BARUCH COLLEGE MATH 05 SPRING 006 MANUAL FOR THE UNIFORM FINAL EXAMINATION Joseph Collison, Warren Gordon, Walter Wang, April Allen Materowski The final examination for Math 05 will consist of two parts. Part I: Part II: This part will consist of 5 questions similar to the questions that appear in Problem Section A. No calculator will be allowed on this part. 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 questions that appear in Problem Section C (Calculator allowed). The remaining 4 questions will be similar to questions that 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 similar to the problems in Sections A and C. If such problems appear, they will be 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 05 Textbook Sections Corresponding to Sample Uniform Final Exam Questions Spring 006 Textbook: Applied Calculus, Gordon, Wang, Materowski, Baruch College, CUNY Section Problems 1.1 (Extrema) 1, 0, 59, 8, 86, 99, 101, 19, 130, 151, 154, 155, 156, 158, C4, C5, C19 st 1. (1 Der. Test) 1,, 60, 145, 160 1.3 (Concavity) 11, 16, 17, 89, 90, 95, 100, 10, 10, 11, 1, 16, 140, 141, 144, 146, 147, 150, 153, 159, C, C8, C9, C1 1.4 (Geom. Apps.) 31, 61, 65, 103, 165, 168 1.5 (Business Apps.) 3, 6, 9, 97, 104, 118, 13, 17, 137, 143, 148, 149, 15, 166, C35 1.6 (Linearization) 9, 5, 93, 105, 115, 116, 117, 139, 161, 16, 167, C3, C0, C1, C9, C30.1 (Inverses) 5, 38, 66. (Exponent. F cns) 8, 19, 50.3 (Number e) 15, 57, 96, C7, C11, C13, C3.4 (Derivative e^x) 1, 10, 6, 7, 41, 63, 64, 107, 119, C10, C3, C5, C6, C7.5 (Logarithms), 6, 4, 43, 51, C1, C14.6 (Log. Props/Der) 7, 36, 39, 40, 44, 47, 48, 53, 67, 68, 85, 106, 114, 18, 131, 14, 157, C6, C15, C16, C4, C31, C34.7 (Applications) C, C8 3.1 (Antiderivatives) (See 3. Problems) 3. (Apps. Antider.) 4, 84, 14, 15, 138 3.3 (Substitution) 3, 49 3.4 (Approx. Area) 18, C6, C18 3.5 (Sigma and Area) 81, 88, C17 3.6 (Definite Integral) 13, 4, 45, 80 3.7 (Subst. Def. Int.) 37, 5, 54, 55, 58, 78, 79, 83, 87, 94 3.8 (Applications) 14, 46, 56, C33 4.1 (f(x,y)) 8, 69, 109, 13 4. (Partial Derivs.) 9, 30, 3, 70, 71, 73, 74, 110, 11, 133, 134, 164 4.3 (Extrema) 33, 34, 7, 76, 77, 108, 113, 136 4.4 (Lagrange Mult.) 35, 111, 135 4.5 (Econ. Apps) 75, 91, 98, 163
TI-89 FACTS FOR THE UNIFORM FINAL EXAMINATION The following information represents the minimal calculator usage students should be familiar with when they 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 whose syntax is limit(expression, variable, value). This function is found on the Home screen by using the F3 Calc key. The calculus menu that appears when F3 is pressed is shown in Figure 1. The limit function is selected by either pressing 3 or using the down arrow cursor to highlight choice 3:limit and then pressing ENTER. Figure 1 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. Figure Example : Find Solution: In order to use the calculator for this problem it is best to think of x as a single letter such as t. That is, Now enter on the command line the following: Figure 3 limit((1/(x+t)-1/x)/t,t,0) The result is -1/x as shown in Figure 3. Note that example could have been done without a calculator if the limit requested was recognized as the definition of the derivative of f(x) = 1/x for which f (x) = -1/x. The syntax for the solve command and for finding the first or second derivative is: solve(equation, variable) d(expression, variable) d(expression, variable, ) (first derivative, f (x), where f (x) = expression) (second derivative, f (x), where f (x) = expression)
The solve command is obtained by pressing the F (Algebra menu) key in the Home Screen as shown in Figure 4 and then pressing ENTER to select choice 1:solve. The differentiation operator d is choice 1 on the Home screen calculus menu shown in Figure 1. It can be more easily accessed by pressing the yellow nd key and then 8 (the d appears above the 8 in yellow). The purple D appearing above the comma key cannot be used for this purpose. Figure 4 3 If an answer such as e or 15/36 appears when a decimal answer is desired, the command can be repeated with 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 of the 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 by using the calculator command d(x*(10-ln(x)),x) obtained by pressing the keys nd nd 8 ( x * ( 1 0 - x x ) ), x ) ENTER as 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. Figure 5 Press F (Algebra) ENTER as shown in Figure 4. Then complete the command line as solve(9-ln(x)=0,x) 9 Pressing ENTER produces the result e. Since a decimal answer is desired, repeat the command by first pressing the green diamond key before pressing ENTER. The final