Advanced Calculus. Summer Assignment. Name: Grade: Period: 1

Transcription

1 Advanced Calculus Summer Assignment Name: Grade: Period: 1

2 MERCER ISLAND HIGH SCHOOL 9100 SE 42nd Street Mercer Island, Washington (206) FAX (206) Vicki Puckett, Principal Mark Zmuba, Associate Principal Jamie Prescott, Associate Principal Dear Students, Welcome! Congratulations on your decision to take this challenging ocurse. Below is a note to help you be successful, or may make you decide to drop What makes this course interesting? The use of calculus and its consequences cuts across many disciplines, ranging from biology to business to engineering to the social sciences. At the risk of oversimplifying, calculus provides powerful tools to study the rate of change. For example, we might want to study how fast a disease is spreading through a population, by studying the number of diagnosed cases per day. We hope that seeing how calculus can be used to solve real world problems will be interesting. Many practical applications of calculus involve functions that depend on ore than one variable. You will learn about the geometry of curves and surfaces and get an introduction to differentiation and integration of functions in two variables. What makes this course difficult? The hardest thing about calculus is precalculus. The hardest thing about precalculus is algebra. You all know from previous math classes how one course will build upon the next, and calculus is no exception. This course will not only use material from precalculus and algebra, but it will use material you learned lats year. Very few of you will go on to major in mathmatics or computer science or engineering, but most of you will eventually see how calculus is appliedin your chosen field of study. For this reason, we aim for the ability to solve application problems using calculus. Some of the homework problems are quite lenthy and building up your mathematical problem solving stamina is just one of the aims of this course. This is one key place where this course differs from aa typical high school math course. In addition, it is important to note that the ability to apply calculus requires more than computational skill; it requires conceptual understanding. As you work through the homework, you will find two general types of problems: calculation/skill problems and multi-step/work problems. A good rule of thumb is to work enough of the skill problems to become proficient, then spend the bulk of your time working on the longer multi-step problems. Five common misconceptions Misconception#1: Theory is irrelevant and the instruction should be aimed at just showing you how to do the problems. The issue here is that we want you to be able to do ALL problems not just particular kinds of problems to which the methods of the course apply. For that level of command, you must attain some conceptual understanding and develop judgment. Thus, a certain amount of theory is very relevant, indeed essential. A student who has been trained only to do certain kinds of problems has acquired very limited expertise. 2

3 Misconception #2: The purpose of the classes and assignments is to prepare the student for the exams. The real purpose of the class and homework is to guide you in achieving the aspiration of the course; command of the material. If you have command of the material, you should do well on the exams. Misconception #3: It is the teacher s job to cover the material. As covering the material is the role of the textbook, and the textbook is to be read by the student, the teacher should be doing something else, something that helps the student grasp the material. The instructor s role is to quide the students in their learning: to reinforce the essential conceptual points of the subject, and to show their relation to the solving of problems. Misconception #4: Since you are supposed to be learning from the book, there s no need to pay attention in class. The discussions, classes, reading, and homework should combine to produce the true comprehension of the material. For most students, reading a math text won t be easy. The class should serve to orient the student in learning the material. Misconception #5: Since I did well in math in the past, I ll have no trouble with this class. There is a different standard at the college level. Students will have to put in more effort in order to get a good grade than in a typical high school math course (or equivalently, to learn the material sufficiently well by college standards). How do I succeed? Most people learn mathematics by doing mathematics. That is you learn it by active participation; it is very unusual for someone to learn calculus by simply watching the teacher perform. For this reason, the homework and classwork is THE heart of the course and more than anything else, study time is the key to success. I advise an average of 5 hours of study time per week. If you were taking this course on a college campus you would be spending 15 hours of study time per week OUTSIDE of class. In addition, it is much better to spread your studying time evenly as possible across the week; cramming 5 hours of homework into the day before an assignment is due does not work. Pacing yourself, using a time schedule throughout the week, is a good way to insure success; this applies to any course at the college level, not just math. It is critical you take this class seriously. While you may have earned A s in past courses, there is more material we will cover that is in a typical college single variable course that we will cover in the first few months before we even begin to cover the topics in Math 126. Hence, it is imperative you begin in the fall with solid skills and understanding from all you learned last year. I look forward to seeing you in the fall. Have a fabulous summer, and study, study, study Sincerely, Ms Schjelderup Kim.Schjelderup@mercerislandschools.org 3

