(a) What is the half-life of the element? 1.(1 pt) Find an equation of the curve that satisfies. dy dx 10yx 4

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1 Tom Robbins MATH 6- Fall Homework Set due 9// at : PM Here are some hints on how to use WeBWorK effectively: After first logging into WeBWorK, change your password and set your address. Find out how to print a hard copy on the computer system that you are going to use. Contact me if you have any problems. Print a hard copy of this assignment. Find a place and time when you can work on these problems away from the computer for a sustained period without distractions. Work the problems and record your answers. If you get stuck on one problem, set it aside and later read the relevant section in the textbook, talk to friends, go to the tutoring lab, or send me to find out why you are stuck. Then finish the problem. When you have the answers to the problems go to a computer and enter them. If the system tells you that your answer is wrong try to figure out what happened and reenter the answer. Maybe you mistyped something. If it isn t clear what s wrong answer the remaining questions. Regarding those questions that you could not answer correctly go back to step. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. The primary purpose of the WeBWorK assignments in this class is to give you the opportunity to learn by having instant feedback on your active solution of relevant problems. Make the best of it!.( pt) Find an equation of the curve that satisfies. dy dx yx and whose y intercept is. y x (function of x).( pt) An unknown radioactive element decays into non-radioactive substances. In days the radioactivity of a sample decreases by 8 percent. (a) What is the half-life of the element? half-life: (days) (b) How long will it take for a sample of mg to decay to 88 mg? time needed: (days).( pt) Solve the inital value problem for y x ; xy y x with the initial condition: y. y x Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

2 Tom Robbins MATH 6- Fall Homework Set due 9/9/ at : PM.( pt) It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation (what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear:?. t d y?.?.?. dt y d y dt t dy dt y sint d y dt d y dt t dy d y dt dt y e t dy dt d y dt sin t y sint.( pt) Find the function satisfying the differential equation y y e t and y..( pt) Find y as a function of t if 6y 9y y 9 y y t.( pt) Find y as a function of t if y 8y y y 6 y.( pt) Find y as a function of t if y 9y y y 8 y y t Note: This problem cannot interpret complex numbers. You may need to simplify your answer before submitting it. 6.( pt) Find a particular solution to the differential equation y 7y y t y p 7.( pt) Find a particular solution to the differential equation 6y y y t t e t y p 8.( pt) Find a particular solution to y 9y sin t. y p 9.( pt) Find the solution of y y y 6exp t with y 8 and y y.( pt) Find y as a function of x if y y y y y y 6 y 9 y x Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

3 Tom Robbins MATH 6- Fall Homework Set due 9// at : PM.( pt) Let a = (, -7, ) and b = (6,, -) be vectors. Compute the following vectors. A. a + b = (,, ) -a= (,, ) a - b= (,, ) a =.( pt) If a = (-7,, ) and b = (-6, -, ), find a b =..( pt) What is the angle in radians between the vectors a = (,, -) and b = (-, -, 8)? Angle: (radians).( pt) Let a = (, 8, -) and b = (, -8, 7) be vectors. Find the scalar, vector, and orthogonal projections of b onto a. Scalar Projection: Vector Projection: (,, ) Orthogonal Projection: (,, ).( pt) Let a = (8,, 7) and b = (9, 8, ) be vectors. Compute the cross product a b. (,, ) 6.( pt) If a i j k and b i j k, find a unit vector with positive first coordinate orthogonal to both a and b. i + j + k 7.( pt) Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P Q 6 and R (,, ) 8.( pt) Find a vector equation for the line through the point P = (, -, ) and parallel to the vector v = (, -, -). Assume r i j k and that v is the velocity vector of the line. r(t) = i + j + k 9.( pt) Find the angle in radians between the planes x z and y z.( pt) Find the distance from the point (,, ) to the line x y t z t Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

4 Tom Robbins MATH 6- Fall Homework Set due 9// at : PM.( pt) For the given position vectors r t compute the tangent velocity vector r t for the given value of t. A.) If r t cost sint π Then r = ( ) If r t t t, ) Then r = (, ) ) If r t e t i e t j tk. Then r = i j k..( pt) For the given position vectors r t compute the unit tangent vector T t for the given value of t. A.) If r t cost sint π Then T = (, ) ) If r t t t Then T = (, ) ) If r t e t i e t j tk. Then T = i j k..( pt) Find parametric equations for the tangent line at the point π cos 6 sin π 6 π 6 on the curve x cost y sint z t x t = y t = z t =.( pt) Find the length of the given curve: r t t sint cost where t..( pt) If r t cos t i sin t j 9tk, compute: A. The velocity vector v t i j k The acceleration vector a t i j k Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. cos(t) 6.( pt) Find the curvature κ t of the curve r t sint i sint j cost k 7.( pt) Find the curvature of y sin x at x π. 8.( pt) Given that the acceleration vector is a t cos t i sin t j t k, the initial velocity is v i k, and the initial position vector is r i j k, compute: A. The velocity vector v t i j k The position vector r t i j k Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. cos(t) Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