answer is 8103.08 as shown in Figure 6 and can be rounded off to 8103. Figure 6 Example 4: The function has exactly one inflection point. Find it. Solution: Recall that an inflection point occurs if h (t) = 0 and the concavity changes sign. So first the second derivative is found with the command d(30/(10+e^(-0.1t+5)),t,) where e^( is obtained by pressing x x (e appears in green over the x key) Figure 7 is the result. Recall that to see the entire result the up cursor arrow button must be pressed and then the right cursor arrow button must be pressed until the rest of the result on the right can be seen as shown in Figure 8. Thus, Figure 7 Move the cursor down to the command line again. The easiest way to solve h (t) = 0 is as follows. First press F (Algebra) and select choice 1 so that Figure 8
the command line now only shows solve( Press the up cursor arrow to highlight the expression for h (t) and press ENTER. The command line now shows the end of solve(above expression Complete the command by entering =0,t)ENTER The 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 = 6.9741. It only remains to find the value of the original function at t = 6.9741. To do so, enter 30/(10+e^(-0.1*6.9741+5)) on the command line to obtain h(6.9741) = 1.5 as shown in Figure 10. Therefore, the inflection point is (6.9741, 1.5). Figure 9 Figure 10 5x Example 5: 0.0559 and 1.788 are the only critical numbers for f (x) = e ln(x/). Determine if 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 relative minimum, if it is negative the point is a relative maximum, and if it is 0 the first 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/),x,) x=0.0559 Since f (0.0559) = -304.911 is negative as shown in Figure 11, the critical point is a relative maximum. Solution : A graph that clearly showed the point (0.0559, -4.73) would reveal what was true for the critical point. Press F1 for the y= screen. Enter the function y1=e^(5x)*ln(x/) as shown in Figure 1. Figure 11 Figure 1 If F (Zoom) and choice 6:ZoomStd are selected (so that the x and y values go from -10 to 10), the result is Figure 13. Notice that this reveals nothing about the 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 the one 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 y between -6 and -4. So press F (Window) and enter the following values. xmin=-1 xmax=1 xscl=1 ymin=-6 ymax=-4 yscl=1 as shown in Figure 14. Figure 13 Figure 14
Then press F3 (Graph). Figure 15 is the result. Now press F3 (Trace) and then press the right cursor arrow a few times and observe values of x and y for each point on the graph shown. Clearly the high point is the critical point and therefore the critical point is a relative maximum. Figure 15 Students should also be familiar with using F5 (Math) in the graph window to determine the exact location of a relative maximum or relative minimum displayed on the graph. For the graph shown in Figure 15 press F5 (Math) to obtain the menu shown in Figure 16. Select choice 4:Maximum. Figure 16 In response the Lower Bound? request, just use the cursor movement arrows to 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 of the maximum and press ENTER. Figure 17 is the result and indicates that (0.05591, -4.7309) is a relative maximum. Figure 17 4 3 Example 6: Given f (x) = x + 14x - 4x - 16x + 135 i) Find the critical numbers. ii) Find the critical points iii) 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.)? d) e) Solution: i) The critical numbers are the solutions to 3 0 = f (x) = 4x + 4x - 48x - 16. Enter the command solve(4x^3+4x^-48x-16=0,x) into the calculator as shown in the screen on the right. The critical numbers are x = -11.3145, -1.3106 or.1479
ii) For the critical points the value of y is needed for each of the critical numbers found. Enter the command x^4+14x^3-4x^-16x+135 x=-11.3145 into the calculator. Repeat this for the other critical numbers. (Pressing the right cursor arrow removes the command line highlight and positions the cursor at the end of the line. Then the backspace key can be used to eliminate the -11.3145. Then enter the next critical number.) The screen shown indicates the critical points are (-11.3145, -5401.64), (-1.3106, 30.345) and (.1479, -86.3943). iii) A graphing screen should be used that includes the critical points found. The x values shown should include all values between -1 and 3. The y values shown should include all values between -540 and 31. A reasonable window to choose would thus be xmin=-15 xmax=5 xscl=1 ymin=-5500 ymax=500 yscl=500 The graph shown on the right is the result. So the answer is d. Integration and the TI-89 Example 7: Evaluate In the HOME screen, Integration is under F3(calc)->: 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 extra arguments: F3(calc)->: integration (The last arguments are the lower limit, followed by the upper limit.) Figure 19 shows the entry on the TI-89. Figure 19
