Calculus Math 21B, Winter 2009 Final Exam: Solutions
|
|
- Buck Greer
- 6 years ago
- Views:
Transcription
1 Calculus Math B, Winter 9 Final Exam: Solutions. (a) Express the area of the region enclosed between the x-axis and the curve y = x 4 x for x as a definite integral. (b) Find the area by evaluating the definite integral in (a). Solution. (a) The area A is given by A = x 4 x dx. (b) Using the substitution u = 4 x, with du = x dx, we get A = = = 3 43/ = u du [ 3 u3/ ] 4
2 . Evaluate the following indefinite integrals: (a) x 4 x dx; (b) x tan x dx; (c) (9 x ) 3/ dx. Solution. (a) We use the substitution u = 4 x, with du = dx and x = 4 u. This gives x 4 x dx = (4 u) u du = (4u / u 3/) du = 8 3 u3/ + 5 u5/ + C = 8 3 (4 x)3/ + 5 (4 x)5/ + C. (b) We use integration by parts to differentiate the inverse tangent. Setting u = tan x, dv = x dx, du = + x dx, v = x, we get x tan x dx = x tan x x + x dx = x tan x ( ) dx + x = x tan x x + tan x + C. (c) We use the trigonometric substitution x = 3 sin θ (since the integral involves a square root of the form 3 x ). This gives dx = (9 x 3/ ( ) 9 9 sin θ ) 3 cos θ dθ 3/ = cos θ ( 9 sin θ ) dθ 3/
3 Using the identity sin θ = cos θ to simplify the integrand, we get (9 x ) dx = 3/ 9 cos θ dθ = sec θ dθ 9 = tan θ + C. 9 We have tan θ = sin θ cos θ = sin θ sin θ = = 3 sin θ 9 9 sin θ x 9 x. Thus, (9 x ) dx = x 3/ 9 + C. 9 x 3
4 3. For a >, let R be the region of the (x, y)-plane with x a that lies above the x-axis and below the curve (a) Find the area of R. y = x. (b) Let S be the solid formed by rotating R about the x-axis. volume of S. Find the (c) How do the area of R and the volume of S behave as a? Comment briefly on your answer. Solution. (a) The area A(a) of R is given by A(a) = a x dx = [ln x] a = ln a. (b) Splitting R in to vertical strips of height y and width dx, which give discs of radius y and thickness dx when rotated about the x-axis, we find that the volume V (a) of S is given by V (a) = a a πy dx = π x dx [ = π ] a x ( = π a ). (c) As a, we see that A(a) diverges to, whereas V (a) π converges to a finite limit. It may seem a bit odd that one can rotate an infinite area about an axis and get a finite volume; this is possible because areas closer to axis of rotation contribute less to the volume. 4
5 4. Consider the curve with parametric equation x = t sin t, y = cos t for t π. (a) Write down a definite integral for the length of the curve. (Don t evaluate the integral.) (b) Write down a definite integral for the area of the surface formed by rotating the curve about the x-axis. (Don t evaluate the integral.) Solution. (a) The arclength L is L = = = = = π π π π ds (dx ) ( ) dy + dt dt dt ( cos t) + sin t dt cos t + cos t + sin t dt cos t dt (b) The area A of the surface, with radius of revolution about the x-axis equal to y, is A = πy ds = π π = π ( cos t) cos t dt π ( cos t) 3/ dt. 5
6 Remark. This parametric curve is one loop of the cycloid shown on the next page. We can in fact evaluate these integrals. Use of the double angle formula ( ) t cos t = sin gives: π L = = 4 = 8; π A = 4 π = 6 π = 6 π = 3 π = 64π 3. cos t dt ( ) t sin dt π π π ( cos t) 3/ dt ( ) t sin 3 dt [ ( )] t cos sin ( u ) du ( ) t dt 6
7 π π π π 3π 4π Figure : A cycloid. 7
8 5. Define the improper integral x 3 e x dx as a limit of Riemann integrals, and evaluate it. Solution. The improper integral is defined by x 3 e x dx = lim b b x 3 e x dx. The Riemann integral on the right hand side exists because x 3 e x continuous on the interval x b for any b. is To evaluate the integral, we first use the substitution t = x, with dt = x dx, which gives where c = b. b x 3 e x dx = = b c Next, we use integration by parts with which gives c u = t, x e x x dx te t dt dv = e t dt, du = dt, v = e t, te t dt = [ te t] c c + e t dt = ce c + [ e t] c = e c ce c 8
9 It follows that x 3 e x dx = lim b b = lim c =. x 3 e x dx ( e c ce c) Here, we use the fact that ce c as c. This follows, for example, from l Hôspital s rule: ( lim ) ce c c = lim c c e = lim c c e =. c 9