5 Tom Robbins MATH 6- Fall Homework Set due // at : PM.( pt) The equation of the ellipse that has a center at 7 9, a focus at 9, and a vertex at 9, is x C A y D B where A B C D.( pt) Find the area of the region bounded by the given curve: r e θ on the interval π θ π..( pt) Find the limit, if it exists, or type N if it does not exist. lim x y x x y.( pt) Find the limit, if it exists, or type N if it does not exist. (Hint: use polar coordinates.) 8x y lim x y x y.( pt) Find the first partial derivatives of f x y at the point (x,y) = (, ). f x f y x y x y 6.( pt) Find the first partial derivatives of f x y sin x y at the point (-6, -6). A. f x 6 6 f y ( pt) If sin x y z, find the first partial derivatives z z x and y at the point (,, ). A. z x z y 8.( pt) Find all the first and second order partial derivatives of f x y sin x y! cos x y. A. f x f x f y f y f f x xx f f y yy f x y f yx F. f y x f xy 9.( pt) Find the equation of the tangent plane to the surface z y 6x at the point. z = Note: Your answer should be an expression of x and y; e.g. x - y + 6.( pt) Find the linearization L x y of the function f x y #" 6x 6y at. L x y Note: Your answer should be an expression in x and y; e.g. x - y + 9.( pt) Find an equation of the tangent plane to the parametric surface x r cosθ, y r sinθ, z r at the point $% & & ' when r, θ π(. z = Note: Your answer should be an expression of x and y; e.g. x - y Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

6 Tom Robbins MATH 6- Fall Homework Set 6 due /8/ at : PM.( pt) Suppose f x y xy ax by. (A) How many local minimum points does f have in R? (The answer is an integer). (B) How many local maximum points does f have in R? (C) How many saddle points does f have in R? ) ).( pt) Suppose f x y xy x 9y. f x y has critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x z or if x z and y w. Also, describe the type of critical point by typing MA if it is a local maximum, MI if it is a local minimim, and S if it is a saddle point. First point (, ) of type Second point (, ) of type Third point (, ) of type Fourth point (, ) of type.( pt) Each of the following functions has at most one critical point. Graph a few level curves and a few gradiants and, on this basis alone, decide whether the critical point is a local maximum (MA), a local minimum (MI), or a saddle point (S). Enter the appropriate abbreviation for each question, or N if there is no critical point. (A) f x y e x y Type of critical point: (B) f x y e x y Type of critical point: (C) f x y x y Type of critical point: (D) f x y x y Type of critical point:.( pt) You are to manufacture a rectangular box with dimensions x, y and z, and volume v 96. Find the dimensions which minimize the surface area of this box. x = y = z =.( pt) Find the coordinates of the point (x, y, z) on the plane z = x + y + which is closest to the origin. x = y = z = 6.( pt) Find the maximum and minimum values of f x y 9x y on the disk D: x y. maximum value: minimum value: 7.( pt) Find the maximum and minimum values of f x y x y on the ellipse x 6y maximum value: minimum value: 8.( pt) Find the maximum and minimum values of f x y z x y z on the sphere x y z. maximum value = minimum value = 9.( pt) Find the maximum and minimum values of f x y xy on the ellipse x y. maximum value = minimum value = Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

7 - - Tom Robbins MATH 6- Fall Homework Set 7 due // at : PM.( pt) Evaluate the iterated integral * * x y dxdy.( pt) Evaluate the iterated integral y dydx.( pt) Calculate the double integral * * *+* R x x y da where R is the region: x y. *,*.( pt) Calculate the double integral xcos x y da where R is the region: R π π x 6 y.( pt) Calculate the volume under the elliptic paraboloid z x y and over the rectangle R./.. 6.( pt) Evaluate the iterated integral I * * x x x y dydx 7.( pt) Evaluate the double integral I ** xyda D where D is the triangular region with vertices. 8.( pt) Find the volume of the solid bounded by the planes x =, y =, z =, and x + y + z =. 9.( pt) Evaluate the integral by reversing the order of integration. * * 7 e x dxdy 7y Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