4 Please read carefully:! This packet is designed to review (a) limits and continuity, (b) other key Precalculus topics and (c) topics learned last year. You will be assessed in September on any/all of the topics focused on in this packet, except vectors.! Problems in this packet will also introduce you to some new notation. Please see the next page for some clarifications.! You are encouraged to complete this packet on your own with minimal outside help. It is intended to call your attention to any material you will need additional help on over the course of next year. You may use the Formula packet available on my website. Please print (or download onto your ipad in the fall) out a copy of this packet and use it as a reverence throughout the school year. You will also see a link to notes from Precalculus should you need clarification on a topic.! The Limits and Continuity questions should be completed without a calculator. You may use a Nspireon the remainder of the assignment, but use it as a scientific calculator for manipulations. You will only have a scientific calculator, no graphing devices will be permitted on any assessments. The Nspire will be used to check work and to use the 3D graphing capabilities.! The last 18 pages are from old UW final exams. These are a complete review from AP Calculus last year, except for the work and center of mass problems (we will cover these this year).! YOU MUST HAVE THIS PACKET COMPLETED IN ITS ENTIRETY ON THE FIRST FULL DAY OF CLASS! --Sept. 6! I will be grading your summer packet on your level of effort/completeness. You will have an opportunity to ask questions on the packet before you are formally assessed on the material. We will go over all the UW final exam questions that are giving you difficulty.! Answer will be posted on the course webpage by the end of August. Please download the answer key and check your own work. Try to correct any of the problems you get wrong. Make a list of questions that are challenging that you cannot solve correctly.! Pace yourself It is lengthy! The idea is to complete a little bit each day/week so your math brains are ready to go come September. Show all work and box your final answer(s). 4

5 Terminology You May Have Forgotten and Alternative Notations: Hence: Otherwise: Sketch: Graph: Show: Factorize: you must utilize your answer from the preceding part to solve, it should give you a hint in how to solve the problem you should utilize your answer from the preceding part to solve (think of it as a hint, but if you can solve it a different way that is okay too) this is a rough graph of the function, it should have the correct shape and key points labeled, but does not have to be to scale this is a graph to scale I give you the answer, now show me the steps to get there factor Singular Matrix: a matrix whose determinant is 0 Mutually Exclusive: disjoint, the intersection of the two sets is empty Alternative Notations: f : x x 2 is equivalent to f ( x) = x 2 ( ) ( f! g) ( x) = f g( x) n( A) : the number of elements (items) in set A Calculators We will use both scientific calculators and your Nspire. Students who plan to attend the University of Washington will not be able to use a graphing calculator for any math class. Universities across the nation permit scientific calculators or graphing. Most professors do not know what a Nspire is, not do they know the capabilities of the CAS Nspire. 5

6 Limits and Continuity (Non-calculator) Rates of Change and Tangents to Curves 1. Find the average rate of change of the function over the given interval. a. f ( x) = x 2 +1 [ 2,3] b. f ( x) = x 2 +1 [ 1,1] c. R( θ ) = 4θ +1 [ 0,2] 2. Let f ( x) = x 2 2x 3 a. Find the slope of the line through the points P 2, 3 ( ) and Q( 2 + h, f ( 2 + h) ). b. Find the equation of the tangent line to f ( x) at P. 3. The accompanying graph shows the total distance s traveled by a bicyclist after t hours. a. Estimate the bicyclist s average speed over the time intervals [0,1], [1,1.25], and [2.5, 3.5]. b. Estimate the bicyclist s instantaneous speed at the times t = 1, t = 2, t = 3. c. Estimate the bicyclist s maximum speed and the specific time at which it occurs. 6

7 Limit of a Function and Limit Laws 4. For the function g( x) graphed here, find the following limits, or explain why they do not exist. a. lim g( x) x 1 b. lim g( x) x 2 c. lim g( x) x 3 d. lim g( x) x Which of the following statements about the function y = f ( x) graphed here are true, and which are false? a. lim f x x 0 ( ) exists b. lim f x x 0 ( ) = 0 c. lim f ( x) = 1 x 0 d. lim f x x 1 ( ) = 1 e. lim f x x 1 ( ) = 0 f. lim f ( x) exists at every point x 0 in ( 1,1) x x 0 g. lim f x x 1 ( ) does not exist 6. Calculate the limit. a. lim 2x + 5 x 7 ( ) b. x + 3 lim x 2 x + 6 c. lim ( 5 x) 4/3 x 3 7

8 d. lim x 5 x 5 x 2 25 g. lim x 1 x 1 x e. x 2 + x 2 lim x 1 x 2 1 h. lim( 2sin x 1) x 0 f. lim x 9 x 3 x 9 i. lim x π x + 4 cos( x + π ) 7. Suppose lim x c ( ) = 5 and lim f x a. lim f x x c ( )g x x c g( x) = 2, find ( ) c. lim x c ( f ( x) + 3g( x) ) b. lim 2 f ( x)g( x) d. x c lim x c ( ) ( ) g( x) f x f x 8. Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form f ( x + h) f ( x) lim h 0 h occur frequently in calculus. Evaluate this limit for x = 1 for the function f x ( ) = x 2. 8

9 9. If 5 2x 2 f ( x) 5 x 2 for 1 x 1, find lim f ( x). Justify your answer with the Squeeze Theorem. x If lim x 4 f ( x) 5 = 1, find lim f ( x). x 2 x If lim x 2 f ( x) 5 = 3, find lim f ( x). x 2 x If lim x 2 f ( x) 5 = 4, find lim f x x 2 x 2 ( ). Onen-Sided Limits 13. Which of the following statements about the function y = f x false? a. lim f ( x) = 1 g. lim f x x 1 + x 0 ( ) = 1 b. lim f ( x) = 0 h. lim f x x 0 x 1 ( ) = 1 c. lim f ( x) = 1 i. lim f x x 0 x 1 ( ) = 0 d. lim f ( x) = lim f ( x) j. lim f ( x) = 2 x 0 x 0 + x 2 e. lim f x x 0 ( ) exists k. lim x 1 f. lim f x x 0 ( ) = 0 l. lim f ( x) = 0 x 2 + f ( x ) does not exist ( ) graphed here are true, and which are 9

10 14. Let f ( x) = 0, x 0 sin 1 x, x > 0. a. Does lim f ( x) exist? Is so, what is it? If not, why not? x 0 + b. Does lim f ( x) exist? Is so, what is it? If not, why not? x 0 c. Does lim f ( x) exist? Is so, what is it? If not, why not? x 0 In 15 and 16, graph f ( x), then answer the following: a. What are the domain and range of f? b. At what points c, if any, does lim f ( x) exist? x c c. At what points does only the right-hand limit exist? d. At what points does only the left-hand limit exist? 15. f ( x) = 1 x 2, 0 x < 1 1, 1 x < 2 2, x = f ( x) = x, -1 x < 0, or 00 < x 1 1, x = 0 0, x < 1 or x > Calculate the limit. x 2x + 5 a. lim x 2 + x +1 x 2 + x b. lim h 0 + h 2 + 4h h 10

11 c. lim ( x + 3) x + 2 x 2 + x + 2 f. lim θ 0 tan2θ θ d. lim ( x + 3) x + 2 x 2 x + 2 g. lim θ 0 θ cosθ sin e. lim θ 0 2θ 2θ h. lim θ 0 tan 3θ sin8θ Continuity 18. State whether the function graphed is continuous on [-1,3]. If not, where does it fail to be continuous and why? Use the definition of continuity to justify your answer. a. c. b. d. 19. at what points are the functions continuous? a. y = 1 x 2 3x c. y = ( 2x 1) 1/3 b. y = cos x x d. f ( x) = x 2 x 6, x 3 x 3 5, x = 3 11

12 20. Is the function continuous at the point being approached? Explain. lim t 0 π 19 3sec2t 21. Define g( 3) in a way that extends g( x) = x2 9 x 3 to be continuous at x = For what value of a is f x 2ax, x 3 ( ) = x2 1, x < 3 continuous at every x? 23. For what values of a and b is f ( x) = 2, x 1 ax b, -1 < x < 1 3, x 1 continuous at every x? 24. Show that the equation x 3 15x +1 = 0 has three solutions in the interval [-4,4]. (Do NOT solve! Hint: The Intermediate Value Theorem). Limits Involving Infinity; Asymptotes of Graphs 25. Calculate the limit. a. 2 lim x x 3 b. 2 lim x x 3 c. ( ) 3 1/ x / x lim x ( ) 12

13 d. ( ) 3 1/ x / x lim x ( ) e. lim x 2 x + sin x x + cos x 26. Find the limit of each ration function (a) as x and (b) as x. a. f ( x) = 7x 3 x 3 3x 2 + 6x b. f ( x) = 10x5 + x x Calculate the limit. a. lim x 8x 2 3 2x 2 + x b. lim x 3 5 x x c. 3 5 x + x lim x x 3 4x x d. lim x 7 4 e. lim tan x ( x 7) 2 x ( π /2) 28. Find lim x2 3x + 2 x 3 2x 2 as (briefly explain your answers) a. x 0 + c. x 2 b. x 2 + d. x 2 e. What, if anything, can be said about the limit as x 0? 13

14 29.. Find lim 1 x + 2 2/3 x 1 ( ) 2/3 as (briefly explain your answers) a. x 0 + c. x 1 + b. x 0 d. x Graph the rational function. Label the x- and y-intercepts and the asymptotes. (No calculators!) y = x + 3 x Sketch a labeled graph of a function y = f ( x) that satisfies the given conditions. f ( 0) = 0 lim f ( x) = 0 x ± lim f ( x) = lim f ( x) = x 1 x 1 + lim f ( x) = lim f ( x) = x 1 + x Find lim x x 2 1 x ( ) 14

15 33. Graph the rational function. Label the x- and y-intercepts and the asymptotes. (No calculators!) y = x2 4 x 1 Algebra and Functions (Calculator Allowed) 34. Find the value of c for which the coefficient of x 4 in the expansion of ( 2x + c) 7 is Let f ( x) = x and g( x) = 4 x 2. a. Find: (i) f g( 2) ( ) (ii) g( f ( 2) ) b. State all values of x for which f ( g( x) ) is defined. c. State all values of x for which g( f ( x) ) is defined. 36. Solve each of the following equations for x, giving exact values in terms of the natural logarithm, ln, or in terms of e if necessary. a. 3 x = 6 b. log 3 ( x +17) 2 = log 3 2x 15

16 37. Find the equation of the line which passes through both the intersection of x + y = 2 and 2x + 3y = 8 and the point (0,0). 38. Find k such that the equation 2x 2 + kx + 2k = 0 has exactly one solution. 39. Consider the polynomial P( x) = 2x 3 + x 2 13x + 6. a. Factorize P( x). b. Find { x P( x) > 0} (Read: Find x such that P( x) > 0 ) 40. Find a and b such that x 4 + 4x 2 + ax + b is exactly divisible by x 2 + x + 2. x + y z = Solve the system of equations 3x + 4y + 3z = 2 4x + 5y + 3z = 5 16

17 42. Given that 2 + i is a root of the equation x 3 6x 2 +13x 10 = 0, find the other two roots. 43. The function f is given by f ( x) = x 2 6x +13 for x 3. a. Write f ( x) of ( x a) 2 + b. b. Find the inverse function of f ( x). c. State the domain and range of the inverse. 44. Find the three cube roots of 8i. give your answers in the form x + yi. (Hint: Convert to trig form and use DeMorive s Theorem). Note: Look this up online if you don t know DeMorive s Theorem. 17

18 Sequences and Series (Calculator Allowed) 45. Find the following sums a. 6 n=1 ( n 2 n) b. 4 n= n 3 2 n2 46. Find the sum of the infinite geometric series: In an arithmetric sequence, the first term is -2, the fourth term is 16 and the n th term is 11,998. Find the common difference, d, and the value of n. 48. A geometric sequence has a first term 2 and a common ratio 1.5. Find the sum of the first 11 terms. 49. For what values of x will the sum 1+ ( 1 x) + ( 1 x) 2 + ( 1 x) exist? 18

19 50. Prove by mathematical induction that the sum of the series ( 1 3) + ( 2 4) n( n + 2) = 1 6 n( n +1) ( 2n + 7) 19

20 Trigonometry (Calculator Allowed) 51. the lengths of the sides of a triangle ABC are x 2, x, and x + 2. The largest angle is 120!. a. Find the value of x. b. Show that the area of the triangle is (Hint: Heron s Formula) c. Find sin A + sin B + sinc give your answer in the form p q r, where p,q,r!. 52. Given f ( x) = 3sin 2 x 11sin x + 6. a. Factorize the expression. b. Find the two values of sin x which satisfy the equation. c. Solve the equation for 0! x 180!. 20

21 Vectors (Calculator Allowed) Note: It is ok if you do not know how to do these 53. The vectors i and j are unit vectors along the x-axis and y-axis respectively. a. Given the vectors u = i + j and v = 3i + 5j, find u + 2v in terms of i and j. b. A vector w has the same direction as u + 2v, and has a magnitude of 26. Find w in terms of i and j. 54. The line L passes through the origin and is parallel to the vector 2i + 3j. Write down a vector equation for L. 55. Find the size of the angle between the two vectors degree and 3 1. Give your answer to the nearest 56. The vectors 2x x 3 and x +1 5 are perpendicular for two values of x. Find the two values of x. 57. Find the equation of the plane that passes through (0,0,0), (1,2,3) and (-2,3,3). 21

22 Probability (Calculator Allowed) 58. In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20 males. a. Use the information to complete the table below. Unemployed Male Female Totals Employed Totals 200 b. If a person is selected at random from this group of 200, find the probability that this person is i. An unemployed female ii. A male, given that the person is employed. 59. The following Venn diagram shows a sample space U and events A and B where n( U ) = 36, n( A) = 11, n( B) = 6 and n( A B)' = 21. a. On the diagram, shade the region ( A B)'. b. Find n( A B). A B c. Find p( A B). d. Determine if the events are mutually exclusive. Explain your answer. 22

23 60. For the events A and B, p( A) = 0.6, p( B) = 0.8, and p( A B) = 1.Find a. p( A B) b. p( A' B' ) 61. Two fair 6-sided dice are rolled. a. Complete the tree diagram by entering the probabilities. 6 6 Not 6 Not 6 6 Not 6 b. Find the probability of getting one or more sixes. 62. A factory makes staplers. Over a long period of time 4% of them are faulty. A random sample of 200 staplers is tested. a. Find the expected number of faulty staplers in the sample. b. Find the probability that exactly 5 staplers are faulty. c. Find the probability that one or more are faulty. 23

24 Item Analysis: Please indicate below by a check mark on the line provided which of the questions you were unable to solve and need further clarification in order to complete. Do this after you have checked your work and made at least one additional attempt at the problem The next few pages are old exams from the University of Washington s single variable courses. There are great practice as well and will help you prepare for the test in the fall. At the University of Washington you are only permitted to use a scientific calculator. Hence, when doing these next exams, only use the scientific calculator features of your Nspire to complete these problems. this will REALLY help you to get better at manipulation of algebraic expressions/equations. On the Integral Calculus portion you may have challenges with i) problem 3, but you should be able to use limits and get through the question. ii) problem 6, we will learn/practice work in the fall iii) problem 8, we will cover center of mass with uniform density (single variable) in the fall and complete center of mass with changing densities (multivariable) in the spring. 24

25 Differential Calculus 1. (15 Points) Find the derivatives of the following functions. You do not have to simplify. Your final answers must give the derivative in terms of x. (a) f(x) = 5x2 +2x 1 e x ( ) (b) f(x) =cos e x3 + π arctan ( x 2) (c) f(x) =6 x + x 6 + x x 25

26 2. (10 Points) Find the equations of the tangent line and the normal line to the curve x 2 +6xy + y 2 =17 at the point P (1, 2). ThenormallinetoacurveatthepointP is perpendicular to the tangent line at the point P. 26

27 3. (12 Points) In this problem, consider the polynomial y(x) =x 4 2x 3 x 2 +2x (a) Use the linear approximation of y(x) at x = 1 to estimate a root of the polynomial. (b) Is the exact root of y(x) bigger or smaller than your estimate in (a)? Explain. 27

28 4. (12 Points) A particle starts moving in the xy-plane at time t =0.Itspositionattime t 0 is given by x(t) = ln(t +1) 2t +2, y(t) =2 t 3 t 2. (a) Express dy in terms of t. dx (b) Find all times t 0 at which the path of the particle has a horizontal tangent. (c) At time t = 1,theparticledepartsfromtheparametriccurveandcontinues along the tangent line. Find the coordinates at which this tangent line crosses the y-axis. 28

29 5. (12 Points) Let r be a positive number. Consider the circle of radius r centered at the origin and the line with equation y = x.letp be the positive y-intercept of the circle 2r and Q the intersection point of the line and circle in the first quadrant. Draw the line through P and Q, whichcrossesthex-axis at the value S. P Q x y= 2r (S,0) (a) Find a formula for S in terms of r. (b) Calculate lim S (No credit without work justifying your answer.) r

30 6. (12 Points) A swimming pool is 60 feet wide and 82 feet long. It is 10 feet deep at its deep end. A cross section is shown below. If the pool is being filled at a rate of 2ft 3 /min, how fast is the water level rising when the depth at its deep end is 6 feet? 82 Cross section of the pool

31 7. (15 Points) Let f(x) be the function y = f(x) =2 3 x + 4 x 2 on the domain of all non-zero real numbers. (a) Find all intervals over which f(x) is increasing. (b) Find all intervals over which f(x) is concave down. 31

32 7. (continued) Recall the function y = f(x) =2 3 x + 4 x 2 (c) Calculate the following limits: i. lim x f(x) ii. lim f(x) x iii. lim f(x) x 0 + iv. lim f(x) x 0 (d) Sketch the graph f(x) using the grid below (e) Clearly label the (x, y) coordinates of all critical points and all points of inflection. 32

33 8. (12 Points) The cost of fuel per hour in running a locomotive isproportionaltothe square of the speed and is $25 per hour for a speed of 25 mi/hr. Other costs amount to $100 per hour, regardless of speed. (a) Find the total cost per hour as a function of the speed. (b) Find the speed which will make the cost per mile aminimum. Makesureto justify why your answer is a minimum. (Note: Be careful with the units.) 33

34 Integral Calculus 1. (10 total points) Evaluate the following indefinite integrals. 1 (a) (5 points) x x dx 2 (b) (5 points) x + 1 x 3 + 4x 2 + 5x dx 34

35 2. (10 total points) (a) (5 points) Evaluate the definite integral (1 + dx. Give your answer in exact form. x) 3 (b) (5 points) Find the average value of the function Give your answer in exact form. f (x)= ln(2x) (2x) 1/3 on the interval [ ] 1 2,

36 3. (10 points) Does the improper integral xe x dx converge or diverge? Justify your answer. 3 If it converges, evaluate the integral and give your answer in exactform. 36

37 4. (10 total points) Starting at time t = 0, a car moves along the x-axis with velocity v(t)=20t 2 30t miles/hr, where t is time in hours. Answer the following questions. Include units in your answers. (a) (5 points) At time t = 3hours,howfaristhecarfromitsstartinglocation? (b) (5 points) Unfortunately, the car s gas mileage is only 6 miles per gallon. How many total gallons of gas are used from time t = 0totimet = 3hours? 37

38 ( π ) 5. (10 points) The region R is bounded by the curve y = 1 + tan 3 x,thelines x = 0 and x = 1, and the x-axis. A solid is obtained by rotating R around the x-axis and then carving out a hole. For 0 x 1theverticalcross-sectionoftheholeatx is a square whose diagonal joins the points (x,x 2 ) and (x, x 2 ).Findthevolumeofthesolid.Giveyouranswerinexactform. y cross section at x x x x 2 2 x 38

39 6. (10 total points) Two laborers dig a hole in the ground 10 feet deep, shoveling the dirt up to the top of the hole. The horizontal cross-section of the hole is a rectangle of length 8 feet and width 3 feet. The dirt weighs 100 pounds per cubic foot. (a) (5 points) What is the total work (in ft-lb) done by the two laborers? (The dirt removed from the hole is cleared away by other laborers.) (b) (5 points) Suppose the first laborer digs the hole part of the way, and then the second laborer finishes digging the hole. How deep (in ft) should the first laborer dig in order to do half of the total work? Give your answer in decimal form with at least three digits after the decimal point. 39

40 7. (10 total points) Consider the Lissajous curve given by the parametric equations { x = cos(5t), y = sin(2t). (a) (5 points) Set up a definite integral for the arc length of the part of the curve for 0.10 t DO NOT EVALUATE THE INTEGRAL. (b) (5 points) Use the Trapezoid Rule with n = 3subintervalstoapproximatethedefiniteintegralin part (a). Give your answer in decimal form with at least four digits after the decimal point. 40

41 8. (10 points) Consider a uniform flat plate in the first quadrant that is bounded by the y-axis, the line y = 1/2, and the curve y = sinx. Findthey-coordinate y of the center of mass of this plate. Give your answer in exact form. y x 41

42 9. (10 points) Find the solution of the initial value problem yy = e 2x (1 + y 2 ), y(0)= 1. Give your answer in the form y = f (x). 42

43 10. (10 total points) A 167 Fcupofcoffeeisbroughtintoa59 Froom. Aftertwominutes,the temperature of the coffee is 149 F. Assume the temperature of the room is constant, and that therate of change of the temperature of the coffee is proportional to the difference between the temperature of the coffee and the temperature of the room. (a) (3 points) Set up a differential equation for the temperature T (t) of the coffee (in F) as a function of time t (in minutes) after it has been brought into the room. (b) (5 points) Find T (t) by solving the differential equation in part (a). Use separation of variables and solve for any unknown constant(s). SHOW ALL OF YOUR STEPS. DO NOT QUOTE A FORMULA. (c) (2 points) Find T (4). Giveyouranswerinexactformandsimplifyyouranswer. 43