SECTION A 1. Evaluate d) e) none of these. Evaluate a) 1 b) 3/ c) d) 5/ e) 9 3. Evaluate d) e) 4. The marginal cost, in dollars is given by the equation C ' (x) = x + 00. If the fixed overhead cost is $000, then the total cost of producing 10 items is a) $000 b) $100 c) $3500 d) $4100 e) $4500 5. Suppose y = f(x) and y = g(x) are inverse functions with f(3) = 9 and is a) 7 b) -7 c) 1/7 d) -1/7 e) does not exist 6. Evaluate a) b) - c) 8 d) -8 e) 7/4 7. Evaluate d) e) 8. A population of bacteria is decaying continuously at a rate of 7% every hour. If there are 3. million bacteria present at 9AM, which of the following gives the population of bacteria t hours after 9AM? d) e) 9. Given y = 3x, find the value of the differential dy if x changes from 4 to 6. a) 48 b) 4 c) 60 d) 1 e) 30 10. The equation of the tangent line to the curve defined by at the point at which x = 1 is a) y = 4x + 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/ c) 0 d) 5/6 e) /3 4 1. Find all of the points at which the graph of f (x) = x - 4x + 5 has horizontal tangent lines. a) (,0) only b) (1,0) only c) (1,) and (-1,10) d) (1,0) and (,13) e) (1,) only 13. Evaluate. a) 8 b) 3 c) 4 d) 16 e) 10 14. The consumer s surplus for the demand and supply functions x + y = 4 and -x + y = is a) ½ b) 3/ c) 5/ d) 7/ e) 4 15. Which of the following best represents the graph of? (All x- and y- scales are 1.) d) e) 6 4 16. (0, 5) (i.e. x = 0 and y = 5) is the only critical point of f (x) = x + 3x + 5. You do not have to verify this. 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 point d) (0, 5) is not a relative extremum e) no conclusion is possible 3 17. Solve f (x) = 0 for the function f (x) = 4x - 9x + 1. a) x = 4 b) x = 1 c) x = -9 d) x = 1 e) x = 0 x 18. The area of the region bounded by f(x) = 3, the lines x = 1 and x = and the x-axis is to be approximated by 10 rectangles using the right endpoint of each rectangle. The sum is d) e) none of these
19. When writing the formula for an exponential function passing through the points and (, 36) using the form, what would be the base, b? a) 4/9 b) 81 c) 1/81 d) 3 e) 1/3 4 0. Find the critical numbers for g(x) = x - x. a) 0, ± b) 0, ±1 c) ± (3)/3 d) only 0, -1 e) only 0 1. If the derivative of g(x) is g (x) = (x - 1) (x - ) then the function is a) decreasing on (-, ) b) decreasing on (-, 1) and increasing on (1, ) c) decreasing on (-, ) and increasing on (, ) d) increasing on (-, 1) and decreasing on (1, ) e) decreasing on (-, 1), increasing on (1, ) and decreasing on (, ) 3. Find the absolute maximum of f (x) = 1x - x on the closed interval [-4, 1]. a) -4 b) - c) 16 d) 11 e) 0 3. Find the number of units to be produced in order to maximize revenue if the demand function is given by p = 600x - 0.0x. a) 15,000 b) 0,000 c) 5,000 d) 30,000 e) 35,000 4. Evaluate a) 7/5 b) c) -5/7 d) e) none of these 5. Let the profit function for a particular item be P(x) = -10x + 160x - 100. Use differentials to approximate the increase in profit when production is increased from 5 to 6. a) 30 b) 40 c) 55 d) 60 e) 70 6. If, then f (x) = x - 3 x - 3 x - 3 a) b) e c) e (x - 3) d) (x - 3) e) e (x - 3x) y 7. Find dy/dx if xe + 1 = xy a) 0 b) c) d) e) 3 8. For f (x, y) = 3x - y find f (3, -4). a) 37 b) 91 c) 18 d) 6x e) 6x - 3y
3 9. For f (x, y) = 3x - y find f x a) 6x b) 6x - 3x c) -6y d) 0 e) cannot be determined 3 xy 30. For f (x, y) = 4x y + e find f x. xy 3 xy xy 3 xy xy a) 4xy + e b) 8xy + e c) 1x y + xe d) 8xy + ye e) 1x y + ye 31. The sum of a number and twice another number is 10. Find the largest possible product. (Notice that your answer is not one or both of the numbers, it is the product.) a) 1.5 b) 8 c) 1 d) 1.6 e) 13 3 xy 3. For f (x, y) = 4x y + e find f y. xy 3 xy xy 3 xy xy a) 4xy + e b) 8xy + e c) 1x y + xe d) 8xy + ye e) 1x y + ye 33. For f (x, y) = x + y + 4x - 6y + 10, the critical point is a) (-3, 4) b) (3, -) c) (-, 3) d) (, -3) e) (4, -3) 34. Suppose a function of two variables has a critical point. If, at that critical point, D = fxxf yy - (f xy) > 0 and f < 0. Then that critical point is xx a) a relative maximum b) a relative minimum c) a saddle point d) an absolute maximum e) cannot be determined 35. To find the maximum of f (x, y) = x + 4xy - 8y subject to the constraint x + 4y = 1 by the method of Lagrange multipliers, one must solve the equations a) 4x + 16y + = 0 b) 4x + 16y + = 0 c) 4x + 4y + = 0 d) 4x + 16y + = 0 e) 4x + 4y + = 0 4x + 4y + 4 = 0 4x + 4y + 4 = 0 4x - 16y + 4 = 0 4x + 4y + 4 = 0 4x - 16y + 4 = 0 x + 4y - 1 = 0 x + 4y + 1 = 0 x + 4y - 1 = 0 x + 4y + 1 = 0 -x - 4y - 1 = 0 36. a) ln b) c) d) ln 7 e) 37. Evaluate a) 4/5 b) -ln5 c) ln5 d) 4/5 e) 1 38.. Find. a) 11 b) 1/11 c) 411 d) 1/411 e) -1/6
