 6y + 10, the critical point is2 2a) (-3, 4) b) (3, -2) c) (-2, 3) d) (2, -3) e) (4, -3)

xx yy xy34. Suppose a function of two variables has a critical point. If, at that critical point, D = f f - (f ) > 0 and2xxf < 0. Then that critical point isa) a relative maximum b) a relative minimum c) a saddle pointd) an absolute maximum e) cannot be determined35. To find the maximum of f (x, y) = 2x + 4xy - 8y subject to the constraint 2x + 4y = 12 by the method of2 2Lagrange multipliers, one must solve the equationsa) 4x + 16y +2" = 0 b) 4x + 16y + 2" = 0 c) 4x + 4y + 2" = 0 d) 4x + 16y + 2" = 0 e) 4x + 4y + 2" = 0 4x + 4y + 4" = 0 4x + 4y + 4" = 0 4x - 16y + 4" = 0 4x + 4y + 4" = 0 4x - 16y + 4" = 0 2x + 4y - 12 = 0 2x + 4y + 12 = 0 2x + 4y - 12 = 0 2x + 4y + 12 = 0 -2x - 4y - 12 = 036. a) ln 2 b) 2 c) d) ln 72 e)

37. Evaluate a) 4/5 b) -ln5 c) ln5 d) 24/25 e) 138. . Find .a) 11 b) 1/11 c) 411 d) 1/411 e) -1/6

39. Evaluate a) log 10 + log 20 b) c) 1 + log 2 d) e) none of these40. a) 5 b) 5/2 c) 2/5 d) -5 e) 5/441. =a) 10e b) e c) 10e d) e) none of these9x 10x 10x

42. Consider the function defined by y = f(x), whose graph is give in the accompanying figure. Suppose the area of the region indicated by A is 5, the area of B is 3 and 4, then the area of the Region C is A C -3 -2 0 1 B y =f(x) a) 1 b) 2 c) 3 d) 4 e) none of these43. Evaluate a) 15 b) 12e c) 125 d) 3/5 e) 24344. Evaluate a) b) c) d) e) 45. The area of the region bounded by f(x) = 3x , g(x) = 4 - x and the x-axis from x = ! to x = 3 is given by2a) b) c) d) e) none of these

46. A probability density function on the interval [0, 4] is given by the equation f(x) = then Pr(2 & x & 4) =a) 1/6 b) 1/5 c) 1/4 d) 1/3 e) ! 47. If 3 = e then k =2x kxa) b) c) ln 3 - ln 2 d) e) 48. Given log(2 + x) + log(x - 3) = log14, then x =a) -4 b) 5 c) 4 d) - 5 e) -4 and 549. a) b) c) d) e) 50. An exponential graph containing the point (2, 5) and (4, 12) has the equationa) b) c) d) e) none of these51. Given the function , which of the following would be the graph of ? (All x- and y-scales are 1.)a) ` b) c)

d) e)

52. a) 65/ 76 b) 64/25 c) 3 d) 4/3 e) !

53. a) 0 b) 5/4 c) 3 d) 1 e) 12/554. Evaluate a) -80 b) -5 c) 7 d) -1/16 e) -1/455. Evaluate a) b) ! c) 8 d) 4 e) 156. The producer’s surplus for the demand and supply functions x + y = 4 and -x + y = 2 isa) ! b) 3/2 c) 5/2 d) 7/2 e) 357. Which of the following represents the amount in an account t years after investing $3000 at a 6% annualrate compounded monthly?a) b) c) d) e) 58. Use your knowledge of the definite integral (and your knowledge of the graph of the function) in order to evaluate .a) -8 b) 1024/3 c) d) e) 59. Find all of the critical numbers for .a) 0 b) -3, 3 c) 3 d) -3, 0, 3 e) -360. Find the open intervals on which is decreasing.a) (-%, -2) and (2, %) b) (0, %) c) (-2, 2) d) (-%, 0) e) (-%, %)