10 6. Consider the Riemann integral + x dx, 4 and partition the interval x into n equal subintervals. (a) Write down the Riemann sum for the integral obtained by evaluating the integrand at the left endpoint of each subinterval. Does this Riemann sum provide a lower bound or an upper bound of the integral, or neither? Why? (b) Write down the Riemann sum for the integral obtained by evaluating the integrand at the right endpoint of each subinterval. Does this Riemann sum provide a lower bound or an upper bound of the integral, or neither? Why? Solution. The length of each subinterval in the partition is x = /n. We denote the endpoints of the intervals in the partition by {x, x, x,..., x n } where x k = k for k =,,,..., n. n A Riemann sum for the integral has the form n f(c k ) x = n k= n + c 4 k k= where x k c k x k. (a) If we choose c k = x k to be the left endpoint of the subinterval, the corresponding Riemann sum I n is given by I n = n n k= = n 3 n k= + [(k )/n] 4 n 4 + (k ) 4. The function /( + x 4 ) is decreasing on x, so its value at the left endpoint of a subinterval is larger than in the rest of the subinterval. Therefore this Riemann sum provides an upper bound for integral.
11 (b) If we choose c k = x k to be the right endpoint of the subinterval, the corresponding Riemann sum J n is given by J n = n n k= = n 3 n k= + (k/n) 4 n 4 + k 4. Since the function is decreasing, its value at the right endpoint of a subinterval is smaller than in the rest of the subinterval. Therefore this Riemann sum provides a lower bound for the integral.
12 7. Suppose that an empty reservoir has the shape of a circular cone of radius a at its surface and depth h, and the surface is at a height h above sea level. Calculate the work done against gravity required to fill the reservoir by pumping water from sea level into the reservoir. Express your answer in terms of a, h, and the weight density w = ρg of water. Solution. Let z be the height measure from above the bottom of the reservoir. (It would be equally correct to use height y = z + h above sea level instead.) The reservoir cross section at height z is a circle of radius r = az/h (since the radius varies linearly from r = at z = to r = a at z = h). The volume dv of a slice of the reservoir of thickness dz at height z is A(z) dz where A = πr is the cross-sectional area. The mass dm of water required required to fill this slice is ρa(z) dz, where ρ is the mass-density of water. Water at height z has to be pumped a vertical distance (z + h) from sea level. The work dw required to pump a mass dm of water is g(z + h) dm, or dw = ρg(z + h)a(z) dz = w(z + h)a(z) dz. Thus, the total work W required is W = = h h = πwa h = πwa h w(z + h)a(z) dz w(z + h)π h = 7 πwa h. ( az ) dz h ( z 3 + hz ) dz [ 4 z4 + 3 hz3 ] h
13 Figure : The Ben Cruachan reservoir. Remark. Although this problem is obviously artificial, there are technological uses of pumping water into reservoir. For example, the Ben Cruachan power station in Scotland is a hydroelectric power storage facility. During periods of low power demand, water is pumped a vertical distance of approximately 36 m from sea level to a reservoir in the mountain. During periods of peak demand, the water is released and its gravitational potential energy is converted back into electrical power by turbines located in a cavern excavated in the bottom of the mountain. 3
14 8. Define two functions f(x), g(x) by f(x) = x ln e t t dt, g(x) = x ln t dt. (a) For what x-values does the Riemann integral defining f(x) exist? Why? (b) For what x-values does the Riemann integral defining g(x) exist? Why? (c) If x >, show that f (ln x) = g(x). Solution. (a) The Riemann integral exists for any x >, since ln > and e t /t is continuous on any interval that does not contain t =. Since e t /t is discontinuous and unbounded at t =, the integral is not defined for x. (b) The Riemann integral exists for any x >, since / ln t is continuous on any interval in < t <. Since ln =, the function / ln t is discontinuous and unbounded at t =, so the integral is not defined for x. (c) Replacing x by ln x in the definition of f(x) we get f (ln x) = ln x ln e t t dt. Note that ln x > if x >, so f(ln x) is well-defined. Making the substitution t = ln u in this integral, with dt = u du, observing that u = when t = ln and u = x when t = ln x, and simplifying the result, we get f (ln x) = = = x x x = g(x). 4 e ln u ln u u ln u ln u du u du u du
15 Remark. The function f(x) is called an exponential integral, while g(x) is called a logarithmic integral, usually denoted by Li(x) = x ln t dt. Neither integral can be expressed in terms of elementary functions. The logarithmic integral is important in number theory. If π(x) denotes the number of primes less than or equal to x (for example, π(8) = 4 since, 3, 5, 7 are prime numbers), then π(x) is approximately equal to Li(x) for large values of x. This means that the density of prime numbers among the natural numbers near x is approximately equal to (/ ln x) when x is large. How π(x) fluctuates about Li(x) is not well understood, and is related to some of the outstanding unsolved problems in mathematics, such as the Riemann hypothesis. 5
