Integrated Calculus II Exam 2 Solutions 3/28/3
|
|
- Sophie Harris
- 5 years ago
- Views:
Transcription
1 Integrated Calculus II Exam 2 Solutions /28/ Question 1 Solve the following differential equation, with the initial condition y() = 2: dy = (y 1)2 t 2. Plot the solution and discuss its behavior as a function of time. This equation is separable. We separate, writing it as an equation of differentials: dy (y 1) = 2 t2. Now we integrate both sides: dy (y 1) = 2 t 2, (y 1) 1 = t + C. Now we put in the initial conditions that y = 2, when t =, determine C and solve for y: (2 1) 1 = + C, C = 1, (y 1) 1 = t + 1, y 1) = t, y = t = 2 + t 1 + t. We have y = t2 (1+t ) 2 and y = 6t(1 2t ) (1+2t ), which gives a critical point at (, 2), no local maxima or minima and inflection points at (, 2) and at (2 1, 5 ).
2 The solution curve is monotonically decreasing in t. As t, the solution curve approaches the horizontal asymptote y = 1, from above. As t 1 +, the solution curve approaches the vertical asymptote x = 1, with y going to infinity. The curve is concave up in the intervals ( 1, ] and [2 1, ). The curve is concave down in the interval [, 2 1 ]. Question 2 A spring of rest length cm is stretched to a length of 5cm by a force of 5 Newtons. How much work is done in stretching the spring from a length of 4cm to a length of one meter? Let x denote the extension of the spring in meters from its rest length. By Hooke s law, if a force F extends the spring an amount x, then we have F = kx, where k is the spring constant. The spring is stretched by a force of F = 5 Newtons a distance x = 5 =.2 meters. So we have: 5 = k(.2), or k = 5 = 25 in units of 1.2 Newtons per meter. So Hooke s law for the spring now reads: F = 25x. Then the work W Joules done in stretching the spring from a length of 4cm to a length of 1 meter, which is the work done in streching the spring from an extension of x = (4 ) =.1m to an extension of (1 ) =.7m is: 1 1 W =.7.1 F dx = xdx = [125x 2 ].7.1 = 125(.49.1) = =
3 Question A region R is bounded by the curves y = x 2 6x + 5 and y = 2x 2. Sketch the region R. The region R is bounded below by the concave up parabola, which has vertex at (, 4) and crosses the x-axis at (1, ) and (5, and above by the straight line of slope 2 through the point (1, ). The two curves meet where both equations hold, so where x 2 6x + 5 = 2x 2, or x 2 8x + 7 =, or (x 1)(x 7) =, so at the points (1, ) and (7, 12). Find the area of the region R. We use a vertical strip at position x of wih dx and height h = 2x 2 (x 2 6x + 5) = 8x x 2 7. So the strip area is da = hdx = (8x x 2 7)dx. To cover the region R, x ranges over the interval [1, 7]. Then the total area A of R is: A = x=7 x=1 da = 7 1 (8x x 2 7)dx = [4x 2 x 7x]7 1 = 4(49 1) 1 (4 1) 7(7 1) = = 6. If the region R is rotated about the y-axis, what is the volume of the solid it traces out? We rotate the vertical strip about the y-axis, giving a cylindrical shell of wih dx, height h = 8x x 2 7 and radius x, so it has volume: dv = 2πxdA = 2πx(8x x 2 7)dx = 2π(8x 2 x 7x)dx. To cover the region R, x ranges over the interval [1, 7]. Then the total volume V of the solid is: V = 2π 7 1 (8x 2 x 7x)dx = 2π[ 8x x4 4 7x2 2 ]7 1 = 2π(7 )( ) 2π( ) = 2π(4)( 1 12 )(2 21 6) 2π( 1 )(2 42) 12 = π 6 (4(5)+1) = π 6 (1715+1) = π (1728) = 288π =
4 Question 4 A circuit has connected in series: A resistance of 1Ω. An inductance of 2H. A voltage source 5t Volts at time t seconds. Given that initially there is no current flowing, find the current at time t. The circuit will burn up when there are 2 Amperes flowing in it. Approximately when will that be? The differential equation governing the current I Amps at time t seconds is: H di + RI = E. Here H Henry is the inductance, E Volts is the voltage and R Ohms is the resistance. Putting in the numbers, the differential equation is: 2 di + 1I = 5t. Dividing by 2, the differential equation is put in standard linear form: di + 5I = 25t. We have P (t) = 5 and Q(t) = 25t, so the integrating factor is J(t) = e R P (t) = e R 5 = e 5t. The equation is now: d (IJ) = QJ, or: d (IJ) = 25te5t, Integrating both sides, we get: IJ = 25te 5t. 4