39. Evaluate a) log 10 + log 0 b) c) 1 + log d) e) none of these 40. a) 5 b) 5/ c) /5 d) -5 e) 5/4 41. = 9x 10x 10x a) 10e b) e c) 10e d) e) none of these 4. 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 -3-0 1 B y =f(x) a) 1 b) c) 3 d) 4 e) none of these 43. Evaluate a) 15 b) 1e c) 15 d) 3/5 e) 43 44. Evaluate d) e) 45. The area of the region bounded by f(x) = 3x, g(x) = 4 - x and the x-axis from x = ½ to x = 3 is given by d) e) none of these
46. A probability density function on the interval [0, 4] is given by the equation f(x) = then Pr( x 4) = a) 1/6 b) 1/5 c) 1/4 d) 1/3 e) ½ x kx 47. If 3 = e then k = ln 3 - ln d) e) 48. Given log( + x) + log(x - 3) = log14, then x = a) -4 b) 5 c) 4 d) - 5 e) -4 and 5 49. d) e) 50. An exponential graph containing the point (, 5) and (4, 1) has the equation d) e) none of these 51. Given the function, which of the following would be the graph of? (All x- and y-scales are 1.) a) ` b) c) d) e) 5. a) 65/ 76 b) 64/5 c) 3 d) 4/3 e) ½
53. a) 0 b) 5/4 c) 3 d) 1 e) 1/5 54. Evaluate a) -80 b) -5 c) 7 d) -1/16 e) -1/4 55. Evaluate a) b) ½ c) 8 d) 4 e) 1 56. The producer s surplus for the demand and supply functions x + y = 4 and -x + y = is a) ½ b) 3/ c) 5/ d) 7/ e) 3 57. Which of the following represents the amount in an account t years after investing $3000 at a 6% annual rate compounded monthly? 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) 104/3 c) d) e) 59. Find all of the critical numbers for. a) 0 b) -3, 3 c) 3 d) -3, 0, 3 e) -3 60. Find the open intervals on which is decreasing. a) (-, -) and (, ) b) (0, ) c) (-, ) d) (-, 0) e) (-, )
61. Of all numbers (positive, negative and zero) whose difference is 4, find the two that have the minimum product. One of the numbers is: a) 4 b) -3 c) - d) 1 e) -1 6. Find the maximum profit for the profit function. a) 18 b) 3 c) 1 d) 0 e) 1-3x 63. Find the derivative of f (x) = 4e. 3x -3x -3x -3-3x a) -1e b) 1e c) -1e d) 4e e) 4e 64. Find the derivative of f (x) = x 4 e x 3 x 4 x 3 x 3 x 4 x 4 x a) 4x e b) xe + 4x e c) 4x + e d) x + e e) 1x e 65 and 66. The height in feet above the ground of a ball thrown upwards from the top of a building is given by s = -16t + 160t + 00, where t is the time in seconds 65. What is the maximum height (in feet) of the ball? a) 5 b) 600 c) 1400 d) 00 e) 160 66. What is? a) 3 ft/sec b) 3 sec c) -864 ft/sec d) 4 ft/sec e) 4 sec 67. Find the derivative of f (x) = -6 ln x a) 6/x b) -6/x c) -6/(ln x) d) 6/x e) -6/x 68. Find the derivative of. d) e) 69. Suppose that f (x, y) = 30 - (y + 4x). Find f (3, -). a) 9 b) 38 c) 16 d) 13 e) 14
3 70. Find f x for f (x, y) = 5x - 4xy + y a) 10x - 4y + 3y b) 10x - 4y c) -4x + 3y d) 6x - 4y + 3y e) 3y 71 and 7. Given f (x, y)= x - xy + 4y - 7: 71. Find f x (x, y). a) 6 - y b) - y c) 6 - x - y d) - x - y e) -x + 4 7. Find the critical points. a) (4, ) b) (6, 6) c) (3, 3) d) (4, 4) e) (, ) 3 z 73. Find w y at the point (3,, ) if w(x, y, z) = xye. a) 108e b) 1e c) 1 d) 7e e) 48e x 74. Find f xy if f (x, y) = e ln y. x d) e) e ln y 75. The demand function d(p, n) for umbrellas is a function of their price, p, and the number of rainy days per 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. 3 76. Find all critical points (x, y) for f (x, y)= x + 6x + y - 6y - 15y. a) (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) 4 77. Suppose that f (x, y) = x - y - x + y - 7. Use the second partials test to determine the type of extrema for 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. p 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? d) e) 80. Evaluate a) 54 b) 108 c) 0 d) 18 e) 36 81. Evaluate a) 60 b) 630 c) 160 d) 410 e) 100 8. Find any relative extrema of a) y = 0 b) y = 7/5 c) y = -7/3 d) y = 7 e) there is no relative extremum 83. represents the cost in millions of dollars of producing x thousand items. Find the average value of C on [0, 3]. a) $14 million b) $7 million c) $8 million d) $ million e) $1 million 84. The marginal cost (in thousands of dollars) of producing x hundred items is given by. If overhead is $500, find the cost function, C(x). a) C(x) = 504x + Co b) C(x) = 4x - 500 c) d) C(x) = 4x + 500 e) C(x) = 496x 85. a) 8 b) c) 48 d) 18 e) 1 3 86. Find the critical numbers for f (x) = x + 3x + 1. a) - only b) 0 and - c) 0 only d) 3x + 6x e) There are no critical numbers.