16 9. Suppose that a function g(y) is given for < y < by g(y) = y t t dt. (a) Use partial fractions to evaluate the integral and find g(y). (b) Find the inverse function f of g by solving the equation x = g(y) for y = f(x). (You can assume that < y <.) Solution. (a) We use a partial fractions expansion of the form t t = t( t) = A t + B t. Putting the right-hand side of this equation over a common denominator, we get A t + B (B A)t + A =. t t( t) We therefore choose B A =, A =. The solution is A = B = /, meaning that t( t) = ( t + t ) It follows that g(y) = y ( t + ) dt t = [ln t ln t ]y = (ln y ln y ) = ln y y. 6
17 (b) For < y <, we have y/( y) >, so we may omit the absolute values in the logarithm, and write g(y) = ( ) y ln. y If x = g(y), then e g(y) = e x, or y y = ex. Multiplying this equation by ( y) and collecting the terms proportional to y, we get ( + e x ) y = e x. Thus, y = f(x) where f(x) = ex + e x. Remark. This integral arises in solving the logistic equation dy dx = y y. This nonlinear ordinary differential equation (ODE) describes the evolution of a population y(x) as a function of time x whose growth rate dy/dx decreases with increasing population y (e.g. due to the effects of overcrowding or limited resources). There are two constant solutions of this ODE, y = and y =. The solution y = corresponds to extinction, and y = corresponds to the maximum sustainable population. We can find general solutions of the ODE by the method of separation of variables. Formally, we write dy y y = dx and integrate to get y y dy = dx. 7
18 This integral gives If y(x) satisfies the initial condition then C =, and it follows that x = g(y) + C. y(x) = y() = ex + e x. Thus, the population grows from y = at x =, but the growth rate slows as y(x) increases, and y(x) approaches the maximum sustainable population as x. 8
19 . As a simplified model of the earth s atmosphere, consider a stationary atmosphere in which the pressure p(z) and mass density ρ(z) vary with height z <. Let g be the gravitational acceleration (assumed constant). (a) Explain briefly why p(z) = g z ρ(s) ds. (You can assume that the improper integral converges at infinity.) (b) Explain briefly why it follows that dp dz = gρ(z). (c) Suppose that ρ(z) = cp(z) for some constant c, meaning that the density is proportional to the pressure, and p() = p where p is the atmospheric pressure at the ground z =. Find p(z). (d) At what height H is the pressure p(h) = p /e reduced by a factor e from its value at the ground? Solution. (a) In hydrostatic equilibrium, the pressure at some height in a fluid is equal to the gravitational force per unit horizontal area exerted by the fluid above. This force is equal to g times the mass per unit area of the fluid above, or g z ρ(s) ds. (b) This equation follows from the fundamental theorem of calculus, and the fact that the integral changes sign when we exchange the limits: d dz z ρ(s) ds = d dz z ρ(s) ds = ρ(z). (c) Using the formula ρ = cp to eliminate the density ρ from the equation in (b), and the condition at z =, we get dp dz = cgp, p() = p. 9
20 The solution of this initial value problem is the exponential function p(z) = p e cgz. (d) The pressure decreases by a factor of e over a height H = cg. Remark. The pressure p, specific volume V, and absolute temperature T of a unit mass of an ideal gas are related by the equation pv = RT, where R is a gas constant. (For dry air, R = 87 JKg K, where the absolute temperature T is measured in Kelvins K.) The density of the gas is given by ρ = /V. Thus, ρ is proportional to p if the temperature T is constant, and in that case the constant of proportionality is given by c = RT. In an isothermal atmosphere in hydrostatic equilibrium, the density and pressure decrease exponentially with height, p(z) = p e z/h, ρ(z) = ρ e z/h. The height H over which they decrease by a factor of e is called the scale height of the atmosphere, and it is given by H = RT g. Although the earth s atmosphere is not exactly isothermal, its temperature in the first 7 km above the surface is approximately within 5% of 5 K. The corresponding scale height is H = 7. km. Humans require an air pressure of at least.5 atmospheres to survive on a long-term basis. This sets an upper limit on habitable altitudes of about 5.8 km or 9, ft e.g. for some communities in the Andes or Tibet. The peak of Mt. Everest, at an altitude of 8.85 km or 9, 8 ft, is not habitable.
Sample Final Questions: Solutions Math 21B, Winter y ( y 1)(1 + y)) = A y + B
Sample Final Questions: Solutions Math 2B, Winter 23. Evaluate the following integrals: tan a) y y dy; b) x dx; c) 3 x 2 + x dx. a) We use partial fractions: y y 3 = y y ) + y)) = A y + B y + C y +. Putting
More informationPractice Exam 1 Solutions
Practice Exam 1 Solutions 1a. Let S be the region bounded by y = x 3, y = 1, and x. Find the area of S. What is the volume of the solid obtained by rotating S about the line y = 1? Area A = Volume 1 1
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
More informationMATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
More informationCalculus II. Philippe Rukimbira. Department of Mathematics Florida International University PR (FIU) MAC / 1
Calculus II Philippe Rukimbira Department of Mathematics Florida International University PR (FIU) MAC 2312 1 / 1 5.4. Sigma notation; The definition of area as limit Assignment: page 350, #11-15, 27,
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationLesson Objectives: we will learn:
Lesson Objectives: Setting the Stage: Lesson 66 Improper Integrals HL Math - Santowski we will learn: How to solve definite integrals where the interval is infinite and where the function has an infinite
More informationMATH 101 Midterm Examination Spring 2009
MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationx 2 y = 1 2. Problem 2. Compute the Taylor series (at the base point 0) for the function 1 (1 x) 3.
MATH 8.0 - FINAL EXAM - SOME REVIEW PROBLEMS WITH SOLUTIONS 8.0 Calculus, Fall 207 Professor: Jared Speck Problem. Consider the following curve in the plane: x 2 y = 2. Let a be a number. The portion of
More informationMath 113/113H Winter 2006 Departmental Final Exam
Name KEY Instructor Section No. Student Number Math 3/3H Winter 26 Departmental Final Exam Instructions: The time limit is 3 hours. Problems -6 short-answer questions, each worth 2 points. Problems 7 through
More informationMath Makeup Exam - 3/14/2018
Math 22 - Makeup Exam - 3/4/28 Name: Section: The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all work.
More informationMath 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
More informationCalculus II Practice Test Problems for Chapter 7 Page 1 of 6
Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on
More informationx+1 e 2t dt. h(x) := Find the equation of the tangent line to y = h(x) at x = 0.
Math Sample final problems Here are some problems that appeared on past Math exams. Note that you will be given a table of Z-scores for the standard normal distribution on the test. Don t forget to have
More informationMATH 162. Midterm 2 ANSWERS November 18, 2005
MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able
More informationMath Exam III - Spring
Math 3 - Exam III - Spring 8 This exam contains 5 multiple choice questions and hand graded questions. The multiple choice questions are worth 5 points each and the hand graded questions are worth a total
More informationt 2 + 2t dt = (t + 1) dt + 1 = arctan t x + 6 x(x 3)(x + 2) = A x +
MATH 06 0 Practice Exam #. (0 points) Evaluate the following integrals: (a) (0 points). t +t+7 This is an irreducible quadratic; its denominator can thus be rephrased via completion of the square as a
More informationPrelim 2 Math Please show your reasoning and all your work. This is a 90 minute exam. Calculators are not needed or permitted. Good luck!
April 4, Prelim Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Trigonometric Formulas sin x sin x cos x cos (u + v) cos
More informationLearning Objectives for Math 166
Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the
More informationMath 107H Fall 2008 Course Log and Cumulative Homework List
Date: 8/25 Sections: 5.4 Math 107H Fall 2008 Course Log and Cumulative Homework List Log: Course policies. Review of Intermediate Value Theorem. The Mean Value Theorem for the Definite Integral and the
More informationMATH 31B: MIDTERM 2 REVIEW. sin 2 x = 1 cos(2x) dx = x 2 sin(2x) 4. + C = x 2. dx = x sin(2x) + C = x sin x cos x
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate sin x and cos x. Solution: Recall the identities cos x = + cos(x) Using these formulas gives cos(x) sin x =. Trigonometric Integrals = x sin(x) sin x = cos(x)
More informationCalculus I Sample Final exam
Calculus I Sample Final exam Solutions [] Compute the following integrals: a) b) 4 x ln x) Substituting u = ln x, 4 x ln x) = ln 4 ln u du = u ln 4 ln = ln ln 4 Taking common denominator, using properties
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, one should study the past exams and practice midterms (and homeworks, quizzes, and worksheets), not just this practice final. A topic not being on the practice
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationMATH 2300 review problems for Exam 1 ANSWERS
MATH review problems for Exam ANSWERS. Evaluate the integral sin x cos x dx in each of the following ways: This one is self-explanatory; we leave it to you. (a) Integrate by parts, with u = sin x and dv
More informationChapter 7: Applications of Integration
Chapter 7: Applications of Integration Fall 214 Department of Mathematics Hong Kong Baptist University 1 / 21 7.1 Volumes by Slicing Solids of Revolution In this section, we show how volumes of certain
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationMath 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3
Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationMath 113 Fall 2005 key Departmental Final Exam
Math 3 Fall 5 key Departmental Final Exam Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.. Fill in the blanks with the correct answer. (a) The integral
More informationFinal Exam Solutions
Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed
More informationIntegration Techniques
Review for the Final Exam - Part - Solution Math Name Quiz Section The following problems should help you review for the final exam. Don t hesitate to ask for hints if you get stuck. Integration Techniques.
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.13 INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) by A.J.Hobson 13.13.1 Introduction 13.13. The second moment of a volume of revolution about the y-axis 13.13.3
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More information1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2
Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationMATH 1242 FINAL EXAM Spring,
MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More informationMath 190 (Calculus II) Final Review
Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES We have seen how to represent curves by parametric equations. Now, we apply the methods of calculus to these parametric
More informationSolutions to Math 41 Final Exam December 10, 2012
Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)
More informationMATH 162. Midterm Exam 1 - Solutions February 22, 2007
MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [
More informationPart I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
More informationExaminer: D. Burbulla. Aids permitted: Formula Sheet, and Casio FX-991 or Sharp EL-520 calculator.
University of Toronto Faculty of Applied Science and Engineering Solutions to Final Examination, June 216 Duration: 2 and 1/2 hrs First Year - CHE, CIV, CPE, ELE, ENG, IND, LME, MEC, MMS MAT187H1F - Calculus
More informationMore Final Practice Problems
8.0 Calculus Jason Starr Final Exam at 9:00am sharp Fall 005 Tuesday, December 0, 005 More 8.0 Final Practice Problems Here are some further practice problems with solutions for the 8.0 Final Exam. Many
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More informationFinal Exam 2011 Winter Term 2 Solutions
. (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L
More informationFinal Exam. Math 3 December 7, 2010
Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.
More informationMTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation
MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationSolutions to Math 41 Second Exam November 5, 2013
Solutions to Math 4 Second Exam November 5, 03. 5 points) Differentiate, using the method of your choice. a) fx) = cos 03 x arctan x + 4π) 5 points) If u = x arctan x + 4π then fx) = fu) = cos 03 u and
More informationMath 113 (Calculus II) Final Exam KEY
Math (Calculus II) Final Exam KEY Short Answer. Fill in the blank with the appropriate answer.. (0 points) a. Let y = f (x) for x [a, b]. Give the formula for the length of the curve formed by the b graph
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationMath 181, Exam 2, Fall 2014 Problem 1 Solution. sin 3 (x) cos(x) dx.
Math 8, Eam 2, Fall 24 Problem Solution. Integrals, Part I (Trigonometric integrals: 6 points). Evaluate the integral: sin 3 () cos() d. Solution: We begin by rewriting sin 3 () as Then, after using the
More informationMath 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
More informationMA 126 CALCULUS II Wednesday, December 14, 2016 FINAL EXAM. Closed book - Calculators and One Index Card are allowed! PART I
CALCULUS II, FINAL EXAM 1 MA 126 CALCULUS II Wednesday, December 14, 2016 Name (Print last name first):................................................ Student Signature:.........................................................
More informationMathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.
Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the
More informationMath 113 Winter 2005 Key
Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple
More informationMathematics 111 (Calculus II) Laboratory Manual
Mathematics (Calculus II) Laboratory Manual Department of Mathematics & Statistics University of Regina nd edition prepared by Patrick Maidorn, Fotini Labropulu, and Robert Petry University of Regina Department
More informationMath. 151, WebCalc Sections December Final Examination Solutions
Math. 5, WebCalc Sections 507 508 December 00 Final Examination Solutions Name: Section: Part I: Multiple Choice ( points each) There is no partial credit. You may not use a calculator.. Another word for
More information4x x dx. 3 3 x2 dx = x3 ln(x 2 )
Problem. a) Compute the definite integral 4 + d This can be done by a u-substitution. Take u = +, so that du = d, which menas that 4 d = du. Notice that u() = and u() = 6, so our integral becomes 6 u du
More information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
More informationMath 142, Final Exam. 12/7/10.
Math 4, Final Exam. /7/0. No notes, calculator, or text. There are 00 points total. Partial credit may be given. Write your full name in the upper right corner of page. Number the pages in the upper right
More informationFINAL EXAM CALCULUS 2. Name PRACTICE EXAM SOLUTIONS
FINAL EXAM CALCULUS MATH 00 FALL 08 Name PRACTICE EXAM SOLUTIONS Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of
More informationInfinite series, improper integrals, and Taylor series
Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions
More informationParametric Curves. Calculus 2 Lia Vas
Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider
More informationMat104 Fall 2002, Improper Integrals From Old Exams
Mat4 Fall 22, Improper Integrals From Old Eams For the following integrals, state whether they are convergent or divergent, and give your reasons. () (2) (3) (4) (5) converges. Break it up as 3 + 2 3 +
More informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationTo calculate the x=moment, we multiply the integrand above by x to get = )
. ( points) The region shown below is the area between the curves y = 3x and y = x. Find the center of mass of this region. 8 6 3 We must calculate the area, x-moment, and y-moment to find the center of
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationSolutions to Final Exam
Name: ID#: Solutions to Final Exam Math b Calculus, Series, Differential Equations 27 January 24 Show all of your work. Full credit may not be given for an answer alone. You may use the backs of the pages
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationOld Math 220 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 0 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Spring 05 Contents Contents General information about these exams 4 Exams from 0
More informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
3 Differentiation Rules 3.1 The Derivative of Polynomial and Exponential Functions In this section we learn how to differentiate constant functions, power functions, polynomials, and exponential functions.
More information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationPurdue University Study Guide for MA Credit Exam
Purdue University Study Guide for MA 60 Credit Exam Students who pass the credit exam will gain credit in MA60. The credit exam is a twohour long exam with 5 multiple choice questions. No books or notes
More informationAssignment 13 Assigned Mon Oct 4
Assignment 3 Assigned Mon Oct 4 We refer to the integral table in the back of the book. Section 7.5, Problem 3. I don t see this one in the table in the back of the book! But it s a very easy substitution
More informationFinal Exam SOLUTIONS MAT 131 Fall 2011
1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as
More informationMath 1272 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Exam ) This fraction appears in Problem 5 of the undated-? exam; a solution can be found in that solution set. (E) ) This integral appears in Problem 6 of the Fall 4 exam;
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More informationCalculus II Study Guide Fall 2015 Instructor: Barry McQuarrie Page 1 of 8
Calculus II Study Guide Fall 205 Instructor: Barry McQuarrie Page of 8 You should be expanding this study guide as you see fit with details and worked examples. With this extra layer of detail you will
More informationFall 2016, MA 252, Calculus II, Final Exam Preview Solutions
Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and
More informationLecture 7 - Separable Equations
Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only
More informationMath Practice Exam 2 - solutions
C Roettger, Fall 205 Math 66 - Practice Exam 2 - solutions State clearly what your result is. Show your work (in particular, integrand and limits of integrals, all substitutions, names of tests used, with
More informationCALCULUS II MATH Dr. Hyunju Ban
CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Exam 4 Review 1. Trig substitution
More information