120 Final Exam Packet
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University of Maryland, College Park *
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120
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Mathematics
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Apr 3, 2024
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MATH 120 FINAL EXAM SPRING 2022 page 1 @63) DIRECTIONS ¢ Use exactly ONE answer sheet per numbered question. Use the back of the sheet if needed. ¢ There are 7 numbered questions, so you should use 7 answer sheets. Number the sheets 1 through 7 where it says “Prob. # e Put your name, your TA’s name, and your section number on each page. e No books, no notes, no cell phones, no calculators, no electronic devices. e Answers should be exact and may include fractions, exponentials, or logarithms ¢ Show enough work that we can follow your thinking. Make sure your work is legible. Any unclear work will be marked wrong. ¢ You must show all appropriate calculus work in order to receive full credit for an answer. o Before handing in your test: on your first answer sheet only, copy and sign the Honor Pledge. Answer Question 1 on page 1 1. (a) (10 points) Find the derivative of f(z) = (2/T + €*)4. (b) (10 points) The value (in dollars) of your investment after ¢ years is given by P(£) = 100et*~3t. Is the value increasing or is it decreasing when ¢ = 1? You must use calculus concepts to justify your answer. (c) (10 points) Alex wants to build a rectangular wooden fence to enclose an area of 80 square feet. Inside, there is a partition made of wire mesh. The wooden fencing costs y $2 per foot and the wire mesh costs $1 per foot. Find the dimensions of the fence that minimizes the cost. You do not need to verify that this is a minimum. Answer Question 2 on page 2 2. (a) (10 points) The number of people (in thousands) showing symptoms of a certain disease after t weeks is P(t) = §t* — ;Jgt°. At what time ¢ > 0 will the maximum number of people be showing symptoms? {You do not need to show it is a maximum.) (b) (10 points) Given the function f(z) = 5z + Inz, find the equation of the tangent line at the point where £ = 1. You may use either point-slope form or slope-intercept form. Answer Question 3 on page 3 3. (a) (10 points) Find the z-coordinates of all critical points (that is, points that are a possible relative maximum or minimum) for f(z) = 2% — 75z. {b) (10 points) One of the values you should have obtained in Question 3(a) is £ = 5. Determine whether it is a relative maximum or a relative minimum. Justify your answer using calculus techniques. (c¢) (10 points) How much should you invest at 5% annual interest, compounded continuously, if you want to have $1000 after 12 years? (Do not evaluate logarithms and exponentials.) (d) (10 points) Find the function y = f(z) that has the following properties: f'(z) = 32 + 2z + 2 with the initial condition f(1) = 10. Questions 4 through 7 are on page 2
Answer Question 4 on page 4 4. (a) (10 points) Find all values of z that satisfy the equation € €2-3% = 1. (b) (10 points) Find il (@ + 4z +7)° dz2 (c) (20 points) The graph of a function f(x) is to the right. Copy the r chart below onto your answer paper. Then fill in the chart with r a symbol 4, —, or 0, according to whether the first or second T derivative (given by the row label) is positive, negative, or zero 2 " : at the z-coordinate given by the column label. z==-25|z2=-1|x2=0|z=2|x2=25 | f(2) i f'(z) b Answer Question 5 on page 5 5. (a) {10 points) Assume the population of a certain species as a function of time is given by 120 P() = 14 e~05¢° (i) What is the initial population? (ii) Find the value of t when P(t) = 100. (Do not evaluate logarithms and exponentials.) (b) (10 points) You study how long it takes for students to finish an assignment, and you find that it is described by the probability density function f(t) = (¢t +1)~2 for 0 < ¢ < oo, where the time ¢ is measured in hours. What is the probability that a randomly chosen student takes between 2 and 5 hours to finish the assignment? Answer Question 6 on page 6 1 1 , . 3 6. (a) (15 points) Evaluate the following: / (\/E +8z° + prs + 752 + 8) dx (b) (10 points) Find the area enclosed by the curves ¥ = 23 + 4 and y = % + 2 between their points of intersection. Answer Question 7 on page 7 7. (&) (10 points) For the function f(z,y) = 3z + 4zy + €*¥, find 7 g OL Oy Ox0y (b) (10 points) For the function f(z,y) = 22 — y® + 6z + 12y + 1, use the First Derivative Test to find the (z,y) coordinates of all the points at which the function possibly has a relative maximum or minimum. (c) (5 points) One of the points you found in Question 7(b) should be (z,y) = (-3,2). Use 8f 8%f [ 8%f \? Dy =gz g2 - (B:cay) to determine whether the point {—3, 2) is the location of a relative maximum, relative minimum, or neither maximum nor minimum (saddle point). Before handing in ezam: (1) make sure that you have answered every part of every problem; (2) check the directions at the beginning of the exam to make sure you followed them. After handing in ezam: Enjoy your break.