5 We calculate the integral on the right-hand side by integration by parts: 25te 5t = udv, u = t, du =, dv = 25e 5t, v = Ie 5t = 25te 5t = = 5te 5t dv = udv = uv 5e 5t = 5te 5t e 5t + C = e 5t (5t 1) + C. 25e 5t = 5e 5t, vdu Putting in the initial conditions, I =, when t =, we get: = 1 + C, C = 1. Ie 5t = e 5t (5t 1) + 1. Multiplying both sides of the equation by e 5t gives the solution for the current as: I = 5t 1 + e 5t. As t increases, the exponential term falls rapidly until it is very close to zero and the term 5t 1 dominates. In particular, we see that I is approximately 2, when 5t 1 = 2, so t = 21 = 4.2 seconds. 5 In fact if we use Maple to solve the equation 5t 1+e 5t = 2, numerically, we get: t =
6 Question 5 Consider the integral J = 5 ln(1 + x)dx. Determine the trapezoidal rule approximation, T 1 to J with ten intervals. The data for T 1, with f(x) = ln(1 + x) and the weights w are as follows: x f ln( ) ln(2) ln( 5) ln() ln( 7) ln(4) ln( 9) ln(5) ln( 11) ln(6) w wf 2 ln( ) 2 ln(2) 2 ln( 5) 2 ln() 2 ln( 7) 2 ln(4) 2 ln( 9) 2 ln(5) 2 ln( 11) ln(6) We add the weighted data and multiply by one half of the interval wih 1 2 to give T 1 : T 1 = 1 ( 4 ln ( 9 ) 4 )(4)(25 4 )(9)(49 4 )(16)(81 4 )(25)(121 4 )(6) = 1 ( ) 9 4 ln = 1 4 ln( ) 8 = Is the estimate T 1 too high or too low? Explain your answer. For f(x) = ln(1 + x), we have f (x) = 1 = (1 + 1+x x) 1 and f (x) = (1 + x) 2 <, so the graph of the function f is everywhere concave down. So the trapezoids of the trapezoidal rule lie below the curve, so T 1 is an underestimate of the true integral. Determine the Simpson s rule approximation, S 1 to J with ten intervals. The data for S 1, with f(x) = ln(1 + x) and the weights w are as follows: x f ln( ) ln(2) ln( 5) ln() ln( 7) ln(4) ln( 9) ln(5) ln( 11) ln(6) w wf 4 ln( ) 2 ln(2) 4 ln( 5) 2 ln() 4 ln( 7) 2 ln(4) 4 ln( 9) 2 ln(5) 4 ln( 11) ln(6) 6
7 We add the weighted data and multiply by one third of the interval wih 1 2 to give S 1 : S 1 = 1 ( 6 ln ( 81 ) 16 )(4)( )(9)( )(16)( )(25)( )(6) = 1 6 ln ( ) = 1 ( ln 8192 = Use appropriate error formulas to estimate the errors in T 1 and S 1. The error formula E T for the trapezoidal rule is: E T = K 2(b a) 12n 2. The error formula E S for Simpson s rule is: E S = K 4(b a) 5 18n 4. Here n is the number of intervals, [a, b] is the interval I of integration, K 2 is the maximum of f (x) on the interval I and K 4 is the maximum of f (x) on the interval I. For the given integral approximations, we have n = 1, a = and b = 5. We have f (x) = (1 + x) 2, so f (x) = (1 + x) 2, which decreases on the interval [, 5], with its maximum value of 1 at x =. So K 2 = 1. Also, we have f (x) = 2(1 + x), so f (x) = 6(1 + x) 4 and f (x) = 6(1 + x) 4, which decreases on the interval [, 5], with its maximum value of 6 at x =. So K 4 = 6. So we get: 1(5 ) E T = 12(1 2 ) = = Also we get: E S = 6(5 )5 18(1 4 ) = 1 (2 5 ) = )
8 Question 6 A storage tank filled with water has a V -shaped cross-section. The tank is 8 meters long. The cross-section is an isosceles triangle with one vertex at a depth of 2 meters and the other two level with the surface of the water and meters apart. How much work is done in removing the water from the tank, through a pipe level with the top of the tank? We consider the work dw done in removing a horizontal layer of water of thickness dx, at a height of x above the bottom of the tank. The length of the layer is 8 meters. If its wih is y, then by similar triangles, we have: Then we have: y x = 2 = 2, y = x 2. dw = (2 x)dw, where dw is the weight of the layer, since 2 x is the distance moved by the layer against gravity. dw = ρgdv, where ρ is the density of water, g is the acceleration due to gravity and dv is the volume of the layer. dv = x (8)(dx) = 12xdx. 2 So dw = (2 x)ρg(12)xdx = k(2x x 2 )dx, k = 12ρg. Then the total work W Joules done in removing the water from the tank is: W = x=2 x= dw = 2 k(2x x 2 )dx = k(1x 2 x ]2 = k(4 8 ) = k 4 = 12ρg 4 = 16ρg = 16(9.81) = = (1 9 ). 8