8 Tom Robbins MATH 6- Fall Homework Set 8 due // at : PM.( pt) ** Using polar coordinates, evaluate the integral sin x y da where R is the region 9 x R y..( pt) Electric charge is distributed over the disk x y so that the charge density at (x,y) is σ x y 9 x y coulombs per square meter. Find the total charge on the disk..( pt) A lamina occupies the part of the disk x y in the first quadrant and the density at each point is given by the function ρ x y x y. A. What is the total mass? What is the moment about the x-axis? What is the moment about the y-axis? Where is the center of mass? (, ) What is the moment of inertia about the origin?.( pt) Find the surface area of the part of the plane x y z that lies inside the cylinder x y..( pt) The vector equation r u v ucosvi usinvj vk, v π, u, describes a helicoid (spiral ramp). What is the surface area? 6.( pt) Evaluate the triple integral *** xyzdv E where E is the solid: z, y z, x y. 7.( pt) Use *** cylindrical coordinates to evaluate the triple integral " x y dv, where E is the E solid bounded by the circular paraboloid z 9 x y and the xy -plane. 8.( pt) Use*** spherical coordinates to evaluate the triple integral x y z dv, where E is the E ball: x y z 8. ** 9.( pt) Evaluate R x y da where R is the region in the first quadrant bounded by the curves: xy, xy, y x and y 7x. [Hint: Make the substitution u xy and v y( x.] Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

9 * * * Tom Robbins WeBWorK problems. WW Prob Lib Math course-section, semester year WeBWorK assignment 9 due /8/ at : PM..( pt) Let C be the curve which is the union of two line segments, the first going from (, ) to (, ) and the second going from (, ) to (, ). Computer the line integral dy C dx..( pt) Let F be the radial force field F xi yj. Find the work done by this force along the following two curves, both which go from (, ) to (, 9). (Compare your answers!) A. If C is the parabola: x t y t t, then F dr C If C is the straight line segment: x t y 9t t, then F dr C.( pt) Compute the total mass of a wire bent in a quarter circle with parametric equations: x 8cost y 8sint t π and density function ρ x y x y..( pt) Consider a wire in the shape of a helix r t costi sintj 7tk t π with constant density function ρ x y z. A. Determine the mass of the wire: Determine the coordinates of the center of mass: (,, ) Determine the moment of inertia about the z- axis:.( pt) Find the work done by the force field F x y z 6xi 6yj k on a particle that moves along the helix r t cos t i sin t j tk t π. 6.( pt) For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, f F). If it is not conservative, type N. A. F x y x y i x y j f x y F x y 7yi 6xj f x y F x y z 7xi 6yj k f x y z F x y 7siny i y 7xcosy j f x y F x y z 7x i y j z k f x y z Note: Your answers should be either expressions of x, y and z (e.g. xy + yz ), or the letter N 7.( pt) If C is the curve given by r t sint i sin t j sin t k, π t and F is the radial vector field F x y z xi yj zk, compute the work done by F on a particle moving along 8.( pt) Suppose C is any curve from to and F x y z z y i z x j y x k. Compute the line integral C F dr. 9.( pt) Let F xi 9yj 9zk. Compute the divergence and the curl. A. div F curl F i j k.( pt) Let F yz i xz j xy k. Compute the following: A. div F curl F i j k div curl F Note: Your answers should be expressions of x, y and/or z; e.g. xy or z or Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

10 Math 6- online Fall WeBWorK Questionnaire Tom Robbins due // at 7: AM We are very interested in your opinion about WeBWorK, and we appreciate your willingness to complete this questionnaire. The information you provide will not be traced back to you, and will be used excl;usively to evaluate and improve our WeBWorK services. Thank you for your time!.( pt) Dear Students, We would very much appreciate your help in evaluating WeBWorK. We would like to hear about both the strengths and weaknesses of WeBWorK as you see them and any suggestions that you have for improving the program or the way it is used. WeBWorK has already benefited from past comments by students and we want to continue to improve it and make it as useful as possible as an educational tool. All questions are optional, but we appreciate all the information you give us. Thank you very much for your time. Please select the best choice for each question. A. For which mathematics course are you filling out this questionnaire? A. Math Math Math Math 6 Math 7 F. Math 9 G. Math H. Math I. Math J. Math K. Other (please specify)

11 What is your gender? A. Female Male Is this your first time taking this course? A. Yes No What was the most recent previous math course, if any, that you were enrolled in at this university? A. Math Math Math Math 6 Math 7 F. Math 9 G. Math H. Math I. Math J. none K. Other (please specify) What was the content of your most recent high school math course? A. Calculus Pre-Calculus Algebra Geometry

12 Other (please specify) F. What is your ethnicity? A. Asian American/Pacific Islander Black/African American Caucasian/White Latino/Hispanic American Native American/Alaskan Native F. Other (please specify) G. What is your academic status? A. Freshman Sophomore Junior Senior Other (please specify) H. What is your intended major? A. Social Sciences (i.e. Psychology, Ecomonics, Political Science, History, etc.) Humanities (i.e. English, Religion and Classics, Languages, etc.) Natural Sciences (i.e. Physics, Chemistry, Biology, etc.) Engineering Mathematics F. Other (please specify) I. Do you plan to enroll in other mathematics courses at this university? A. Yes