MATH 120 FINAL EXAM FALL 2021 page 1 DIRECTIONS e Use exactly ONE answer sheet per numbered question. Use the back of the sheet if needed. e There are 9 numbered questions, so you should use 9 answer sheets. Number the sheets 1 through 9 where it says “Prob. # ¢ Put your name, your TA’s name, and your section number on each page. e No books, no notes, no cell phones, no calculators, no electronic devices. ¢ Answers should be exact and may include fractions, exponentials, or logarithms e Show enough work that we can follow your thinking. Make sure your work is legible. Any unclear work will be marked wrong. e You must show all appropriate calculus work in order to receive full credit for an answer. o Before handing in your test: on your first answer sheet only, copy and sign the Honor Pledge. Answer Question 1 on page 1 1. (a) (10 points) Find the derivative of f(z) = (z + In(x))3. (b) (10 points) The manager of a department store wants to build a 600-square foot rectangular enclosure on the store’s parking lot to display some equipment. Three sides of the enclosure will be built of wood fencing at a cost of $2 per foot, and the fourth side (the side with the x in 7 the picture) will be built of cement blocks at a cost of $4 per foot. Find the dimensions of the enclosure that will minimize the total cost of the building materials. Youdo oo not need to verify that this is a minimum. Answer Question 2 on page 2 2. (a) (10 points) Find % (e"’z"'l) (b) (20 points) The graph of a function f(z) is to the right. Copy the chart below onto your answer paper. Then fill in the chart with a symbol 4+, —, or 0, according to whether the first or second derivative (given by the row label} is positive, negative, or zero at the z-coordinate _ given by the column label. ’ r=-4|lz=-83|z=-2|xz=0|2=1 f'(z) f"(z) Answer Question 3 on page 3 3. (a) (10 points) Find the value of z that satisfies the equation 4e3*+2 +5 = 13. (b) (10 points) The number of people (in millions) infected by a disease after ¢ weeks is P(t) = 4¢t/20 — ¢t/10 At what time ¢ will the maximumn number of people be infected? Questions 4 through 9 are on page 2
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3] Answer Question 4 on page 4 1 . 1 4. 1 i ine: - . & , (a) (15 points) Evaluate the following / ( 5211 3vz +8z° + o + 8) dz (b) (10 points) You invest $100 at 3% annual interest, compounded continuously. After how many years will you have $500? (Do not evaluate logarithins and exponentials.) Answer Question 5 on page 5 5. (a) (10 points) Initially, 6 pigs are put on a large island. The pig population grows cxponentially. After 3 years, there are 24 pigs on the island. After 5 years, how many pigs will be on the island? (b) (10 points) Given the function f(z)} = z + \/, find the cquation of the tangent line at the point where z = 4. You may use either point-slope form or slope-intercept form. Answer Question 6 on page 6 6. (a) (10 points) Find the z-coordinates of all critical points (that is, points that are a possible relative maximum or minimum) for f(z) = 2z* — 1622 + 18. (b) (10 points) One of the values you should have obtained in part (a) in £ = 0. Determine whether it is a relative maximum or a relative minimum. Justify your answer. Answer Question 7 on page 7 7. (a) (10 points) Find the function y = f(z) that has the following properties: @)=+ % + 2z with the initial condition f(1) =0. (b) (10 points) Find the area under the curve y =9z2 4+ 5for 1 <z < 2. Answer Question 8 on page 8 8. (a) (10 points) Find the area enclosed by the curves y = z2 and y = 2x + 3 between their points of intersection. (b) (10 points) You study how long it takes for students to finish a final exam, and you find that it is described by the probability density function f(t) = 3t2/8 for 0 < t < 2, where the timne t is measured in hours. What is the probability that a randomly chosen student takes between 1 and 2 hours to finish the exam? Answer Question 9 on page 9 62 9. (a) (10 points) For the function f(z,y) = y(3z + 4y)%, find % and Wgy (b) (10 points) For the function f(z,y) = 23 + y% — 3z + 6y + 12, use the First Derivative Test to find the (z,y) coordinates of all the points at which the function possibly has a relative maximum or minimum. (c) (5 points) One of the points you found in part (b) should be (z,y) = (-1, 3). Use Rf 8f _( PF\° Do) = 55 7~ (72 to determine whether the point (—1,3) is the location of a relative maximum, relative - minimum, or neither maximum nor minimum (saddle point). Before handing in exam: (1) make sure that you have answered every part of every problem; (2) check the directions at the beginning of the exam to make sure you followed them. After handing in exam: Enjoy your break.
MATH 120 FINAL EXAM FALL 2019 page 1 DIRECTIONS o Use exactly ONE answer sheet per numbered question. Use the back of the sheet if needed. e There are 9 numbered questions, so you should use 9 answer sheets. Number the sheets 1 through 9 where it says Prob. # e Put your name, your TAs name, and your section number on each page. ¢ No books, no notes, no cell phones, no calculators, no electronic devices. . ¢ Answers should be exact and may include fractions, exponentials, or logarithms e Show enough work that we can follow your thinking. Make sure your work is legible. Any unclear work will be marked wrong. ¢ You must show all appropriate calculus work in order to receive full credit for an answer, ¢ Before handing in your test: on your first answer sheet only, copy and sign the Honor Pledge. Answer Question 1 on page 1 1. (a) (12 points) Given the function y = (4z2 + 1)3, find @ dx?’ (b) (12 points) A rectangular box with square base and no top is to be made to have a total volume of 4 cubic feet. Find the dimensions of the box that minimize the surface area. [Hint: The surface of this box consists of five rectangles.] You must label your constraint and your objective function that you use in the process of answering this question. You do not need to verify that this is a minimum. The picture to the right is not drawn to scale! Answer Question 2 on page 2 2. (15 points) The graph of the function f(z) is to the right. Copy the chart below onto your answer paper. Then, fill in the chart with a symbol 4, —, or 0, according to whether the first or second derivative (given by the row label) is positive, negative, or zero at the z-coordinate given by the column 5 label. _ z=-2|z=-1|z=0|2z=1]|2z=2 : Pt f'(z) T f(z) Answer Question 3 on page 3 3. An announcement is broadcast by mass media to a potential audience of 5000 students. After ¢t days, f(t) = 5000 (1 — e~%-%) students will have heard the announcement. (a) (6 points) How many students initially heard the announcement? Simplify your answer. (b) (10 points) At what rate is the announcement spreading after 10 days? Answer Question 4 on page 4 4. (a) (12 points) Find Ed&;' (ln(:c2 +1) — %45 4 oz — 7). (b) (12 points) Emily currently has $1000 to invest in her retirement fund, and she wants to have $4000 in her fund 20 years from now. What annual interest rate (continuous compounding) does she need to reach her goal? Questions 5 through 9 are on page 2 h
Answer Question 5 on page 5 5. (a) (12 points) The population describing the population of a city satisfies the differential equation P’ = 0.03P. This question has two parts: (i) Determine how fast the population will be growing when the size of the population reaches 100,000 people. (ii) Suppose that the initial population of this city (time ¢ = 0) was 70,000. Write the exponential growth function that describes the population of the city. (b) (12 points) Find the slope of the line tangent to the curve y = ¢ (1+ 2:4v4)_l at the point where z = 1. Answer Question 6 on page 6 6. (a) (12 points) A lemon tree that was 11 inches tall at time ¢ = 0 was given a nitrogen-rich fertilizer. After the fertilizer was applied, the tree started growing at a rate of 0.5v/% inches per week in week t. What is the height of the tree 9 weeks after the fertilizer was applied? (b) (14 points) Evaluate / (264’ + -31; + eg) dz. Answer Question 7 on page 7 7. (a) (12 points) (i) The price-demand function for widgets is p(z) = 16 — 2z. Find the revenue function. (ii) Suppose, after costs are taken into account, that the profit function is P(z) = —z2 + 8z — 10. What value of z maximizes profit? (b) (12 points) Set up the necessary