1 DE MATH 127 : Calculus 1 for the Sciences Instructor: B. Forrest INDEPENDENT TERM PROJECT Weight: 10% Due: RECEIVED at the Centre for Extended Learning (CEL) Office or UW Campus Drop Box before NOON (Waterloo, Ontario time) on MONDAY, November 28, 2016 INSTRUCTIONS: IMPORTANT: Write your answers on the PRINTED COPY of this project and the requested Maple lab exercise pages. Ensure you have completed the weekly tasks up to and including Week 9, including the Self Checks and the Maple Labs. Print and complete the following project INDEPENDENTLY. NO GROUP WORK IS PERMITTED. You should consider this project as similar to an open book mid-term exam. You must show enough work to justify your solution. Marks will be awarded for showing and justifying your work, not for the final solution. You can use Maple to check your solutions, if applicable. WEIGHT The weight of this assignment is 10% toward your final grade. No late assignments are accepted (no exceptions). Projects received at the CEL Office or UW Campus Drop Box after the due date/time will be marked for the student's feedback, but not included in the final grade calculation. SUBMITTING YOUR PROJECT It is the student's responsiblity to ensure that projects are RECEIVED at the Centre for Extended Learning (CEL) Office or in the Campus Drop Box BEFORE NOON on the due date. Any project received after the due date/time will not be included in the student's final grade calculation. All Projects must be dropped off in person at the dropbox at the University of Waterloo or sent to the Centre for Extended Learning (CEL) Office (not your instructor) by mail, courier, or fax. (See instructions on the next page.) IMPORTANT: REFERENCE DECLARATION You must complete the Reference Declaration section below in order for your assigment to be graded. Part 0: Reference Declaration (weight: ZERO GRADE on the project if not completed ) All of the solutions you submit must be your original work. NO GROUP WORK IS PERMITTED. You cannot discuss or exchange solutions with any other person. The objective of this project is to determine if you have mastered the course material. You may only use the course materials stated in the course outline (text, course notes, lectures, course website, or Maple). Indicate which of these resources you used to complete this project on the lines below. IMPORTANT: Your submitted work must be your original work. By submitting this assignment you understand and agree to the University's policies regarding academic honesty. Any violation of the University of Waterloo's Policy 71- Student Academic Discipline Policy (see http://www.adm.uwaterloo.ca/infosec/Policies/policy71.htm) on academic honesty will result in an academic penalty. Family Name: First Name: I.D. Number: Signature: Declared Aids (check all that apply): Lectures Course Notes Course Text: edition # Other Declared Aids: I did not use any aids or seek any help to complete this project. Maple Software: version # 2 SUBMISSION METHODS IMPORTANT: The CEL office will send your projects to me just after NOON on the due date. Any projects received after this time will not be included in the final grade calculation. To submit your project, use one of the following methods: 1. CANADA POST or COURIER: Address your envelope as: Centre for Extended Learning (CEL) c/o Learner Support Services University of Waterloo 200 University Avenue West Waterloo, Ontario, Canada N2L 3G1 Attention: DE Math 127 Re: Your Name, Your UW ID IMPORTANT: Projects will be sent to me after NOON on the due date. It is the STUDENT'S responsibility to ensure their project arrives before the due date. Any projects received after this time will not be included in the final grade calculation. 2. DROP OFF - Campus Drop Box: You can drop off your project in a designated drop box in the (old) Math Building at the University of Waterloo campus before NOON (Waterloo, Ontario time) on the due date. The location of the drop box will be posted under the NEWS section of the LEARN course website during the term. IMPORTANT: Dropped off projects will be sent to me at NOON on the due date. Any projects received after this time will not be included in the final grade calculation. The University is not responsible for lost submissions (e.g. projects placed in the wrong box). If you are uncomfortable using the campus drop-boxes, please send your project to the CEL Office as noted above. 