87. Evaluate. a) 4 b) c) d) e) 88. Evaluate. a) 3/ b) ½ 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 extrema 3 90. Given y = 1x - x, find the critical points. Then determine whether they are relative extrema (the second 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 are demanded for the first product and is the per unit price when y units are demanded for the second product. These two products are: a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices 9. Find the Elasticity of Demand when the price is 4 for the demand function a) ½ b) c) - d) -5/8 e) - ½ 93. The linearization of near is: d) e) 94. Evaluate d) e)
4 95. If f (x) = 8x - 10x + 3x - 1, find the value of f (x) (the second derivative) at x =. a) 19 b) 364 c) 93 d) 376 e) 384 x y 96. If e = 43 and e = 3 then a) 6/5 b) 30 c) 864 d) 66 e) 43 97. An apartment complex has 500 units available. At a monthly rent of $1000 all the apartments are rented. For each $10 increase in rent, apartments become vacant. If x is the number of $10 increases in rent, the monthly Revenue will be: d) e) 98. The demand equations for two related products are given, where is the per unit price when x units are demanded for the first product and is the per unit price when y units are demanded for the second product. These two products are: a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices 99. 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 only 3 100. If R = -x + 3x - is the revenue function, where x is the amount spent on advertising, then the point of diminishing returns (point of inflection) occurs when a) x = 1 b) x = -1 c) x = 6 d) x = - e) Not enough information is given to decide 4 101. The critical numbers for y = x - x are a) x = 0 only b) x = -1, 0, 1 c) x = -1, 1 only d) x = - only e) x = -4, 4 only 10. The graph of f (x) = -x - x + 3 resembles d) e)
103. A farmer wishes to construct 3 adjacent enclosures alongside a river as shown. Each enclosure is x feet wide and y feet long. No fence is required along the river, so each enclosure is fenced along 3 sides. The total enclosure area of all 3 enclosures combined is to be 900 square feet. What is the least amount of fence required? a) 0 feet b) 10 feet c)10 feet d) 70 3 feet e) 360 feet 104. For a production level of x units of a commodity, the cost function in dollars is C = 00x + 4100. The demand equation is p = 300-0.05x. What price p will maximize the profit? a) $100 b) $50 c) $900 d) $1500 e) $6000 105. If y = x + x, find the value of the differential dy corresponding to a change in x of dx = 0.1 when x =. a) 0.11 b) 0.5 c) 0.51 d) 1. e) 6.5 106. Find the derivative of y = ln (x + 3) 7 6 7 a) 14 ln (x + 3) b) 14 ln (x + 3) c) d) e) x - 4 107. Find the equation of the line tangent to the graph of y = xe at the point where x =. a) y = ex + b) y = x - c) y = 3x - 4 d) y = 4x - 6 e) y = 5x - 8 108. Find the critical points of the function f (x, y) = xy - x. a) (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) 5 e) 7 x 3 3 110. If f (x, y) = ey + x, find the first partial derivative f. x x 3 3 x 3 x 3 a) ey + x b) 3e y c) 3x d) y + 1 e) ey + 3x 111. 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) 11. If f (x, y) = 3x + x y - 7y, find the second partial derivative f xy. a) 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? a) Relative maximum b) Relative minimum c) Saddle point d) Neither a maximum, minimum or saddle point e) No conclusion is possible 114.When expressed as a single logarithm, d) e) none of these 115. For the function, find dy d) e) 116. For the function, find d) e) 117. For the function, find 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) 4-5x + 7 119. Find the derivative of f(x) = 3e. -5-5x + 7-5 -5x + 7-5x + 7-5 -5x + 7 a) -15e b) -15e c) 3e d) -15e + 3e e) 3e + 3e 10. The position of an object at any time t is given by. Find the velocity when a) 18 b) -1 c) -16 d) e) 5/4 11. The position of an object at any time t is given by. Find the acceleration when a) 18 b) -1 c) -16 d) e) 5/4 1. The position of an object at any time t is given by. Find the time when the velocity is 0. a) 18 b) -1 c) -16 d) e) 5/4