definite integral to find the area of the region bounded by the curves y = z%® + 4 and y = z% + 22. DO NOT EVALUATE THE INTEGRAL. Answer Question 8 on page 8 8. (a) (10 points) Given the function f(z,y) = z* + 4z%y° + €¥ + In(z? + 3zy + Ty?), find g oz (b) (10 points) (i) Suppose z units of labor costs $100 per unit and y units of capital costs $200 per unit. Express the total cost of labor and capital combined as a function of z and y. (ii) Let f(z,y) = 9z/y — (2% + y)/2. Find f(4,9). Simplify your answer. Answer Question 9 on page 9 3 H — pTi+3z fl 9. (a) (12 points) Given f(z,y) = Y, find 920y (b) (15 points) The function f(z,y) = z2 — 4z +y° — 3y has two critical points, one of which is (z,y) = (2,1). (A critical point is a point at which the function f(z,y) possibly has a relative maximum or relative minimum.) This question has two parts: (i) Use the first partial derivatives to find the other2critical point. 2 2 (ii) Use the formula D = (%z_—é) (37{) - (%aiy) to determine whether the point (2,1) is the location of a relative maximum, a relative minimum, or neither a maximum nor a minimum (i.e., is a saddle point) Before handing in ezam: (1) make sure that you have answered every part of every problem; (2) check the directions at the beginning of the exam to make sure you followed them. After handing in ezam: Enjoy your break.
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MATH 120 Final Exam Spring 2019 page 1 READ AND FOLLOW DIRECTIONS CAREFULLY: » Use exactly ONE answer sheet per question (use the back of the sheet if needed). o There are 8 questions. Number your answer sheets 1 through 8 where it says “Prob #”. ¢ Put your name, your TA's name, and your section number on EACH page. » No books, notebooks, cell phones, calculator or other electronic devices are allowed. ¢ Answers should be exact, and may include fractions, exponentials or logarithms. ¢ Put a BOX around the final answer to a question. ¢ Show enough work that we can follow your thinking. You must show all appropriate calculus work in order to receive full credit for an answer. e Before handing in your test: on your first answer sheet only, please copy the pledge and sign. Answer question 1 on answer page 1. 2 1.a. [12 points] Find %[ln(x2 + 2)] . You do not need to simplify your answer. 1.b. [12 points] A company has found that when it sells x units of a product, the revenue is given by R(x)= —x%+30x. Their cost of production is given by C(x) = 50 + 10x. First determine the Profit function, then determine the number of units sold which will maximize profit. You do not need to verify that thisis a maximum. Answer question 2 on answer page 2. 2.a. [15 points] The graph of a function f{x) is to the right. Copy the chart below onto your answer paper. Then, fill in each cell in the chart with a symbol +, — e or 0, according to whether the function or derivative (given by the row label) is positive, negative, or zero at the x-coordinate (given by the column label). Make sure your symbols are legible. Any unclear symbols will be marked as wrong. . - - - - [N/ x=-2 x=-1 x=0 x=1 x=2 ! \/ f(x) ] ['(x) /(%) 2.b. [11 points] The size of an insect population is given by P(r)=400e""', where ¢ is measured in days. Part i) Find the time at which there will be 600 insects. Part ii) Find the rate at which the insect population is growing at the moment that there will be 600 insects. (Hint: You can use the differential equation satisfied by P().) Answer question 3 on answer page 3. 3_s 3.a. [12 points] Find the equation of the line tangent to the curve f (x) = e(x * at x = 0. Write your answer in slope-intercept (that is, y = ...) form. 3 3.b. [11 points] Given f(x)= %——sz , find the x-values for all maximum, minimum, and points of inflection. (You must show how you know that these are maximum, minimum, and point of inflection.) Answer question 4 on answer page 4. 4.a. [11 points] A sample of 15 grams of a radioactive mckel decays to 10 grams after 60 years. Part i) Find the exponential function that models the amount of nickel after ¢ years. Part i1) Use your function to determine the half-life. Math 120 Spring 2019 Final Exam There are questions on both sides.
MATH 120 Final Exam Spring 2019 page 2 Answer question 4 on answer page 4. 4.b. [11 points] An epidemic of measles has started in a city, and the number of people infected is modeled by the function P(t) = 1—4_% , where ¢ is measured in days since the beginning of the epidemic. At what rate is o0 the epidemic spreading after 3 days? Answer question 5 on answer page 5. 5.a. [11 points] Alice wants to put money into a savings account so that, after 4 years, there will be $3,000 in the account. If the account has an interest rate of 2% compounded continuously, how much money should she deposit now? 5.b. [11 points] In a factory, the lifetime of one of their production machines has a probability density function f(£)=0.2¢7°* for 0 << 0. What is the probability that their newest production machine will last between 2 and 5 years from now? Answer question 6 on answer page 6. 6.a. [14 points] Evaluate I (2e3‘ -2x'+In (4))dx 6.b. [12 points] The promoters of a small county fair estimate that ¢ hours after the gates open at 9 am, visitors will be entering the fair at a rate of (—tz +8t+ 20) people per hour. How many people do they expect to arrive between 10 am and noon? (Define ¢ in hours, and specify that at 9 am, £ =0.) You do not have to do the final calculations. Answer question 7 on answer page 7. 7.a. [11 points] You want to find the area bounded by the curves y =2x and y=5x? between their points of intersection. Set up the necessary integration (with all values and function formulas in place) but DO NOT EVALUATE. 7.b. [8 points] A manufacturing company produces x of product A and y of product B. The production cost is given by the formula C(x, y)=10x>+4y*+xy-10 (in thousands of dollars). Calculate the cost for producing 4 product A and 5 product B. Give your answer in dollars. 7.c. [11 points] Given f(x,y,z)=e" +2° (4x+ y) determine both =— af ai Answer question 8 on answer page 8. o f dyox 8.a. [12 points] Given f(x,y,z)=xye’", determine 8.b. [15 points] For the function f(x,y)=-x>+3x y—% y? +1, Part i) Use the First Derivative Test to find the (x, y) coordinates of all the points at which the function f{x, y) possibly has a relative maximum or relative aZ f aZ f 62 f 2 minimum. Then use the formula D(x, y)=| —5 = |- to determine whether those points are ox” J\ oy Ox Ay the locations of relative maximums, relative minimums, saddle points, or “undetermined by the Derivative Test™. Math 120 Spring 2019 Final Exam There are questions on both sides.
MATH 120 Final Exam Fall 2018 page 1 READ AND FOLLOW DIRECTIONS CAREFULLY: o Use exactly ONE answer sheet per question (use the back of the sheet if needed). e There are 9 questions. Number your answer sheets 1 through 9 where it says “Prob #”. e Put your name, your TA's name, and your section number on EACH page. « No books, notebooks, cell phones, calculator or other electronic devices are allowed. o Answers should be exact, and may include fractions, exponentials or logarithms. ¢ Put a BOX around the final answer to a question. » Show enough work that we can follow your thinking. You must show all appropriate calculus work in order to receive full credit for an answer. e Before handing in your test: on your first answer sheet only, please copy the pledge and sign. Answer question 1 on answer page 1. 2 1.a. [12 points] Given the function y = e , find % . 1.b. [12 points] You want to build a fish tank in the shape of a rectangular box with square base. The tank should hold a volume of 18 cubic feet. The tank will have glass sides and a glass bottom, and the top will be made of a metal mesh. Glass costs $3 per square foot, and the metal mesh costs $1 per square foot. Find the dimensions that x minimize the cost of the fish tank. You must label your constraint function and your = objective function you used in the process of answering this question. You do not need to verify that this is a minimum. Answer question 2 on answer page 2. 2. [15 points] The graph of a function f{x) is to the right. Copy the chart below onto your answer paper. Then, fill in each cell in the chart with a symbol +, —,or 0, according to whether the function or derivative (given by the row label) is positive, negative, or zero at the x-coordinate 'y (given by the column label). Make sure your symbols are legible. Any unclear symbols will be marked as wrong. x=-2 | x=-1 x=0 x=1 x=2 3 o1 /;\ 3 4 R ) 1) - S (x) Answer question 3 on answer page 3. 3.a. [11 points] Find the equation of the line tangent to the curve f(x)= ln(x'° +e2) atx=0. 3.b. [12 points) The concentration, C, in parts per million, of a medication in the bloodstream ¢ hours after ingestion is given by the function C(¢)=10¢%e™’. Find the time at which the maximum concentration occurs. You do not need to verify that this is a maximum. Answer question 4 on answer page 4. 4.a. [11 points] Find %[\/fi +l+7° _§+e4-\r-l]_ X 4.b. [11 points] 160 grams of a radioactive substance decays to 120 grams in 10 years. First find the exponential function that models this substance’s decay, then use your function to determine the half-life of the substance. Math 120 Fall 2018 Final Exam There are questions on both sides.
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MATH 120 Final Exam Fall 2018 page 2 Answer question 5 on answer page 5. 5.a. [11 points] The function P(t)= 20000 14+19¢7%Y on a university campus ¢ weeks after the outbreak began. At what rate will influenza be spreading after 4 weeks? describes the number of people who became ill with influenza 5.b. [11 points] Following the birth of their granddaughter, two grandparents want to make an initial investment of Py that will grow to $10,000 by the child’s 20th blrthday If interest is compounded contmuously at 6%, what should the initial investment Pg be? Answer question 6 on answer page 6. 6.a. [11 points] In a factory, the average time between accidents has a probability density function F(1)=0.1e™*" for 0 << co. What is the probability that the next accident will happen between 5 and 10 days from now? 6.b. [14 points] Evaluate .[14 [7i +5e* +\/J—chx x Answer question 7 on answer page 7. 7.a. [12 points] You drop a stone from an initial height of 500 feet. Its velocity after # seconds is —32¢. Give a formula for A(f), the height of the stone after ¢ seconds. 7.b. [11 points] You want to find the area bounded by the curves y =4x*>~2x+3 and y =4x’ +x* between their points of intersection. Set up the necessary integration (with all values and function formulas in place) but DO NOT EVALUATE. Answer question 8 on answer page 8. 8.a. [8 points] Given the function f'(x, y)= (x yi+x* y)s, calculate f(-1,1). 8.b. [1 points] A manufacturing company produces x of product A and y of product B (in hundreds of units). The production cost is given by the formula C(x, y)=10x*+4y* +xy—10 (in thousands of dollars). Find the Marginal Cost with respect to product A, at the time when 500 product A and 600 product B have been produced. Answer question 9 on answer page 9. 2 9.a. [12 points] Given f(x, y)=e*""” g ln(xz) , determine both ¥ and o7 . | x oyox 9.b. [15 points] For the function f(x, y)=x*-3xy+2y* +2x—8y+7, first use the First Derivative Test to find the (x, y) coordinates of all the points at which the function f{x, y) possibly has a relative maximum or a2 r\(af) (&f Y relative minimum. Then use the formula D(x, y)=| —3 = |- to determine whether those | o J\ &y Ox Oy points are the locations of relative maximums, relative minimums, saddle points, or “undetermined by the Derivative Test”. Math 120 Fall 2018 Final Exam There are questions on both sides.
MATH 120 (WASHINGTON) FINAL EXAM SPRING 2018 DIRECTIONS e Use exactly one answer sheet per numbered question. Use the back of the sheet if needed. o There are 8 numbered questions, so you should use 8 answer sheets. Number the sheets 1 through 8 where it says Prob. # e Put your name, your TAs name, and your section number on each page. e No books, no notes, no cell phones, no calculators. e You do not need to simplify answers except where indicated otherwise. ¢ You must show your work to receive credit for a problem. ‘o Before handing in your test: on your first answer sheet only, copy and sign the Honor Pledge. (1) (a) (12 points) You invest $1000 at 4% annual interest, compounded continuously. When will the investment be worth $3000 ? (b) (12 points) The depth of a lake after z months of a drought is given by f(z) = 200— 3z for 0 < z < 5. Find the average depth of the lake during this time period. (2) (a) (8 points) For f(z) = (z2 + 1) In(x), evaluate f’(z). (b) (8 points) Evaluate [(3 + 55 )dz. (c) (10 points) Find the equation for the tangent line of f(z) = /z at z = 4. (3) (a) (12 points) Find the volume of the solid of revolution achieved by rotating the function f(z) = vz + 5, between z = —5 and = = 0, around the z-axis. (b) (14 points) Find the area bounded between f(z) = 323 — 12z, z = 2, = 3, and the T-axis . (4) (a) (14 points) Aristotle is a gardener who wants to build a rectangular garden. He has $700 to spend on a fence for this garden. The side closest to the road requires a special fence that costs $3 per foot, whereas the rest of the sides will be built using a fence that costs $1 per foot. What are the dimensions (length and width) of the biggest (greatest area) garden that Aristotle can build, given his monetary constraints? (b) (10 points) Compute fls(f; + z)dz. (5) (a) (10 points) Benazir is driving a car on a straight road. After ¢ seconds she is f(t) = t3 + 2t2 meters away from her starting position. What is her velocity after 2 seconds? (In your answer you need not specify the units.) (b) (12 points) Find two numbers whose sum is 150, and whose product is largest possible. (6) Recall that logs and exponents need not be evaluated as you solve the following questions: (a) (16 points) The Cyrus Charter is an ancient artifact containing the famous declaration of Cyrus the Great of Persia. When scientists found it, they wanted to determine its age. They noticed that it has 74% of the amount of 4C found in living things. (Recall that the decay constant of 14C is 0.00012.) How old is the artifact? (b) (8 points) Solve In(2z — 5) = 3. (7) In these questions, you must use calculus techniques to explain how you’ve reached your conclusions. For f(z) = z3 — 27z: (a) (8 points) At z = 0 is f(z) increasing, decreasing, or neither? (b) (14 points) Find all critical points of f(z), and classify each one as either a relative
(i-e., local) minimum or maximum? (c) (8 points) Is f(z) concave up or concave down at z =17 (8) (a) For f(z,y) = 9z%y + 4¥: (i) (8 points) Evaluate %. (i1) (8 points) Evaluate %5- (b) (8 points) For f(z,y) = 2®+4zy+ 3y2, the point (0, 0) is a critical point. Does f(z,y) have a local Immmum, a local maximum, or neither (saddle point) at (0,0)? (Recall 2 that D = E’éa—y{v (a%a% Before handing in exam: (1) make sure that you have answered every part of every problem; (2) check the directions at the beginning of the exam to make sure you followed them. After handing in exam: Enjoy your break.
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