3. FAX: FAXing should only be used if the other options are not available. If you use this method you must FAX your assignment to the Centre for Extended Learning Office at (519-746-4607) well in advance of the NOON deadline on the due date. Projects that are faxed are often UNREADABLE; portions of the project that are unreadable will not be graded. It is recommended that you use a DARK leaded pencil to complete your project to avoid this problem. It is the STUDENT'S responsibility to ensure the project is received on time. Late submissions due to technical problems with FAX machines will not be accepted. If you intend to FAX your assignment it is recommended that you submit at least one business day before the due date to avoid any difficulties. IMPORTANT: The CEL Office DOES NOT proofread faxed assignments. The time the fax is received is the time that will be used for your project submission to determine if it was submitted before the deadline. 4. EMAIL: Emailing your submission is not an option for this project. If you can not use one of the methods listed above, please contact your instructor immediately (well in advance of the deadline). 3 Part 0: Reference Declaration (weight: ZERO on the project if not completed) Please ensure that you have completed the Reference Declaration section on the front page of this assignment. Students who do not complete and sign the Reference Declaration section will receive ZERO as their grade on this project. Part 1: Short Answer and True or False (weight: 1 mark each) Provide an answer to the following questions in the space provided. For any questions that require a calculation, you must show enough work to justify your answer. 1. Write ONE mathematical statement that means \"The distance from 3 to x is less than 1 4 and x 6= 3\". 2. What is the domain and the range of f (x) = ex ? Draw a sketch of f (x). Label the value of the y-intercept. 3. If a ball is thrown into the air with a velocity of 40 m/s, its height (in meters) after 2 seconds is given by s(t) = 40t16t2 . Find the velocity when t = 2. 4. Find the derivative of f (x) = 4 2 . 5. Find the derivative of f (x) = x2 ecos(x) . 6. Find the derivative of f (x) = ln(x) 1+x2 using the quotient rule. Do not simplify the expression. 7. Find the derivative of f (x) = sin(x2 ). 8. State 3 ways by which a function can fail to be differentiable. 9. What is an antiderivative of a function f(x)? 10. Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related? 4 11. If f (x) = 6x2 +3ex +8x+5 sin(x)+3, find the most general antiderivatives F (x). Check your answer by differentiation (show this). 12. Let v(t) be a continous, positive velocity function. What physical quantity does the area under the curve of v(t) from t = a to t = b represent? True and False Section: For the following questions, circle true or false. If you choose false, correct the given statement, explain why it is not true, or give a counter example. 1. True or False: If a function is continuous at a point a, then the function is differentiable at a. 2. True or False: If a function is differentiable at a point a, then the function is continuous at a. 3. True or False: If f (1) > 0 and f (3) < 0, then there exists a number c between 1 and 3 such that f (c) = 0. 4. True or False: If s(t) is the position function of an object, then s 00 (t) is its velocity function. 5. True or False: If f 0 (c) = 0, then f (x) has a local maximum or minimum at c. 6. True or False: If f 0 (x) < 0 for 1 < x < 6, then f (x) is decreasing on (1, 6). 7. True or False: If f 00 (x) = 0 at the point x = c, then c is a point of inflection. 8. True or False: The most general antiderivative of f (x) = x2 is F (x) = x1 + C. 5 Part 2: Multiple Choice: Circle the best answer. (2 marks each) t 1. The equation of motion for a particle in meters at time t seconds is s(t) = sin( t 6 ) + cos( 6 ), 0 t 2. The acceleration at the instant when the velocity is 0 is: a) 0.14142m/s2 b) +0.14142m/s2 c) 0.3877m/s2 d) +0.3877m/s2 e) None of the above 2. The picture shows the graphs of three functions of a moving car: one is the position function, one is the velocity function, and the other is the acceleration function. Identify which curve is the position function. a) graph A b) graph B c) graph C d) Not enough information to determine 3. A small boy standing in a flat field watches a balloon rise in the distance. The balloon leaves the ground 500 m away from the boy and rises vertically at the rate of 7 m/minute. At what rate is the angle of inclination of the boy's line of sight ( in radians) increasing at the instant when the balloon is exactly 500 m above the ground? a) 7 500 b) 7 1000 c) 14 500 radians/minute d) 14 250 radians/minute radians/minute radians/minute e) None of the above 4. Find the area of the largest rectangle that can be inscribed in a semicircle of radius 2 cm. a) b) 2 cm2 2 cm2 c) 2 cm2 d) 4 cm2 e) None of the above 6 5. A woman standing on a cliff is watching a boat through a telescope as the boat approaches a dock on the shoreline directly below her. If the telescope is 250 meters above the water level and if the boat is approaching at 20 meters/second, at what rate is the angle of the telescope changing when the boat is 100 meters from shore? a) 0.049 radians/second b) 0.049 radians/second c) 0.079 radians/second d) 0.079 radians/second e) None of the above 6. Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. If it takes 2 times as much energy to fly over water as land, what is the minimum amount of total energy that the bird could expend returning to its nesting area? a) 2.88 units b) 21.66 units c) 23.00 units d) 27.85 units e) None of the above 7. Suppose that the body temperature 1 hour after receiving x mg of a drug is given by T (x) = 102 16 x2 (1 x9 ) for 0 x 6. The absolute value of the derivative, | T 0 (x) |, is defined as the sensitivity of the body to the drug dosage. Find the dosage that maximizes sensitivity. a) 0 mg b) 1 2 mg c) 3 mg d) 6 mg e) None of the above 8. A bacterial colony has an initial population of 500. After 3 hours, the population is estimated to be 1800. What would you expect the population to be after 10 hours. Choose the best answer. a) over 20000 b) over 25000 c) over 30000 d) over 35000 e) The population dies out. 7 9. Use the standard trigonometric identities to establish which of the following are identities. I.e., Start with the left-hand side and use known identities to derive the right-hand side. See the handout from your Weekly Tasks on Proving Trigonometric Identities for help if required. Note: there may be more than one correct answer. a) cos(x + 2 ) = sin(x) b) sin(x + 2 ) = cos(x) c) sin(x) sin(2x) + cos(x) cos(2x) = cos(x) d) sin(3x) + sin(x) = 2 sin(2x) cos(x) e) sin( x) = cos(x) 10. Find ln(x50 ) x x100 lim a) 0 b) 1 2 c) 1 d) + e) 11. Find lim 0 sin() + tan() a) 0 b) 1 2 c) 1 d) + e) 12. Find lim ln(sin(x)) x a) 0 b) 1 2 c) 1 d) + e) 8 Part 3: Calculation Marks will be awarded for the showing how you arrived at your solution, not for the final solution. 1. Sketch and label the graphs of the following functions on the axes provided. Use a DIFFERENT colored pen/pencil for each graph in each question. Include extra values on the given x- and y- axis to make your sketch more accurate, where applicable. (Hint: use the first function given and then use translations to help you sketch the remaining function(s).) Marks will be deducted for inaccurate graphs. f (x) = cos(x), g(x) = cos( x2 ), and a. h(x) =| cos(x) | (2 marks) f (x) = ex , g(x) = ex , and b. h(x) = ex 1 (2 marks) c. The graph of y = f (x) is given. Draw a graph of each of the following on the axes provided below. Give reasons for your solutions. (3 marks) (i.) y = 13 f (x) (ii.) y = 2f (x + 6) (iii.) y = f (x + 4) 9 2. Inverse Functions: Let f (x) = ln(x + 3). (10 marks; -1 for each incorrect or incomplete answer) i.) Sketch the mirror line y = x on the axes provided. ii.) Sketch and label the graph of f (x) on the axes provided. iii.) State the domain and range of f (x). iv.) Is f (x) a one-to-one function? What test could you use to graphically show whether f (x) is one-to-one? Find the derivative of f (x) and prove whether this function is one-to-one using the calculus theory you have learned. v.) Find the inverse function of f(x). Show all of your work. Sketch and label the inverse function on the axes given above. vi.) State the domain and range of the inverse function. Compare the domain and range of f (x) and the inverse of f (x); what statement can be made about them? vii.) Let g be the inverse function that you found in part (v). Show that g f (x) = x and f g(y) = y. (Show all steps in your calculations.) 10 3. In the theory of relativity, the mass of a particle m with velocity v is m0 m(v) = q 1 v2 c2 where m0 is the rest mass of the particle and c is the speed of light. Using the theory of limits that you have studied, explain what happens to the mass of a particle as its velocity approaches the speed of light from below (i.e., v c )? (3 marks) 11 4. Let g(x) = x4 cos( x2 ). a. Use the Squeeze Theorem to show that lim g(x) = 0. Justify all of the steps of your solution. (3 marks) x0 b. In Maple, plot the graphs of f (x) = x4 , g(x) = x4 cos( x2 ), and h(x) = x4 on the same set of axes over the interval x = 0.3 to x = 0.3. Print a copy of these plots, including the Maple command you used, and attach it to the NEXT page of this project. Hand label the functions on the printed copy. (2 marks) 12 5 a.) Hand draw a tangent line to the graph at each x-component given below. Use these tangent lines to estimate the value of each derivative. Write your estimate beside each question. i. f 0 (3) ii. f 0 (2) iii. f 0 (1) iv. f 0 (0) v. f 0 (1) vi. f 0 (2) vii. f 0 (3) b.) Using your estimates from part (a), draw the graph of f 0 (x) on the same set of axes given above. Please use a different color of ink from your tangent lines. (3 marks) c.) Shown is the graph of the population function P (t) for yeast cells in a laboratory culture. (7 marks) i. Use the method above to draw the graph of P 0 (t) on the same set of axes given. P 0 (0) P 0 (2.5) P 0 (5) P 0 (7.5) P 0 (10) P 0 (12.5) P 0 (15) ii. What does P 0 (t) represent? iii. Describe how the rate of population increase varies over time from t = 0 to t = 15 hours. iv. When is the rate of population increase the highest? 13 6. 10 marks : -1 mark for each error or missing/incorrect statement (a) In your own words, explain what each of the following mathematical statements mean in regard to the graph of f (x). With your explanation, include a diagram for each part. Indicate whether the statement refers to a vertical asymptote, a horizontal asymptote, or neither. State the equation of the asymptote where applicable. (i) f (1) = 0 Explanation: Asymptote type and equation of asymptote (if applicable): Hand-drawn diagram: (ii) lim f (x) = x0 Explanation: Asymptote type and equation of asymptote (if applicable): Hand-drawn diagram: (iii) lim+ f (x) = x2 Explanation: Asymptote type and equation of asymptote (if applicable): Hand-drawn diagram: (iv) lim f (x) = x2 Explanation: Asymptote type and equation of asymptote (if applicable): Hand-drawn diagram: (v) lim f (x) = 0 x Explanation: Asymptote type and equation of asymptote (if applicable): Hand-drawn diagram: question 8 is continued on the next page... 14 6. (b) Using your answers from part (a), sketch, BY HAND, the graph of this function. Use dotted lines to indicate where there are any asymptotes. Label all parts of your graph, including roots and equations of asymptotes. 6. (c) Find a formula for a rational function f (x) that satisfies the conditions stated in part (a). Explain how you determined this function. 6. (d) Use Maple to check that your function in part (c) matches your plot in part (b). Attach the Maple printout of your plot for this question on the NEXT page of this project. 15 7 a.) What is linear approximation used for? Why would you use the linear approximation of a function rather than the actual function? (2 marks) b.) State the general formula for La (x), the linear approximation ofa function around the point x = a. Use this formula to find the linear approximation L0 (x) of the function f (x) = 1 x at a = 0. (3 marks) c.) Use the linear approximation that you found in part (b) to approximate d.) Use your calculator to find the actual value of 0.99. (1 mark) 0.99 correct to 5 decimal places. (1 mark) e.) On the axes below sketch and label the graphs of both f (x) and L0 (x) and label the point where they touch. Use these graphs to determine if your estimate in part (c) is greater or less than the actual value of 0.99. Explain your choice by referring to the graphs of f (x) and L0 (x). Clearly indicate the error on the graphs you have drawn. (3 marks) f.) Use Maple to estimate the values of x for which the linear approximation is accurate to within 0.1. Attach your plot on the NEXT page of this project. Label, by hand, the interval which represents your answer on the Maple plot. Note: In Maple, click once on the plot and hover your cursor over the points of intersection to see the coordinates in the Maple toolbar. State the interval you determined in the space below. Hint: see 6th, 7th, or 8th edition text section 3.10, example 2. (3 marks) INTERVAL = 16 8. Let f (x) = x4 + x 3. Explain why there must be at least one point c in [1, 2] satisfying f (c) = 0. Indicate which theorem you must use in your explanation. (3 marks) 9. At 2 : 00P M the speedometer on your car reads 30 km/hr. At 2 : 10 P M , it reads 50 km/hr. Show that at some time between 2 : 00 P M and 2 : 10 P M the acceleration is exactly 120 km/hr2 . Indicate which theorem you must use in your explanation. (4 marks) 17 10 a.) An incorrect use of L'Hospital's Rule is illustrated in the following limit computation. Explain what is wrong and find the correct value of the limit. (2 marks) lim x 2 sin x x = lim x 2 cos x 1 =0 10 b.) Find lim xex using L'Hospital's Rule. Using this information, does the function f (x) = xex have a horizontal or x vertical asymptote? State the equation of this asymptote. (4 marks) 18 11.) Let f (x) = x2 ex . Show all of your work for the following questions. State the name of any test you use, if applicable. (15 marks) a.) Find f 0 (x) and f 00 (x). Provide your answers in factored form. b.) Find all critical points of f (x). (Include both x- and y-components.) c.) Find all of the local maxima and minima. Remember to reference and state the name of the test that you used. d.) Find all points of inflection. e.) Determine where f (x) is concave upwards and concave downwards. Cleary state the intervals. 19 f.) Determine where f (x) is increasing and decreasing. g.) Evaluate lim f (x). x h.) Use L'Hospital's Rule to evaluate lim f (x). x i.) Using these limits, determine if there are any horizontal asymptotes. State the equation(s) of any asymptotes that you find. j.) Sketch the graph of f (x) on the axes below. Label all parts of your sketch (roots, maxima, minima, PI). Indicate any asymptotes with a dotted line and include its equation. Clearly show the intervals where the function is increasing or decreasing, and where it is concave up or concave down. 20 12.) (5 marks) Let x1 = 1.5. Using Newton's Method, state the formula for x2 and then find x2 . Show all of your work. Illustrate the procedure on the diagram below. (Hint: See the diagram in your text.) State the formula for x3 and x4 , and then find x3 and x4 . Show all of your work. Illustrate the procedure on the diagram below. Do all of your calculated values for x2 , x3 and x4 seem to be approaching the value of \"c\" where f(x) crosses the x-axis (i.e., the root)? 21 13.) The graph of a function f (x) is shown. Let F (x) be the antiderivative of f (x). (10 marks) ***For parts (i) - (vii), choose the best multiple choice answer(s). Hint: see the text section on Antiderivatives. i. Where is F (x) increasing? (1 mark) a) x < 1 and x > 3 b) 1 < x < 3 c) x < 0 and x > 2 d) 0 < x < 2 e) F (x) is never increasing. ii. Where is F (x) decreasing? (1 mark) v. Where does F (x) have an inflection point? (1 mark) a) x < 1 and x > 3 a) x = 0 b) 1 < x < 3 b) x = 1 c) x < 0 and x > 2 c) x = 2 d) 0 < x < 2 d) x = 3 e) F (x) is never decreasing. e) There is no inflection point. iii. Where is F (x) concave up? (1 mark) vi. Where does F (x) have a maxima? (1 mark) a) x < 1 and x > 3 a) x = 0 b) 1 < x < 3 b) x = 1 c) x < 0 and x > 2 c) x = 2 d) 0 < x < 2 d) x = 3 e) F (x) is never concave up. e) There is no maxima. iv. Where is F (x) concave down? (1 mark) vii. Where does F (x) have a minima? (1 mark) a) x < 1 and x > 3 a) x = 0 b) 1 < x < 3 b) x = 1 c) x < 0 and x > 2 c) x = 2 d) 0 < x < 2 d) x = 3 e) F (x) is never concave down. e) There is no minima. (viii) Using your answers to parts (i) - (vii) make a rough sketch of an antiderivative F (x) on the same set of axes as f (x) shown above, given that F (0) = 0. On your sketch, label where F (x) has maxima, minima, and inflection points. Label the intervals where F (x) is increasing or decreasing, and where F (x) is concave up or concave down. 22 Part 4: Explanation 1. A hiker leaves his cabin at 7:00 a.m. and takes his usual path to the top of the mountain, arriving at 7:00 p.m. The following morning, he starts at 7:00 a.m. at the top and takes the same path back, arriving at his cabin at 7:00 p.m. Use the Intermediate Value Theorem to explain that there is a point on the path that the hiker will cross at exactly the same time of day on both days. NOTE: You can not assume that the hiker travels at a constant velocity. You are encouraged to use a diagram to help explain your solution. (5 marks) 23 Part 5: Maple Labs Use Maple CLASSIC to complete the following labs which can be found in your weekly/daily tasks on the course website. IMPORTANT NOTE: BOTH the completed lab exercise pages with your HAND-written answers and the printed copy of the Maple worksheet file must be submitted for credit on these labs. If either part is missing, NO MARKS WILL BE AWARDED. 1. (5 marks) Lab 6: Linear Approximation in Maple a) Attach the printed copy of the Lab 6 Exercises (page iv) containing your written answers for grading. b) Attach the printed copy of your lab6.mws for grading. 2. (10 marks) Lab 8: Curve Sketching in Maple a) Attach the printed copy of the Lab 8 Exercises (pages xi, xii, and xiii) containing your written answers for grading. b) Attach the printed copy of your lab8.mws for grading