61. Of all numbers (positive, negative and zero) whose difference is 4, find the two that have the minimumproduct. One of the numbers is:a) 4 b) -3 c) -2 d) 1 e) -162. Find the maximum profit for the profit function .a) 18 b) 3 c) 1 d) 0 e) 1263. Find the derivative of f (x) = 4e . -3xa) -12e b) 12e c) -12e d) 4e e) 4e3x -3x -3x -3 -3x64. Find the derivative of f (x) = x e4 xa) 4x e b) x e + 4x e c) 4x + e d) x + e e) 12x e3 x 4 x 3 x 3 x 4 x 4 x65 and 66. The height in feet above the ground of a ball thrown upwards from the top of a building is givenbys = -16t + 160t + 200, where t is the time in seconds2

65. What is the maximum height (in feet) of the ball?a) 5 b) 600 c) 1400 d) 200 e) 16066. What is ?a) 32 ft/sec b) 32 sec c) -864 ft/sec d) 4 ft/sec e) 4 sec67. Find the derivative of f (x) = -6 ln xa) 6/x b) -6/x c) -6/(ln x) d) 6/x e) -6/x2 2

68. Find the derivative of .a) b) c) d) e) 69. Suppose that f (x, y) = 30 - (y + 4x). Find f (3, -2).2a) 29 b) 38 c) 16 d) 13 e) 14

x70. Find f for f (x, y) = 5x - 4xy + y2 3a) 10x - 4y + 3y b) 10x - 4y c) -4x + 3y d) 6x - 4y + 3y e) 3y2 2 2 271 and 72. Given f (x, y)= 2x - xy + 4y - 7:

x71. Find f (x, y).a) 6 - y b) 2 - y c) 6 - x - y d) 2 - x - y e) -x + 472. Find the critical points.a) (4, 2) b) (6, 6) c) (3, 3) d) (4, 4) e) (2, 2)y73. Find w at the point (3, 2, 2) if w(x, y, z) = x y e .2 3 z

a) 108e b) 12e c) 12 d) 72e e) 48e2 2 2xy74. Find f if f (x, y) = e ln y.x

a) b) c) d) e) e ln yx

75. The demand function d(p, n) for umbrellas is a function of their price, p, and the number of rainy dayspper month, n. Which of the following best explains why the partial derivative satisfies d < 0?a) If it rains a lot, you should lower your price.b) If there are a lot of vendors, you should lower your price.c) As the price of umbrellas increases, demand will decrease.d) If you have a bad location, you will not sell many umbrellas.e) During the rainy season, you will sell more umbrellas.76. Find all critical points (x, y) for f (x, y)= x + 6x + y - 6y - 15y.2 3 2a) (3, 1) and (3, 5) b) (5, 3) and (5, 1) c) (-3,-5) and (-3,-1) d) (5, -3) and (5, -1) e) (-3,-1) and (-3,5)77. Suppose that f (x, y) = x - y - 2x + 2y - 7. Use the second partials test to determine the type of extrema4 2 2for the following two critical points: (0, 1) and (-1, 1).a) (0, 1) is a relative maximum and (-1, 1) is neither a relative maximum nor a relative minimum.b) No conclusion is possible for (0, 1) and (-1, 1) is a relative minimum.c) (0, 1) is a relative maximum and (-1, 1) is a saddle point.d) (0, 1) is a saddle point and (-1, 1) is a saddle point.e) (0, 1) is neither a relative minimum nor a relative maximum and (-1, 1) is a relative maximum.78. Find the area under the curve of on the interval [0, 9].a) 18 b) -18 c) 3 d) 9 e) -9

79. Which of the following will give a result of ?a) b) c) d) e) 80. Evaluate a) 54 b) 108 c) 0 d) 18 e) 3681. Evaluate a) 60 b) 630 c) 1260 d) 410 e) 120082. Find any relative extrema of a) y = 0 b) y = 7/5 c) y = -7/3 d) y = 7 e) there is no relative extremum83. represents the cost in millions of dollars of producing x thousand items. Find theaverage value of C on [0, 3].a) $14 million b) $7 million c) $8 million d) $2 million e) $12 million84. The marginal cost (in thousands of dollars) of producing x hundred items is given by . Ifoverhead is $500, find the cost function, C(x).

oa) C(x) = 504x + C b) C(x) = 4x - 500 c) d) C(x) = 4x + 500 e) C(x) = 496x85. a) 8 b) c) 48 d) 18 e) 1286. Find the critical numbers for f (x) = x + 3x + 1.3 2a) -2 only b) 0 and -2 c) 0 only d) 3x + 6x e) There are no critical numbers.2

87. Evaluate .a) 4 b) 2 c) d) e) 88. Evaluate .a) 3/2 b) ! c) 0 d) 3 e) 89. Identify the x-coordinate any relative extrema of .a) Relative min at x = -1 b) Relative max at x = -1 c) Relative min at x = 1/e d) Relative max at x = 1/e e) No relative extrema90. Given y = 12x - x , find the critical points. Then determine whether they are relative extrema (the3second derivative test is easiest). Sketch the graph and pick the sketch that resembles your sketch the best.

91. The demand equations for two related products are given, where is the per unit price when x units aredemanded for the first product and is the per unit price when y units are demanded for the secondproduct. These two products are:a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices92. Find the Elasticity of Demand when the price is 4 for the demand function a) ! b) 2 c) -2 d) -5/8 e) - !93. The linearization of near is:a) b) c) d) e) 94. Evaluate a) b) c) d) e)

95. If f (x) = 8x - 10x + 3x - 1, find the value of f "(x) (the second derivative) at x = 2.4 2a) 219 b) 364 c) 93 d) 376 e) 38496. If e = 243 and e = 32 then x ya) 6/5 b) 30 c) 864 d) 266 e) 43297. An apartment complex has 500 units available. At a monthly rent of $1000 all the apartments are rented. For each $10 increase in rent, 2 apartments become vacant. If x is the number of $10 increases in rent, themonthly Revenue will be:a) b) c) d) e) 98. The demand equations for two related products are given, where is the per unit price when x units aredemanded for the first product and is the per unit price when y units are demanded for the secondproduct. These two products are:a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices99. The graph of the function has as its vertical asymptote(s)a) x = -1 only b) x = 1 only c) x = 1 and x = -1 d) No vertical asymptotes e) y = 1 only100. If R = -x + 3x - 2 is the revenue function, where x is the amount spent on advertising, then the point of3 2diminishing returns (point of inflection) occurs whena) x = 1 b) x = -1 c) x = 6 d) x = -2 e) Not enough information is given to decide101. The critical numbers for y = x - 2x are4 2a) x = 0 only b) x = -1, 0, 1 c) x = -1, 1 only d) x = -2 only e) x = -4, 4 only102. The graph of f (x) = -x - 2x + 3 resembles2a) b) c) d) e)

103. A farmer wishes to construct 3 adjacent enclosures alongside a river asshown. Each enclosure is x feet wide and y feet long. No fence is requiredalong the river, so each enclosure is fenced along 3 sides. The total enclosurearea of all 3 enclosures combined is to be 900 square feet. What is the leastamount of fence required?a) 20 feet b) 120!2 feet c)120 feet d) 70!3 feet e) 360 feet104. For a production level of x units of a commodity, the cost function in dollars is C = 200x + 4100. Thedemand equation is p = 300 - 0.05x. What price p will maximize the profit?a) $100 b) $250 c) $900 d) $1500 e) $6000105. If y = x + x, find the value of the differential dy corresponding to a change in x of dx = 0.12when x = 2.a) 0.11 b) 0.5 c) 0.51 d) 1.2 e) 6.5106. Find the derivative of y = ln (2x + 3)7

a) 14 ln (2x + 3) b) 14 ln (2x + 3) c) d) e) 6 7

107. Find the equation of the line tangent to the graph of y = xe at the point where x = 2.2x - 4a) y = e x + 2 b) y = 2x - 2 c) y = 3x - 4 d) y = 4x - 6 e) y = 5x - 82108. Find the critical points of the function f (x, y) = xy - x.2a) (1, 1) and (0, 0) b) (0, 3) and (0, 7) c) (1, 0) d) (0, 1) and (0, -1) e) (3,7)109. If , find f(1, -6, 5).a) 4 b) c) 9 d) 25 e) 7

x110. If f (x, y) = e y + x , find the first partial derivative f .x 3 3a) e y + x b) 3e y c) 3x d) y + 1 e) e y + 3xx 3 3 x 2 2 3 x 3 2111. Suppose you were asked to use the method of Lagrange multipliers to maximize the function f(x, y) =xy subject to the constraint x + y - 10 = 0. The first step would be to form the new function L(x, y, ") =a) xy" b) x + 7 - 10" c) xy + "(x + y - 10) d) xy(x + y - 10) e) (x + y - 10) + "(xy)

xy112. If f (x, y) = 3x + 2x y - 7y , find the second partial derivative f .2 2 2a) 6x + 4xy b) 4x c) 6 + 4y d) 4xy - 14y e)

113. The function f (x, y) = x + 3xy - y + 10 has a critical point at (0,0). What kind of critical point is it?2 2a) Relative maximum b) Relative minimum c) Saddle pointd) Neither a maximum, minimum or saddle point e) No conclusion is possible114.When expressed as a single logarithm, a) b) c) d) e) none of these115. For the function , find dya) b) c) d) e) 116. For the function , find a) b) c) d) e) 117. For the function , find a) b) c) d) e) 118. Find the Elasticity of Demand when the price is 4 for the demand function a) -1/4 b) -5/4 c) -4 d) 1/4 e) 4119. Find the derivative of f(x) = 3e .-5x + 7a) -15e b) -15e c) 3e d) -15e + 3e e) 3e + 3e-5 -5x + 7 -5 -5x + 7 -5x + 7 -5 -5x + 7120. The position of an object at any time t is given by . Find the velocity whena) 18 b) -12 c) -16 d) 22 e) 5/4121. The position of an object at any time t is given by . Find the accelerationwhen a) 18 b) -12 c) -16 d) 22 e) 5/4122. The position of an object at any time t is given by . Find the time when thevelocity is 0.a) 18 b) -12 c) -16 d) 22 e) 5/4

123. Find the values of x for which the profit P(x) = x - 12x + 45x -13 is a maximum on [0, 5].3 2a) x = 0 b) x = 3 c) x = 5 d) x = 3 and 5 e) x = 0 and 5124. The acceleration of an object is given by the equation . Determine its velocityfunction v(t) if v(0) = 6.a) b) c) d) e) 125. Solve: a) b) c) d) e) 126. If x + y = 24 find 2 2

a) b) c) d) -2 e) -1127. If the demand equation is given by , then the marginal revenue is 0 whena) x = 9 b) x = 2 c) x = 6 d) x = 3 e) x = 5128. If ln x = 3 and ln y = 2, then a) 39 b) 13/3 c) 5/3 d) 18 e) 12129. For what value(s) of x is the derivative of equal to zero or undefined?a) x = 0 only b) x = 2 only c) x = -2 only d) x = -2, 2 only e) x = -2, 0, 2130. What is the absolute maximum value of f (x) = x - 75x on the closed interval [0, 4]?3a) 0 b) -250 c) 250 d) 236 e) 75

131. a) b) c) d) e) none of these132. The value of the function f (x, y) = 3x - 7xy + y at the point (2, t) is2a) 2 b) t c) -2t + t d) 12 - 13t e) -t

x133. If f (x, y) = x e + 3y + x y, then f equals2 2y 2 3a) e + x b) 4xe + 6xy c) 2e + 3xy d) 2e + 3 e) 2xe + 3x y2y 2 y 2y 2y 2y 2

xy134. Given , find f .a) b) c) d) e) 135. If you were to use the method of Lagrange multipliers to minimize f (x, y) = x + y subject to the2 2

"constraint -2x - 4y + 5 = 0, then L =a) 2x + 2" b) 2y + 4" c) x + y d) -2x - 4y + 5 e) x + y - "2 2 2 2136. The function f(x, y) = 2x + y - 3x - 12x - 3y has a critical point at (2, 1). This critical point is3 3 2a) a saddle point b) a relative maximum c) a relative minimumd) a removable discontinuity e) None of these137. Find the Elasticity of demand when the price is 10 for the demand function a) - 2/3 b) - 3/2 c) 2/3 d) - 6/5 e) 3/2138. Solve .a) b) c) d) e) 139. The linearization of near is:a) b) c) d) e)

140. The position of an object at any time t is given by . Find the accelerationwhen a) -30 b) 3 c) 18 d) 24 e) !141. The position of an object at any time t is given by . Find the time when thevelocity is 0.a) -30 b) 3 c) 18 d) 24 e) !142. If y = x ln x (with x > 0), then dy/dx equals2a) x - 2x ln x b) 2 c) 2x ln x d) x + 2x ln x e) 2x - ln x143. Find the maximum profit for the profit function P(x) = -2x + 10x - 3.2a) 10 b) 19/2 c) (5 + !19)/2 d) 7/4 e) 67/8144. Given f (x) = x - 3x + 3x,3 2a) f (x) has a relative maximum at x = 1b) f (x) has a relative minimum at x = 1c) x = 1 is not a critical number of f (x)d) f (x) has no inflection pointse) f (x) has an inflection point at x = 1145. The absolute maximum of the function f(x) = x - 9 on the interval [-3, 3] is2a) 0 b) 3 c) -3 d) 9 e) -9146. The graph of a) is concave up for x > 0. b) is concave down for x > 0. c) is decreasing for x > 0.d) is negative for x > 0. e) has y = 1 as a horizontal asymptote147. On the interval 0 < x < 2, for the graph shown,a) f (x)> 0, f ! (x) < 0 and f "(x) > 0b) f (x) < 0, f ! (x) < 0 and f "(x) > 0c) f (x) < 0, f ! (x) < 0 and f "(x) > 0d) f (x)< 0, f ! (x) < 0 and f "(x) > 0e) f (x) > 0, f ! (x) < 0 and f "(x) > 0

148. The demand function and cost function for x units of a product are and C = 0.65x + 400.Find the marginal profit when x = 100.a) $2.35 per unit b) $4.58 per unit c) $193.50 per unit d) $187.35 per unit e) $3.65 per unit149. Find the number of units that will minimize the average cost function if the total cost function isa) 6 b) 10 c) 20 d) 40 e) 80150. If , then f "(-5) isa) 3 b) -1/6 c) -1/108 d) 1/108 e) undefined151. Determine the x coordinate of the point(s), if any, at which the graph of has a horizontal tangent.a) x = -1, 9 b) None c) x = 11.778, 2.3601 d) x = 0, 1, 9 e) x = 11152. C = 0.5x + 15x + 5000 represents the total cost of producing x units. The production level that2minimizes the average cost isa) x = 5000 b) x = 15 c) x = -100 d) x = 10,000 e) x = 100153 and 154. For the function f (x) = x - 9x + 15x3 2

153. Find the point(s) of inflection (x, y).a) (1, 7) only b) (5, -25) only c) (1, 7) and (5, -25) d) (3, -9) e) (3, -12)154. Find the absolute extrema on the closed interval [0, 3].a) minimum is -9 and maximum is 0 b) minimum is -9 and maximum is 7c) minimum is -25 and maximum is 7 d) minimum is -25 and maximum is 0e) minimum is 0 and maximum is 7155 and 156. For the function f (x) = 4x - x - 22155. Find all critical points (x, y).a) (2, 2) b) (2, -2) c) (2, 0) d) (-2, -14) e) (0, -2)

156. Find the absolute extrema on the closed interval [0, 5] (i.e 0 & x & 5).a) minimum is -2 and maximum is 2 b) minimum is -7 and maximum is -2c) minimum is -7 and maximum is 2 d) minimum is -7 and maximum is 5e) minimum is -10 and maximum is 10157. Find the derivative of y = 3 ln(5x - 9).a) b) c) 3 ln(5) d) e) 158. The graph of y = f(x) appears on the right. Estimatethe points at which the absolute minimum and theabsolute maximum occur on the interval [0, 3],that is, 0 & x & 3.a) absolute minimum: (0, 0); absolute maximum: (3, 2)b) absolute minimum: (0, 0); absolute maximum: (2, 4)c) absolute minimum: (-2, -4); absolute maximum: (2, 4)d) absolute minimum: (4, -5); absolute maximum: (-4, 5)e) absolute minimum: (3, 2); absolute maximum: (2, 4)159. On the interval 0 < x < 1, for the graph shown,a) f ! (x) > 0 and f "(x)> 0b) f ! (x) > 0 and f "(x)< 0c) f ! (x) < 0 and f "(x)> 0d) f ! (x) < 0 and f "(x)< 0e) None of the above160. Find the open intervals on which is increasing.a) (-%, -1) or (1, %) (i.e. x < -1 or x > 1) b) (1, %) only (i.e x > 1 only)c) (-1, 1) (i.e. -1 < x < 1) d) (- %, -1) only (i.e. x < -1 only)e) (-%, %) (i.e. all real x)161. For the function , find dya) b) c) d) e) 162. A square is measured and each side is found to be 5 inches with a possible error of at most .03 inches. Use DIFFERENTIALS to find the approximate error in computing the area of the square.a) .6 b) .06 c) .006 d) .3 e) .03

163. The demand equations for two related products are given, where is the per unit price when x unitsare demanded for the first product and is the per unit price when y units are demanded for the secondproduct. These two products are:a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices

xy164. Given f(x, y) = 2x y find f3 5a) 10x y + 6x y b) 30x y c) 0 d) 6x + 10y e) 10x y y! + 6x y3 4 2 5 2 4 2 4 3 4 2 5165. Select the correct mathematical formulation of the following problem.A rectangular area must be enclosed as shown. The sides labeled xcost $20 per foot. The sides labeled y cost $10 per foot. If at most$300 can be spent, what should x and y be to produce the largest area?a) Maximize xy if 40x + 20y = 300b) Maximize xy if 20x + 10y = 300c) Maximize xy if 2x + 2y = 300d) Maximize 2x + 2y if xy = 300e) Maximize 2x + 2y if 200xy = 300166. For the Profit function with Domain the MAXIMUM PROFIT is:a) 150 b) 200 c) 250 d) 300 e) 350167. If f (12) = 200 and f !(12) = -6, estimate the value of f (14) for the function y = f (x).a) 194 b) 188 c) 206 d) 214 e) 8168. Select the correct mathematical formulation of the following problem.A farmer wishes to construct 4 adjacent fields alongside a river as shown. Each field is x feet wideand y feet long. No fence is required along the river, so each field is fenced along 3 sides. The totalarea enclosed by all 4 fields combined is to be 800 square feet. What is the least amount of fencerequired?(a) Minimize xy if 4x + 5y = 800(b) Minimize xy if 4x + 5y = 200(c) Minimize 4x + 5y if xy = 800(d) Minimize 4x + 5y if xy = 200(e) Minimize 4(x + y) if xy = 800

SECTION CC1. To three decimal places, a) 1.573 b) 2.262 c) 3.145 d) 4.879 e) 5.741C2. Find the relative maximum value for the function f (x) = x - 15x3a) -!15 = -3.87 b) !15 = 3.87 c) -10!5 = -22.36 d) 10!5 = 22.36 e) 12C3. The profit from manufacturing x items is given by P(x) = -0.5x + 46x - 10. Find on [20, 21].2a) $25.00 b) $25.50 c) $26.00 d) $710.00 e) $735.50C4. Find the absolute extrema of on the closed interval [0, 2].a) (0, 1) and (1, 3) b) (2, 4/5) and (-1, -1) c) (2, 4/5) and (0, 0)d) (1, 1) and (-1, -1) e) (1, 1) and (0, 0)C5. The absolute maximum value of the function f (x) = x - 6x + 15 on the closed interval [0, 3] is3 2a) -7 b) -17 c) 4 d) -12 e) 15C6. Given ln(2x - 3) = 4 , then to three decimal places, x =-xa) 1.54 b) 2.031 c) 3 .057 d) 4.201 e) 5.057C7. A job offer consists of a $27,000 starting salary with a 4% increase each year. To the nearest dollar,what will the salary be in 7 years?a) $34,164 (b) $35,530 (c) $36,951 (d) $38,429 (e) $284,616C8. What is the relative maximum of the function ?a) (-1, 4) b) (1, 0) c) (0, 0) d) (1, 2) e) (2, 1.6)C9. Find the relative maximum of on the closed interval [0, 6].a) 6 b) 3 c) d) -3 e) 0C10. The maximum value of f (x) = xe is-xa) e ' 0.368 b) e ' 2.718 c) 0.5 d) -e ' -2.718 e) 0-1

C11. The half-life of a radioactive substance is 1500 years. How much of a 128 lbs. sample of this substancewill remain after 9000 years? a) 2 lbs b) 3 lbs c) 4lbs d) 5 lbs e) 64 lbs C12. Given find f "(1).a) 84 b) 0 c) 529 d) 84/529 e) 1C13. The “best fit” exponential function passing through the points (2, 1), (3, 4) and (4, 7), rounded to threedecimal places is a) y = 2.646(0.164) b) y = 0.164(2.646) c) y = 1.817(1.122) x x xd) 1.122(1.817) e) y = ex x - 2C14. The “best fit” logarithmic function passing through the points (2, 1), (3, 4) and (4, 7), rounded to threedecimal places is a) -5.083 + 8.574 ln x b) 8.574 -5.083 ln x c) 3.623 + 2.57 ln x d) 2.57 + 3.623 ln x e) none of theseC15. Find the inflection point for a) (4, 8) b) (1.678, 4) c) (0, 1.9048) d) (1.5546, 3.8290) e) (10.3782, 7.9808)C16. For which value of x does the graph of the function f(x) = e ln x have a horizontal tangent? Round-xyour answer to the nearest hundredth.a) 1 b) 1.76 c) 2.81 d) 3.93 e) NeverC17. = a) % b) 3/4 c) 2/3 d) 7 e) 26/3C18. The area of the region bounded by f(x) = x , the x-axis and the line x = 1 and x = 5 is approximated by2n = 4 rectangles. If the right endpoint is used for each of these rectangles, then the approximate areaobtained isa)30 b) 41 c) 42 d) 54 e) 64C19. One critical number of the function y = x - 2x - 3x + 1 is4 2a) 1.263 b) -0.148 c) 0.752 d) 1 e) -1.172

C20. For the function , find a) b) c) d) e) C21. For the function , find #a) b) c) d) e) C22. The demand function for a product is p = 100 - 0.1x ln(x) where x is the number of units of theproduct sold and p is the price in dollars. Find the value of x for which the marginal revenue is one dollarper unit.a) 97.45 b) 0.000017 c) 188.9 d) 190.5 only e) 0.01 and 190.5C23. The function has exactly two points of inflection. The value of x for one of thesepoints of inflection isa) 8 b) -16 c) -0.8 d) 0 e) C24. Find the critical numbers for f(x) = e ln(x).10xa) 0.894 only b) 0.028 and 0.894 c) 0.789 only d) 0.789 and 0.894 e) There are no critical numbersC25. X = -1/3 '-0.333 is a critical number for . The critical point (-0.333, -0.497) isa) a relative minimum b) a relative maximum c) neither a relative maximum norminimumd) a point of inflection e) a saddle pointC26 and C27. Given

C26. Find the critical numbers.a) 0.482 only b) 0.482 and 13.94 c) 1 d) 0.824 only e) 0.824 and 15.494

C27. The graphs shown below display different graphical windows. More than one may actually be graphs the function f(x) shown above. Pick the graph that most accurately portrays the function (shows all its relative extrema, asymptotes, etc.).a) b) c)

d) e)

C28. If x dollars are spent on advertising, then the revenue in dollars is given by where 0 & x & 5000. How much is spent on advertising at the point of diminishing returns (point of inflection)?a) $10,576 b) $100,000 c) $50,086.60 d) x = $3000e) There is no point of diminishing returnsC29 and C30. The weekly profit that results from selling x units of a commodity weekly is given byP = 50x - 0.003x - 5000 dollars.2C29. Find the actual weekly change in profit that results from increasing production from 5000units weekly to 5015 units weekly.a) $20.00 b) $300.00 c) $299.33 d) $298.00 e) $8333.33C30. Use the marginal profit function to estimate the change in profit that results from increasingproduction from 5000 units weekly to 5015 units weekly.a) $20.00 b) $300.00 c) $299.33 d) $298.00 e) $8333.33C31. Find the equation of the straight line that is tangent to f(x) = 2 at the point (5, 32).xa) y = 2x + 22 b) y = 22.1807x - 78.9035 c) y = 15.3745x - 44.8725d) y = 32x - 128 e) y = 21.5678x - 75.839

C32. Evaluate a) 1 b) 0 c) % d) undefined e) eC33. The present value of $5,000 per year flowing uniformly over a 8 year period if it earns 3% interestcompounded continuously is, to the nearest dollara) $454 b) $35,562 c) $45.208 d) $55,116 e) $73,205

C34. If then to two decimal places, f ' (1) = a) -3.67 b) 2.69 c) 12.86 d) -4.62 e) none of theseC35. Given the total cost function , find the value of x for which theaverage cost is a minimum.(a) -533.333 (b) 533.333 (c) 378.594 (d) -378.594 (e) 500

ANSWERS1. d 2. b3. c4. d5. c6. d7. e8. d9. a10. c11. e12. e13. d 14. a15. e16. b17. e18. d19. d20. b21. c22. c23. b24. c25. d26. d27. b28. b29. a30. d31. a

32. c33. c34. a35. c36. c37. c38. b39. c40. a41. c42. b43. c44. c45. c46. c47. b48. b49. d50. c51. e52. a53. d54. b55. d56. a57. c58. c59. d60. d61. c62. a

63. c64. b65. b66. e67. e68. d69. e70. b71. b72. a73. a74. d75. c76. e77. c78. a79. d80. b81. b82. e83. c84. d85. e86. b87. d88. a89. c90. b91. a92. e93. c

94. a95. b96. e97. c98. e99. d100. a101. b102. e103. c104. b105. b106. d107. e108. d109. a110. e111. c112. b113. c114. c115. c116. a117. e118. a119. b120. b121. c122. e123. b124. e

125. d 126. c127. c128. b129. e130. a131. d132. d133. e134. e135. d136. c137. b138. a139. a 140. d141. b142. d143. b144. e145. a146. b147. a148. a149. d150. c151. d152. e153. d154. b155. a

156. c157. d158. b159. b160. a161. c162. d163. b164. b165. a166. b167. b168. d

C1. bC2. dC3. bC4. eC5. eC6. bC7. bC8. dC9. bC10. aC11. aC12. dC13. bC14. aC15. bC16. bC17. eC18. dC19. aC20. aC21. eC22. aC23. eC24. bC25. aC26. eC27. cC28. dC29. cC30. bC31. b

C32. aC33. bC34. dC35. c