3 13. Find the values of x for which the profit P(x) = x - 1x + 45x -13 is a maximum on [0, 5]. a) x = 0 b) x = 3 c) x = 5 d) x = 3 and 5 e) x = 0 and 5 14. The acceleration of an object is given by the equation. Determine its velocity function v(t) if v(0) = 6. d) e) 15. Solve: d) e) 16. If x + y = 4 find d) - e) -1 17. If the demand equation is given by, then the marginal revenue is 0 when a) x = 9 b) x = c) x = 6 d) x = 3 e) x = 5 18. If ln x = 3 and ln y =, then a) 39 b) 13/3 c) 5/3 d) 18 e) 1 19. For what value(s) of x is the derivative of equal to zero or undefined? a) x = 0 only b) x = only c) x = - only d) x = -, only e) x = -, 0, 3 130. What is the absolute maximum value of f (x) = x - 75x on the closed interval [0, 4]? a) 0 b) -50 c) 50 d) 36 e) 75
131. d) e) none of these 13. The value of the function f (x, y) = 3x - 7xy + y at the point (, t) is a) b) t c) -t + t d) 1-13t e) -t y 3 133. If f (x, y) = x e + 3y + x y, then f x equals y y y y y a) e + x b) 4xe + 6xy c) e + 3xy d) e + 3 e) xe + 3x y 134. Given, find f xy. d) e) 135. If you were to use the method of Lagrange multipliers to minimize f (x, y) = x + y subject to the constraint -x - 4y + 5 = 0, then L = a) x + b) y + 4 c) x + y d) -x - 4y + 5 e) x + y - 3 3 136. The function f(x, y) = x + y - 3x - 1x - 3y has a critical point at (, 1). This critical point is a) a saddle point b) a relative maximum c) a relative minimum d) a removable discontinuity e) None of these 137. Find the Elasticity of demand when the price is 10 for the demand function a) - /3 b) - 3/ c) /3 d) - 6/5 e) 3/ 138. Solve. d) e) 139. The linearization of near is: d) e)
140. The position of an object at any time t is given by. Find the acceleration when a) -30 b) 3 c) 18 d) 4 e) ½ 141. The position of an object at any time t is given by. Find the time when the velocity is 0. a) -30 b) 3 c) 18 d) 4 e) ½ 14. If y = x ln x (with x > 0), then dy/dx equals a) x - x ln x b) c) x ln x d) x + x ln x e) x - ln x 143. Find the maximum profit for the profit function P(x) = -x + 10x - 3. a) 10 b) 19/ c) (5 + 19)/ d) 7/4 e) 67/8 3 144. Given f (x) = x - 3x + 3x, a) f (x) has a relative maximum at x = 1 b) f (x) has a relative minimum at x = 1 c) x = 1 is not a critical number of f (x) d) f (x) has no inflection points e) f (x) has an inflection point at x = 1 145. The absolute maximum of the function f(x) = x - 9 on the interval [-3, 3] is a) 0 b) 3 c) -3 d) 9 e) -9 146. 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 asymptote 147. On the interval 0 < x <, for the graph shown, a) f (x)> 0, f (x) < 0 and f (x) > 0 b) f (x) < 0, f (x) < 0 and f (x) > 0 c) f (x) < 0, f (x) < 0 and f (x) > 0 d) f (x)< 0, f (x) < 0 and f (x) > 0 e) 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) $.35 per unit b) $4.58 per unit c) $193.50 per unit d) $187.35 per unit e) $3.65 per unit 149. Find the number of units that will minimize the average cost function if the total cost function is a) 6 b) 10 c) 0 d) 40 e) 80 150. If, then f (-5) is a) 3 b) -1/6 c) -1/108 d) 1/108 e) undefined 151. 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,.3601 d) x = 0, 1, 9 e) x = 11 15. C = 0.5x + 15x + 5000 represents the total cost of producing x units. The production level that minimizes the average cost is a) x = 5000 b) x = 15 c) x = -100 d) x = 10,000 e) x = 100 3 153 and 154. For the function f (x) = x - 9x + 15x 153. Find the point(s) of inflection (x, y). a) (1, 7) only b) (5, -5) only c) (1, 7) and (5, -5) d) (3, -9) e) (3, -1) 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 7 c) minimum is -5 and maximum is 7 d) minimum is -5 and maximum is 0 e) minimum is 0 and maximum is 7 155 and 156. For the function f (x) = 4x - x - 155. Find all critical points (x, y). a) (, ) b) (, -) c) (, 0) d) (-, -14) e) (0, -)
156. Find the absolute extrema on the closed interval [0, 5] (i.e 0 x 5). a) minimum is - and maximum is b) minimum is -7 and maximum is - c) minimum is -7 and maximum is d) minimum is -7 and maximum is 5 e) minimum is -10 and maximum is 10 157. Find the derivative of y = 3 ln(5x - 9). 3 ln(5) d) e) 158. The graph of y = f(x) appears on the right. Estimate the points at which the absolute minimum and the absolute maximum occur on the interval [0, 3], that is, 0 x 3. a) absolute minimum: (0, 0); absolute maximum: (3, ) b) absolute minimum: (0, 0); absolute maximum: (, 4) c) absolute minimum: (-, -4); absolute maximum: (, 4) d) absolute minimum: (4, -5); absolute maximum: (-4, 5) e) absolute minimum: (3, ); absolute maximum: (, 4) 159. On the interval 0 < x < 1, for the graph shown, a) f (x) > 0 and f (x)> 0 b) f (x) > 0 and f (x)< 0 c) f (x) < 0 and f (x)> 0 d) f (x) < 0 and f (x)< 0 e) None of the above 160. 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 dy d) e) 16. 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 units are demanded for the first product and is the per unit price when y units are demanded for the second product. These two products are: a) complementary b) substitutes c) supplementary d) unitary e) none of the other choices 3 5 164. Given f(x, y) = x y find f xy 3 4 5 4 4 3 4 5 a) 10x y + 6x y b) 30x y c) 0 d) 6x + 10y e) 10x yy + 6x y 165. Select the correct mathematical formulation of the following problem. A rectangular area must be enclosed as shown. The sides labeled x cost $0 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 + 0y = 300 b) Maximize xy if 0x + 10y = 300 c) Maximize xy if x + y = 300 d) Maximize x + y if xy = 300 e) Maximize x + y if 00xy = 300 166. For the Profit function with Domain the MAXIMUM PROFIT is: a) 150 b) 00 c) 50 d) 300 e) 350 167. If f (1) = 00 and f (1) = -6, estimate the value of f (14) for the function y = f (x). a) 194 b) 188 c) 06 d) 14 e) 8 168. 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 wide and y feet long. No fence is required along the river, so each field is fenced along 3 sides. The total area enclosed by all 4 fields combined is to be 800 square feet. What is the least amount of fence required? (a) Minimize xy if 4x + 5y = 800 (b) Minimize xy if 4x + 5y = 00 (c) Minimize 4x + 5y if xy = 800 (d) Minimize 4x + 5y if xy = 00 (e) Minimize 4(x + y) if xy = 800
SECTION C C1. To three decimal places, a) 1.573 b).6 c) 3.145 d) 4.879 e) 5.741 3 C. Find the relative maximum value for the function f (x) = x - 15x a) - 15 = -3.87 b) 15 = 3.87 c) -10 5 = -.36 d) 10 5 =.36 e) 1 C3. The profit from manufacturing x items is given by P(x) = -0.5x + 46x - 10. Find on [0, 1]. a) $5.00 b) $5.50 c) $6.00 d) $710.00 e) $735.50 C4. Find the absolute extrema of on the closed interval [0, ]. a) (0, 1) and (1, 3) b) (, 4/5) and (-1, -1) c) (, 4/5) and (0, 0) d) (1, 1) and (-1, -1) e) (1, 1) and (0, 0) 3 C5. The absolute maximum value of the function f (x) = x - 6x + 15 on the closed interval [0, 3] is a) -7 b) -17 c) 4 d) -1 e) 15 -x C6. Given ln(x - 3) = 4, then to three decimal places, x = a) 1.54 b).031 c) 3.057 d) 4.01 e) 5.057 C7. A job offer consists of a $7,000 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) $38,49 (e) $84,616 C8. What is the relative maximum of the function? a) (-1, 4) b) (1, 0) c) (0, 0) d) (1, ) e) (, 1.6) C9. Find the relative maximum of on the closed interval [0, 6]. a) 6 b) 3 c) d) -3 e) 0 -x C10. The maximum value of f (x) = xe is -1 a) e 0.368 b) e.718 c) 0.5 d) -e -.718 e) 0
C11. The half-life of a radioactive substance is 1500 years. How much of a 18 lbs. sample of this substance will remain after 9000 years? a) lbs b) 3 lbs c) 4lbs d) 5 lbs e) 64 lbs C1. Given find f (1). a) 84 b) 0 c) 59 d) 84/59 e) 1 C13. The best fit exponential function passing through the points (, 1), (3, 4) and (4, 7), rounded to three decimal places is x x x a) y =.646(0.164) b) y = 0.164(.646) c) y = 1.817(1.1) x x - d) 1.1(1.817) e) y = e C14. The best fit logarithmic function passing through the points (, 1), (3, 4) and (4, 7), rounded to three decimal places is a) -5.083 + 8.574 ln x b) 8.574-5.083 ln x c) 3.63 +.57 ln x d).57 + 3.63 ln x e) none of these C15. Find the inflection point for a) (4, 8) b) (1.678, 4) c) (0, 1.9048) d) (1.5546, 3.890) e) (10.378, 7.9808) -x C16. For which value of x does the graph of the function f(x) = e ln x have a horizontal tangent? Round your answer to the nearest hundredth. a) 1 b) 1.76 c).81 d) 3.93 e) Never C17. = a) b) 3/4 c) /3 d) 7 e) 6/3 C18. The area of the region bounded by f(x) = x, the x-axis and the line x = 1 and x = 5 is approximated by n = 4 rectangles. If the right endpoint is used for each of these rectangles, then the approximate area obtained is a)30 b) 41 c) 4 d) 54 e) 64 4 C19. One critical number of the function y = x - x - 3x + 1 is a) 1.63 b) -0.148 c) 0.75 d) 1 e) -1.17
C0. For the function, find d) e) C1. For the function, find d) e) C. The demand function for a product is p = 100-0.1x ln(x) where x is the number of units of the product sold and p is the price in dollars. Find the value of x for which the marginal revenue is one dollar per unit. a) 97.45 b) 0.000017 c) 188.9 d) 190.5 only e) 0.01 and 190.5 C3. The function has exactly two points of inflection. The value of x for one of these points of inflection is a) 8 b) -16 c) -0.8 d) 0 e) 10x C4. Find the critical numbers for f(x) = e ln(x). a) 0.894 only b) 0.08 and 0.894 c) 0.789 only d) 0.789 and 0.894 e) There are no critical numbers C5. X = -1/3-0.333 is a critical number for. The critical point (-0.333, -0.497) is a) a relative minimum b) a relative maximum c) neither a relative maximum nor minimum d) a point of inflection e) a saddle point C6 and C7. Given C6. Find the critical numbers. a) 0.48 only b) 0.48 and 13.94 c) 1 d) 0.84 only e) 0.84 and 15.494
C7. 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.). d) e) C8. 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 = $3000 e) There is no point of diminishing returns C9 and C30. The weekly profit that results from selling x units of a commodity weekly is given by P = 50x - 0.003x - 5000 dollars. C9. Find the actual weekly change in profit that results from increasing production from 5000 units weekly to 5015 units weekly. a) $0.00 b) $300.00 c) $99.33 d) $98.00 e) $8333.33 C30. Use the marginal profit function to estimate the change in profit that results from increasing production from 5000 units weekly to 5015 units weekly. a) $0.00 b) $300.00 c) $99.33 d) $98.00 e) $8333.33 x C31. Find the equation of the straight line that is tangent to f(x) = at the point (5, 3). a) y = x + b) y =.1807x - 78.9035 c) y = 15.3745x - 44.875 d) y = 3x - 18 e) y = 1.5678x - 75.839
C3. Evaluate a) 1 b) 0 c) d) undefined e) e C33. The present value of $5,000 per year flowing uniformly over a 8 year period if it earns 3% interest compounded continuously is, to the nearest dollar a) $454 b) $35,56 c) $45.08 d) $55,116 e) $73,05 C34. If then to two decimal places, f ' (1) = a) -3.67 b).69 c) 1.86 d) -4.6 e) none of these C35. Given the total cost function, find the value of x for which the average cost is a minimum. (a) -533.333 (b) 533.333 (c) 378.594 (d) -378.594 (e) 500
ANSWERS 1. d. b 3. c 4. d 5. c 6. d 7. e 8. d 9. a 10. c 11. e 1. e 13. d 14. a 15. e 16. b 17. e 18. d 19. d 0. b 1. c. c 3. b 4. c 5. d 6. d 7. b 8. b 9. a 30. d 31. a 3. c 33. c 34. a 35. c 36. c 37. c 38. b 39. c 40. a 41. c 4. b 43. c 44. c 45. c 46. c 47. b 48. b 49. d 50. c 51. e 5. a 53. d 54. b 55. d 56. a 57. c 58. c 59. d 60. d 61. c 6. a 63. c 64. b 65. b 66. e 67. e 68. d 69. e 70. b 71. b 7. a 73. a 74. d 75. c 76. e 77. c 78. a 79. d 80. b 81. b 8. e 83. c 84. d 85. e 86. b 87. d 88. a 89. c 90. b 91. a 9. e 93. c 94. a 95. b 96. e 97. c 98. e 99. d 100. a 101. b 10. e 103. c 104. b 105. b 106. d 107. e 108. d 109. a 110. e 111. c 11. b 113. c 114. c 115. c 116. a 117. e 118. a 119. b 10. b 11. c 1. e 13. b 14. e 15. d 16. c 17. c 18. b 19. e 130. a 131. d 13. d 133. e 134. e 135. d 136. c 137. b 138. a 139. a 140. d 141. b 14. d 143. b 144. e 145. a 146. b 147. a 148. a 149. d 150. c 151. d 15. e 153. d 154. b 155. a 156. c 157. d 158. b 159. b 160. a 161. c 16. d 163. b 164. b 165. a 166. b 167. b 168. d C1. b C. d C3. b C4. e C5. e C6. b C7. b C8. d C9. b C10. a C11. a C1. d C13. b C14. a C15. b C16. b C17. e C18. d C19. a C0. a C1. e C. a C3. e C4. b C5. a C6. e C7. c C8. d C9. c C30. b C31. b C3. a C33. b C34. d C35. c