9 Question 7 A lamina of density kilos/square meter is bounded by the curves y = 6 x 2 and y = x (units are SI). Sketch the lamina. The two curves bounding the lamina are parabolas, one concave up and the other concave down. They meet where both equations hold at once, so where 6 x 2 = x 2 + 1, or 2x 2 = 5, so where x 2 = 25 and x = ±5. Then the meeting points are (±5, 5). Find the centroid of the lamina. We see that the lamina is symmetrical about the y-axis and about the line y = 5. So the centroid is at (, 5). Find the moments of the lamina about the x and y-axes. The moment about the y-axis is zero, since the center of mass lies on the axis. The moment about the x-axis is 5m = 5ρA = 5()A = 15A, where m is the mass of the lamina, ρ is its density and A is its area. We have: A = 2 5 = 2 (6 x 2 (x 2 + 1))dx 5 (5 2x 2 )dx = 2[5x 2x ]5 = 2(25 25 ) = 1. Then the mass is m = ( 1 ) = 1 kilos and the moment about the x-axis is 5(1) = 5 kilo meters. Find the volume of the surface of revolution generated by rotating the lamina around the x-axis. By Pappus Theorem, we have for the volume V : V = 2πxA = 2π5 1 = 7π 9 =
10 Question 8 Consider the following parametrized curve: x = t t, y = t 2. Sketch the part AB of the curve, between the points A = (, ) and B = (2, ) on the curve. The point A has y = t 2 =, so t =. The point B has y = t 2 =, so t 2 = 1, so t = ±1. When t = 1, we get x = 2, which is incorrect for B. When t = 1, we get x = 2, which gives B correctly. So the t interval is [, 1]. If we plot the curve for this interval we find a simple curve, concave up, stretching from A to B, starting with slope zero at A and turning to vertical as it reaches B. Find the length along the curve from A to B. We have: X = [t t, t 2 ], V = dx = [ t2, 6t] = [1 t 2, 2t], ds = V = (1 t 2 ) 2 + (2t) 2 = 1 2t 2 + t 4 + 4t 2 = 1 + 2t 2 + t 4 = (1 + t 2 ). So the length s along the curve from A to B is: s = t=1 t= ds = 1 ds = 1 = [t + t ] 1 = + 1 = 4. (1 + t 2 ) 1
11 Write an integral for the surface area of the curved surface, obtained by rotating the curve AB about the x-axis. At position t we draw a vertical frustum of a cone with slant wih ds and radius y. The frustum has surface area ds = 2πyds. So the total required area S is: B S = 2π yds = 2π A 1 = 18π y ds = 2π 1 1 (t 2 + t 4 ) (t 2 )((1 + t 2 )) = 18π[ t + t5 5 ]1 = 18π( ) = 144π 15 = 48π 5 = Write an integral for the surface area of the curved surface, obtained by rotating the curve AB about the y-axis. At position t we draw a horizontal frustum of a cone with slant wih ds and radius x. The frustum has surface area ds = 2πxds. So the total required area S is: B S = 2π xds = 2π A = 6π 1 1 = 6π 1 x ds = 2π 1 (t + t t t 5 ) (t + 2t t 5 ) (t t )((1 + t 2 )) = 6π[ t2 2 + t4 2 t6 6 ]1 = π(9 + 1) = 11π =
Integrated Calculus II Exam 1 Solutions 2/6/4
Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (
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 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 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 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 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 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 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 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 informationChapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral Section 6.1 More on Area a. Representative Rectangle b. Vertical Separation c. Example d. Integration with Respect to y e. Example Section 6.2 Volume by Parallel
More informationChapter 6: Applications of Integration
Chapter 6: Applications of Integration Section 6.3 Volumes by Cylindrical Shells Sec. 6.3: Volumes: Cylindrical Shell Method Cylindrical Shell Method dv = 2πrh thickness V = න a b 2πrh thickness Thickness
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 informationMA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.
MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.
More information2) ( 8 points) The point 1/4 of the way from (1, 3, 1) and (7, 9, 9) is
MATH 6 FALL 6 FIRST EXAM SEPTEMBER 8, 6 SOLUTIONS ) ( points) The center and the radius of the sphere given by x + y + z = x + 3y are A) Center (, 3/, ) and radius 3/ B) Center (, 3/, ) and radius 3/ C)
More informationAPPLICATIONS OF INTEGRATION
6 APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6.4 Work In this section, we will learn about: Applying integration to calculate the amount of work done in performing a certain physical task.
More informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
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 informationd` = 1+( dy , which is part of the cone.
7.5 Surface area When we did areas, the basic slices were rectangles, with A = h x or h y. When we did volumes of revolution, the basic slices came from revolving rectangles around an axis. Depending on
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationFor the intersections: cos x = 0 or sin x = 1 2
Chapter 6 Set-up examples The purpose of this document is to demonstrate the work that will be required if you are asked to set-up integrals on an exam and/or quiz.. Areas () Set up, do not evaluate, any
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 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }
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 7.1 Area of a Region Between Two Curves 7.2 Volume: The Disk Method 7.3 Volume: The Shell Method 7.5 Work 7.6 Moments, Centers of Mass, and Centroids 7.7 Fluid Pressure
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 informationExam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.
Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x +
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 informationAP CALCULUS AB Study Guide for Midterm Exam 2017
AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed
More informationA = (cosh x sinh x) dx = (sinh x cosh x) = sinh1 cosh1 sinh 0 + cosh 0 =
Calculus 7 Review Consider the region between curves y= cosh, y= sinh, =, =.. Find the area of the region. e + e e e Solution. Recall that cosh = and sinh =, whence sinh cosh. Therefore the area is given
More informationMA Spring 2013 Lecture Topics
LECTURE 1 Chapter 12.1 Coordinate Systems Chapter 12.2 Vectors MA 16200 Spring 2013 Lecture Topics Let a,b,c,d be constants. 1. Describe a right hand rectangular coordinate system. Plot point (a,b,c) inn
More informationCHAPTER 3 APPLICATIONS OF THE DERIVATIVE
CHAPTER 3 APPLICATIONS OF THE DERIVATIVE 3.1 Maxima and Minima Extreme Values 1. Does f(x) have a maximum or minimum value on S? 2. If it does have a maximum or a minimum, where are they attained? 3. If
More informationCalculus I
Calculus I 978-1-63545-038-5 To learn more about all our offerings Visit Knewton.com/highered Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Gilbert Strang, Massachusetts Institute
More informationSingle Variable Calculus, Early Transcendentals
Single Variable Calculus, Early Transcendentals 978-1-63545-100-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax
More informationAP Calculus BC Syllabus Course Overview
AP Calculus BC Syllabus Course Overview Textbook Anton, Bivens, and Davis. Calculus: Early Transcendentals, Combined version with Wiley PLUS. 9 th edition. Hoboken, NJ: John Wiley & Sons, Inc. 2009. Course
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Solutions to Assignment 7.6. sin. sin
Arkansas Tech University MATH 94: Calculus II Dr. Marcel B. Finan Solutions to Assignment 7.6 Exercise We have [ 5x dx = 5 ] = 4.5 ft lb x Exercise We have ( π cos x dx = [ ( π ] sin π x = J. From 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 informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationBozeman Public Schools Mathematics Curriculum Calculus
Bozeman Public Schools Mathematics Curriculum Calculus Process Standards: Throughout all content standards described below, students use appropriate technology and engage in the mathematical processes
More informationChapter 7 Applications of Integration
Chapter 7 Applications of Integration 7.1 Area of a Region Between Two Curves 7.2 Volume: The Disk Method 7.3 Volume: The Shell Method 7.4 Arc Length and Surfaces of Revolution 7.5 Work 7.6 Moments, Centers
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 informationMATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS
MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if
More information( afa, ( )) [ 12, ]. Math 226 Notes Section 7.4 ARC LENGTH AND SURFACES OF REVOLUTION
Math 6 Notes Section 7.4 ARC LENGTH AND SURFACES OF REVOLUTION A curve is rectifiable if it has a finite arc length. It is sufficient that f be continuous on [ab, ] in order for f to be rectifiable between
More informationUniversidad Carlos III de Madrid
Universidad Carlos III de Madrid Question 1 3 4 5 6 Total Grade Economics Department Final Exam, Mathematics I January 14, 011 Total length: hours. SURNAME: NAME: DNI: Degree: Group: { 4(x 4) if 3 x 4
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationDepartment of Mathematical 1 Limits. 1.1 Basic Factoring Example. x 1 x 2 1. lim
Contents 1 Limits 2 1.1 Basic Factoring Example...................................... 2 1.2 One-Sided Limit........................................... 3 1.3 Squeeze Theorem..........................................
More informationMath 76 Practice Problems for Midterm II Solutions
Math 76 Practice Problems for Midterm II Solutions 6.4-8. DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You may expect to
More informationAP Calculus Free-Response Questions 1969-present AB
AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions
More informationAP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40
Extreme Values in an Interval AP Calculus BC 1. The absolute maximum value of x = f ( x) x x 1 on the closed interval, 4 occurs at A) 4 B) C) 1 D) 0 E). The maximum acceleration attained on the interval
More informationLimits and Continuity. 2 lim. x x x 3. lim x. lim. sinq. 5. Find the horizontal asymptote (s) of. Summer Packet AP Calculus BC Page 4
Limits and Continuity t+ 1. lim t - t + 4. lim x x x x + - 9-18 x-. lim x 0 4-x- x 4. sinq lim - q q 5. Find the horizontal asymptote (s) of 7x-18 f ( x) = x+ 8 Summer Packet AP Calculus BC Page 4 6. x
More information(a) The best linear approximation of f at x = 2 is given by the formula. L(x) = f(2) + f (2)(x 2). f(2) = ln(2/2) = ln(1) = 0, f (2) = 1 2.
Math 180 Written Homework Assignment #8 Due Tuesday, November 11th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,
More informationApplied Calculus I Practice Final Exam Solution Notes
AMS 5 (Fall, 2009). Solve for x: 0 3 2x = 3 (.2) x Taking the natural log of both sides, we get Applied Calculus I Practice Final Exam Solution Notes Joe Mitchell ln 0 + 2xln 3 = ln 3 + xln.2 x(2ln 3 ln.2)
More informationf (x) = 2x x = 2x2 + 4x 6 x 0 = 2x 2 + 4x 6 = 2(x + 3)(x 1) x = 3 or x = 1.
F16 MATH 15 Test November, 016 NAME: SOLUTIONS CRN: Use only methods from class. You must show work to receive credit. When using a theorem given in class, cite the theorem. Reminder: Calculators are not
More informationCalculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016
Name: Class: Date: Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Short Answer 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required.
More informationHonors Calculus Quiz 9 Solutions 12/2/5
Honors Calculus Quiz Solutions //5 Question Find the centroid of the region R bounded by the curves 0y y + x and y 0y + 50 x Also determine the volumes of revolution of the region R about the coordinate
More informationAB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2
AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6
More informationAP Calculus BC Syllabus
AP Calculus BC Syllabus Course Overview and Philosophy This course is designed to be the equivalent of a college-level course in single variable calculus. The primary textbook is Calculus, 7 th edition,
More informationSample 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 information1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.
. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r
More information5 Distributed Forces 5.1 Introduction
5 Distributed Forces 5.1 Introduction - Concentrated forces are models. These forces do not exist in the exact sense. - Every external force applied to a body is distributed over a finite contact area.
More informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More information(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)
1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of
More informationAP Physics C Mechanics Objectives
AP Physics C Mechanics Objectives I. KINEMATICS A. Motion in One Dimension 1. The relationships among position, velocity and acceleration a. Given a graph of position vs. time, identify or sketch a graph
More informationMatrix Theory and Differential Equations Homework 2 Solutions, due 9/7/6
Matrix Theory and Differential Equations Homework Solutions, due 9/7/6 Question 1 Consider the differential equation = x y +. Plot the slope field for the differential equation. In particular plot all
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
More informationMATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010)
Course Prerequisites MATH 100 and MATH 180 Learning Objectives Session 2010W Term 1 (Sep Dec 2010) As a prerequisite to this course, students are required to have a reasonable mastery of precalculus mathematics
More informationENGI Multiple Integration Page 8-01
ENGI 345 8. Multiple Integration Page 8-01 8. Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple integration include
More informationMath3A Exam #02 Solution Fall 2017
Math3A Exam #02 Solution Fall 2017 1. Use the limit definition of the derivative to find f (x) given f ( x) x. 3 2. Use the local linear approximation for f x x at x0 8 to approximate 3 8.1 and write your
More informationSecond Midterm Exam Name: Practice Problems Septmber 28, 2015
Math 110 4. Treibergs Second Midterm Exam Name: Practice Problems Septmber 8, 015 1. Use the limit definition of derivative to compute the derivative of f(x = 1 at x = a. 1 + x Inserting the function into
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationMath 107 Fall 2007 Course Update and Cumulative Homework List
Math 107 Fall 2007 Course Update and Cumulative Homework List Date: 8/27 Sections: 5.4 Log: Review of course policies. The mean value theorem for definite integrals. The fundamental theorem of calculus,
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 informationQuick Review Sheet for A.P. Calculus Exam
Quick Review Sheet for A.P. Calculus Exam Name AP Calculus AB/BC Limits Date Period 1. Definition: 2. Steps in Evaluating Limits: - Substitute, Factor, and Simplify 3. Limits as x approaches infinity If
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationNO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:
AP Calculus AB PRACTICE MIDTERM EXAM Read each choice carefully and find the best answer. Your midterm exam will be made up of 8 of these questions. I reserve the right to change numbers and answers on
More informationMath 75B Practice Midterm III Solutions Chapter 6 (Stewart) Multiple Choice. Circle the letter of the best answer.
Math 75B Practice Midterm III Solutions Chapter 6 Stewart) English system formulas: Metric system formulas: ft. = in. F = m a 58 ft. = mi. g = 9.8 m/s 6 oz. = lb. cm = m Weight of water: ω = 6.5 lb./ft.
More informationSolution to Homework 2
Solution to Homework. Substitution and Nonexact Differential Equation Made Exact) [0] Solve dy dx = ey + 3e x+y, y0) = 0. Let u := e x, v = e y, and hence dy = v + 3uv) dx, du = u)dx, dv = v)dy = u)dv
More informationAP Calculus BC Chapter 4 AP Exam Problems. Answers
AP Calculus BC Chapter 4 AP Exam Problems Answers. A 988 AB # 48%. D 998 AB #4 5%. E 998 BC # % 5. C 99 AB # % 6. B 998 AB #80 48% 7. C 99 AB #7 65% 8. C 998 AB # 69% 9. B 99 BC # 75% 0. C 998 BC # 80%.
More informationFinal exam practice UCLA: Math 3B, Fall 2016
Instructor: Noah White Date: Monday, November 28, 2016 Version: practice. Final exam practice UCLA: Math 3B, Fall 2016 This exam has 7 questions, for a total of 84 points. Please print your working and
More informationAnswer Key. Calculus I Math 141 Fall 2003 Professor Ben Richert. Exam 2
Answer Key Calculus I Math 141 Fall 2003 Professor Ben Richert Exam 2 November 18, 2003 Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem
More information+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent
dx = (A) 3 sin(3x ) + C 1. cos ( 3x) 1 (B) sin(3x ) + C 3 1 (C) sin(3x ) + C 3 (D) sin( 3x ) + C (E) 3 sin(3x ) + C 6 3 2x + 6x 2. lim 5 3 x 0 4x + 3x (A) 0 1 (B) 2 (C) 1 (D) 2 (E) Nonexistent is 2 x 3x
More informationAP Calculus AB Syllabus
AP Calculus AB Syllabus Course Overview and Philosophy This course is designed to be the equivalent of a college-level course in single variable calculus. The primary textbook is Calculus of a Single Variable,
More informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More informationUNIVERSITY OF BOLTON WESTERN INTERNATIONAL COLLEGE FZE BENG (HONS) CIVIL ENGINEERING SEMESTER TWO EXAMINATION 2015/2016
OCD74 UNIVERSITY OF BOLTON WESTERN INTERNATIONAL COLLEGE FZE BENG (HONS) CIVIL ENGINEERING SEMESTER TWO EXAMINATION 015/016 MATHEMATICS AND STRUCTURAL DESIGN MODULE NO: CIE401 Date: Saturday 8 May 016
More informationPuxi High School Examinations Semester 1, AP Calculus (BC) Part 1. Wednesday, December 16 th, :45 pm 3:15 pm.
Puxi High School Examinations Semester 1, 2009 2010 AP Calculus (BC) Part 1 Wednesday, December 16 th, 2009 12:45 pm 3:15 pm Time: 45 minutes Teacher: Mr. Surowski Testing Site: HS Gymnasium Student Name:
More informationMagnetic Fields; Sources of Magnetic Field
This test covers magnetic fields, magnetic forces on charged particles and current-carrying wires, the Hall effect, the Biot-Savart Law, Ampère s Law, and the magnetic fields of current-carrying loops
More informationMAT 132 Midterm 1 Spring 2017
MAT Midterm Spring 7 Name: ID: Problem 5 6 7 8 Total ( pts) ( pts) ( pts) ( pts) ( pts) ( pts) (5 pts) (5 pts) ( pts) Score Instructions: () Fill in your name and Stony Brook ID number at the top of this
More informationAP Calculus AB Section 7.3: Other Differential Equations for Real-World Applications Period: Date: Practice Exercises Score: / 5 Points
AP Calculus AB Name: Section 7.3: Other Differential Equations for Real-World Applications Period: Date: Practice Exercises Score: / 5 Points 1. Sweepstakes Problem I: You have just won a national sweepstakes!
More informationDepartment of Mathematical x 1 x 2 1
Contents Limits. Basic Factoring Eample....................................... One-Sided Limit........................................... 3.3 Squeeze Theorem.......................................... 4.4
More informationReview: Exam Material to be covered: 6.1, 6.2, 6.3, 6.5 plus review of u, du substitution.
Review: Exam. Goals for this portion of the course: Be able to compute the area between curves, the volume of solids of revolution, and understand the mean value of a function. We had three basic volumes:
More informationd = k where k is a constant of proportionality equal to the gradient.
VARIATION In Physics and Chemistry there are many laws where one quantity varies in some way with another quantity. We will be studying three types of variation direct, inverse and joint.. DIRECT VARIATION
More information0.1 Work. W net = T = T f T i,
.1 Work Contrary to everyday usage, the term work has a very specific meaning in physics. In physics, work is related to the transfer of energy by forces. There are essentially two complementary ways to
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 informationMath 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it.
Math 80, Final Eam, Spring 008 Problem Solution. For each of the following limits, determine whether the limit eists and, if so, evaluate it. + (a) lim 0 (b) lim ( ) 3 (c) lim Solution: (a) Upon substituting
More informationBonus Homework and Exam Review - Math 141, Frank Thorne Due Friday, December 9 at the start of the final exam.
Bonus Homework and Exam Review - Math 141, Frank Thorne (thornef@mailbox.sc.edu) Due Friday, December 9 at the start of the final exam. It is strongly recommended that you do as many of these problems
More informationHonors Calculus Homework 1, due 9/8/5
Honors Calculus Homework 1, due 9/8/5 Question 1 Calculate the derivatives of the following functions: p(x) = x 4 3x 3 + 5 x 4x 1 3 + 23 q(x) = (1 + x)(1 + x 2 )(1 + x 3 )(1 + x 4 ). r(t) = (1 + t)(1 +
More informationQuestion. [The volume of a cone of radius r and height h is 1 3 πr2 h and the curved surface area is πrl where l is the slant height of the cone.
Q1 An experiment is conducted using the conical filter which is held with its axis vertical as shown. The filter has a radius of 10cm and semi-vertical angle 30. Chemical solution flows from the filter
More informationProblem Out of Score Problem Out of Score Total 45
Midterm Exam #1 Math 11, Section 5 January 3, 15 Duration: 5 minutes Name: Student Number: Do not open this test until instructed to do so! This exam should have 8 pages, including this cover sheet. No
More informationReview for the Final Exam
Calculus Lia Vas. Integrals. Evaluate the following integrals. (a) ( x 4 x 2 ) dx (b) (2 3 x + x2 4 ) dx (c) (3x + 5) 6 dx (d) x 2 dx x 3 + (e) x 9x 2 dx (f) x dx x 2 (g) xe x2 + dx (h) 2 3x+ dx (i) x
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationFinal practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90
Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x
More information