13 No J. What grade do you expect to receive for this class? A. A B C D E K. Approximately how many days before weekly due dates do you typically begin to work on WeBWorK assignments? A. Other (please specify) L. How many hours per week do you typically spend on WeBWorK problem sets? A. Less than hour hour hours hours More than hours M. How would you rank yourself as a mathematics student? A. Among the best Above average Average

14 Below average Among the worst N. Have you used WeBWorK in courses previous to this one? A. Yes No O. Do you have access to a computer in your dorm room or residence? A. Yes No P. Where do you typically work on WeBWorK problem sets? A. Your dorm room Your off campus residence A campus computing facility The mathematics department The library F. Other (please specify) Please rate the frequency with which you do each of the following: TABLE BORDER CELLSPACING= CELLPADDING= WIDTH=6 TR TD COLSPAN= VALIGN= TOP Scale: all the time, almost all the time, sometimes, almost never, never, no answer /TD /TR TR TD WIDTH= /TD TD WIDTH=. Use the connection to get help with specific problems /TD /TR TR TD WIDTH=

15 /TD TD WIDTH=. Do WeBWorK assignments with other students /TD /TR TR TD WIDTH= /TD TD WIDTH=. Get an entire assignment correct /TD /TR TR TD WIDTH= /TD TD WIDTH=. Seek help from your teaching assistant /TD /TR TR TD WIDTH= /TD TD WIDTH=. Seek help from your instructor /TD /TR TR TD WIDTH= /TD TD WIDTH= 6. Guess at problems you don t understand /TD /TR TR TD WIDTH= /TD TD WIDTH= 7. Get frustrated with and give up on a particular problem due to mathematical difficulty /TD /TR 6

16 TR TD WIDTH= /TD TD WIDTH= 8. Get frustrated with the time it takes WeBWorK to respond to answers you submit to it /TD /TR TR TD WIDTH= /TD TD WIDTH= 9. Get frustrated with the syntactic requirements of answers you submit to WeBWorK /TD /TR /TABLE Please rate the extent to which you agree with each of the following statements: TABLE BORDER CELLSPACING= CELLPADDING= WIDTH=6 TR TD COLSPAN= VALIGN= TOP strongly agree, agree, neutral, disagree, strongly disagree, no answer /TD /TR TR TD WIDTH= /TD TD WIDTH=. I prefer WeBWorK over paper and pencil homework /TD /TR TR TD WIDTH= /TD TD WIDTH=. WeBWorK problems are challenging /TD /TR TR TD WIDTH= 7

17 /TD TD WIDTH=. Class lectures effectively prepare me to complete WeBWorK assignments /TD /TR TR TD WIDTH= /TD TD WIDTH=. The content of WeBWorK problems is consistent with the material taught in lectures /TD /TR TR TD WIDTH= /TD TD WIDTH=. The content of WeBWorK problems is consistent with the material tested on exams /TD /TR TR TD WIDTH= /TD TD WIDTH=. WeBWorK effectively prepares me for course examinations /TD /TR TR TD WIDTH= /TD TD WIDTH= 6. The immediate responses I get from WeBWorK make me more persistent with assignments /TD /TR TR TD WIDTH= 8

18 /TD TD WIDTH= 7. The immediate responses I get from WeBWorK help me learn the course material /TD /TR TR TD WIDTH= /TD TD WIDTH= 8. access to professors is a useful component of WeBWorK /TD /TR TR TD WIDTH= /TD TD WIDTH= 9. The feedback mechanism has made it easy to communicate with my professor /TD /TR TR TD WIDTH= /TD TD WIDTH=. I can successfully access WeBWorK whenever I need to /TD /TR TR TD WIDTH= /TD TD WIDTH=. I know where to go to get help when I am having trouble with course material or WeBWorK problems /TD /TR TR TD WIDTH= /TD TD WIDTH=. WeBWorK makes mathematics courses more enjoyable /TD /TR /TABLE 9

19 Please rate your satisfaction with each of the following: TABLE BORDER CELLSPACING= CELLPADDING= WIDTH=6 TR TD COLSPAN= VALIGN= TOP highly satisfied, satisfied, neutral, dissatisfied, strongly dissatisfied, no answer /TD /TR TR TD WIDTH= /TD TD WIDTH=. Your course instructor /TD /TR TR TD WIDTH= /TD TD WIDTH=. Your course TA /TD /TR TR TD WIDTH= /TD TD WIDTH=. Your course /TD /TR TR TD WIDTH= /TD TD WIDTH= 6. WeBWorK /TD /TR /TABLE 7. Please tell us what you like about WeBWorK. 8. Please tell us what you do not like about WeBWorK

20 9. Please use this space for additional comments regarding WeBWorK Thank you very much. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR