AP Calculus Testbank (Chapter 7) (Mr. Surowski)

Size: px
Start display at page:

Download "AP Calculus Testbank (Chapter 7) (Mr. Surowski)"

Transcription

1 AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. Multiple-Choice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved about the -ais, then the volume of the resulting solid would best be computed b the method of (A) disks/washers (B) shells (C) known cross sections.. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved about the -ais, then the volume of the solid of revolution is given b (A) π (B) π (C) π (D) π (E) π f() d f() d + f() d f() d f() d

2 3. A parabola is drawn having focus (, ) and directri = 4. The definite integral representing the arc length of that portion of the parabola on or above the -ais is given b (A) 4 d (B) 3 4 d = 4 (C) (D) (E) d 4 + d d 4 + F (, ) 4. Consider the solid of revolution formed b revolving the area bounded b the curve = /, the -ais, the line = and the line = a, (a > ) about the -ais. The integral representing the volume of this solid is (A) π (B) π (C) π (D) π (E) π d d d d d = = a

3 5. Consider the surface of revolution formed b revolving the the curve = /, a about the -ais. Then the surface area is given b the definite integral (A) π (B) π (C) π (D) π (E) π d d + 4 d 3 ( + ) + 4 d d 6. Which of the following integrals correctl gives the area of the region consisting of all points above the -ais and below the curve = 8 +? (A) (B) (C) (D) (E) 4 4 ( 8) d (8 + ) d (8 + ) d ( 8) d (8 + ) d.

4 7. A solid is generated with the region in the first quadrant bounded b the graph of = + sin, the line = π, the -ais, and the -ais is revolved about the -ais. Its volume is found b evaluating which of the following integrals? (A) π (B) π (C) π (D) π (E) π π π π ( + sin 4 ) d ( + sin ) d ( + sin 4 ) d ( + sin ) d ( + sin ) d. 8. The volume generated b revolving about the -ais the region above the curve = 3, below the line =, and between = and = is (A) π 4 (B).43π (C) π 7 (D).643π (E) 6π 7 9. Find the distance traveled (to three decimal places) from t = to t = 5 seconds, for a particle whose velocit is given b v(t) = t + ln t. (A) 6. (B).69 (C) 6.47 (D).8 (E) Find the area of the region bounded b the parabolas = and = 6 (A) 9 (B) 7 (C) 6 (D) 9 (E) 8

5 . What is the area of the region in the first quadrant enclosed b the graph of = e 4 and the line =.5? (A).4 (B).56 (C).48 (D).3 (E).349. The base of a solid S is the region enclosed b the graph of =, the -ais, and the -ais. If the cross-sections of S perpendicular to the -ais are semicircles, then the volume of S is (A) 5π 3 (B) π 3 (C) 5π 3 (D) 5π 3 (E) 45π 3 3. The volume of the solid that results when the area between the curve = e and the line =, from = to =, is revolved around the -ais is (A) π(e 4 e ) (B) π (e4 e ) (C) π (e e) (D) π(e e) (E) πe 4. What is the volume of the solid generated b rotating about the -ais the region enclosed b = sin and the -ais, from = to = π? (A) π (B) π (C) 4π (D) (E) 4

6 Part II. Free-Response Questions. Given the velocit function v(t) = t 4 + t, t, () =, (a) determine the terminal position of the particle, and (b) determine the total distance traveled b the particle. The terminal position is given b () = () + v(t) dt = ( (t ) dt = 4 + t ln(4 + t ) ) tan (/) = ln π 8. The total distance traveled b the particle is given b distance = = = = = ( speed dt v(t) dt t dt 4 + 4t ( t) dt t (t ) dt 4 + t tan (/) ) ln(4 + t ) + = π 8 ln(5/4) + ln(8/5) π 8 + tan (/) = ln(3/5) + tan (/) ( ln(4 + t ) ) tan (/). Given the velocit function v(t) = + sin t, t π/6, with () =, (a) determine the terminal position of the particle, and (b) determine the total distance travelled b the particle. (a) The terminal position is (π/6) = ()+ π/6 (+ sin t) dt = (t cos t) π/6 = π

7 (b) Since v(t) on t π/6, we see that velocit and speed are the same on this interval. Therefore total distance = π/6 ( + sin t) dt = π Given the velocit function v(t) = t cos πt, t.5, with () =, (a) determine the terminal position of the particle, and (b) determine the total distance travelled b the particle. (a) The terminal position is (.5) = () +.5 t cos πt dt = =.5 π π. ( t π sin πt + ).5 cos πt π (b) Since v(t) on the interval π t 3π, we have total distance = =.5 speed dt = ( t π sin πt + ).5 cos πt π.5 = π π + 3 π + π = 5 π π.5 t cos πt dt t cos πt dt.5 ( t π sin πt + ).5 cos πt π.5 4. The velocit function of a particle has the graph depicted below. Find the total distance travelled b the particle over the first five seconds. v (cm/sec) v = v(t) t (sec) The total distance traveled is just the total area under the velocit graph. Using simple geometr one discovers that the total distance is cm.

8 5. Suppose that a particle is initiall at rest at the origin, but at time t = a force is applied to the particle which results in an acceleration of + cm/sec. Locate this particle on the -ais after 5 seconds. This is simple: after two simple integrations one has (t) = 5t, so the particle occupies position = 75 cm after 5 seconds. 6. Water is flowing from a faucet into a one-litre bottle at a rate of r(t) = te t l/min. After minutes the water is turned up to a constant rate of of. l/min. (a) Graph the function r = r(t) depicting the rate of flow of water..3 r (l/min).. t (min) 3 4 Equation : =e^( )( )/( ) Equation : =(.)( )/( ) (b) Will the bottle be full after 4 minutes? The total amount of water flowing into the bottle over the first four minutes is ( A = r(t) dt = te t dt+ (.) dt = te t ) 4 e t.4 = e 4 4 e l. Therefore, the bottle will 4 not be full. (Alternativel, ou could have just computed te t dt numericall on our calculator, without having to resort to integration b parts.) +

9 7. Suppose that a particle is resting at the origin and that a force of F = F (t) cm/sec, t, is applied to the particle over the interval t <. Assuming that F (t) > over this interval, compute lim (t) and justif our answer. Since a positive t force is applied, the particle will eperience positive acceleration. This will move the particle off to infinit as t. That is to sa, lim (t) = + t 8. Graph the region bounded b the curves = and + = 3 and compute its area. The points of intersection occur where = ±. Therefore the area between the curves is given b the integration along the -ais (note how smmetr is being used): Area = [(3 ) ] d = (3 3 ) = Graph the region bounded b the curves Equation = : +^=3 + 3 and = 3 5, ( ) and compute itsequation area. 3: =^ Note that the points of intersection of these curves are = and = ±. As we re onl interested in, the area involved is Area = [( +3) ( 3 5)] d = Equation : = ²+3 Equation 3: =³ ² 5 ( 3 +8) d = 4 +4 = 8.

10 . Set up an integral (without evaluating it) that will compute the area of the region 9 + 4,. Whether we integrate along the - or -ais is immaterial. We ll set up both, making full use of smmetr: Area = d = 6 4 d. (Note that the common value is 6π.) Equation : ²/9+^/4

11 . Consider the region bounded b the curve = / p, =, = a, and the -ais. (a) Compute the volume of the solid of revolution obtained b revolving the above region about the -ais. (b) If V (a) represents the volume given in part (a) above, compute lim V (a). a (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p.. Consider the region bounded b the curve = / p, =, = a, and the -ais. (a) Compute the volume of the solid of revolution obtained b revolving the above region about the -ais. (b) If V (a) represents the volume given in part (a) above, compute lim V (a). a (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p.

12 3. Consider the surface generated b revolving the curve = / p, a, about the -ais. (a) Epress the area of the above surface as an integral (ou probabl won t be able to evaluate this integral). (b) If S(a) represents the area given in part (a) above, compute lim a S(a). (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p. 4. The region below is revolved about the -ais to form a solid of revolution. Find the volume of this solid. 4 = /4 - / = / 5. A solid object has a flat base formed b the region enclosed b the parabola with focus having coordinates (, ) and directri = 4 and b the -ais. Each cross section is an equilateral triangle perpendicular to the base and parallel to the directri. Compute the volume of this object.

13 6. Compute the length of that section of the curve = 4 /4 + /8 that joins (3/8, ) to the point (9/3, ). 7. Consider the solid of revolution formed b revolving the area bounded b the curve = /, the -ais, the line = and the line = a about the -ais. Let V (a) represent this volume and compute lim a V (a). 8. If S(a) represents the surface of revolution of problem 5, compute lim a S(a). 9. Use integral calculus to show that the volume of a right circular cone of height h and base area A is 3 Ah.. Suppose that a metal chain weighing newtons/m is hanging over a building. Assuming that the building is 3 m tall, and that the chain is just touching the ground, what is the total work required to pull the chain onto the top of the building?. Suppose that an object rests at the point = on the -ais. We then start pushing this bo in the positive direction, giving the bo a speed of e t/ m/sec. Assume that there is a force due to friction, the magnitude of which is / the speed of the bo. Find the total work needed to push the bo for seconds.. Suppose that a large clindrical drum of height meters and radius 3 meters is full of a fluid whose weight densit is, N/m 3. Find the total force on the side of the clindrical drum. (Recall: the fluid pressure at a depth h is p = wh, where w is the weight-densit in this case, N/m 3.)

14 3. Suppose that we have a lake full of fish the weights of which are modeled b a normal distribution with mean.77 kg and standard deviation of. kg. Epress the probabilit as an integral, written as eplicitel as possible that a randoml-selected fish will have its weight somewhere between.5 kg and. kg. What is the probabilit that a randoml selected fish will have its weight somewhere between.65 kg and.89 kg? 4. Assume that there is a heav bo sitting outside on the pavement. We are going to move this bo a total of feet b sliding it along the pavement. The relevant force here is that of friction, which we shall assume is proportional to the speed at which we slide the bo. Which will result in less work, sliding the bo quickl over the necessar feet or sliding it slowl? Please eplain.

Click here for answers.

Click here for answers. CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

More information

7.3 Volumes Calculus

7.3 Volumes Calculus 7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

More information

SL Calculus Practice Problems

SL Calculus Practice Problems Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

More information

AP Calculus AB 2005 Scoring Guidelines Form B

AP Calculus AB 2005 Scoring Guidelines Form B AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2. INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

More information

AP Calculus BC 2007 Free-Response Questions

AP Calculus BC 2007 Free-Response Questions AP Calculus BC 7 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

Parametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston

Parametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

More information

THE PARABOLA 13.2. section

THE PARABOLA 13.2. section 698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

More information

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

More information

AP Calculus AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

More information

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter

More information

AP Calculus AB 2004 Scoring Guidelines

AP Calculus AB 2004 Scoring Guidelines AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

More information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

DIFFERENTIATION OPTIMIZATION PROBLEMS

DIFFERENTIATION OPTIMIZATION PROBLEMS DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side

More information

AP Calculus AB 2008 Free-Response Questions

AP Calculus AB 2008 Free-Response Questions AP Calculus AB 2008 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

THE PARABOLA section. Developing the Equation

THE PARABOLA section. Developing the Equation 80 (-0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or

More information

AP Calculus AB 2012 Free-Response Questions

AP Calculus AB 2012 Free-Response Questions AP Calculus AB 1 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

AP Calculus BC 2008 Scoring Guidelines

AP Calculus BC 2008 Scoring Guidelines AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

AP Calculus AB 2007 Free-Response Questions

AP Calculus AB 2007 Free-Response Questions AP Calculus AB 2007 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

AP Calculus AB 2012 Scoring Guidelines

AP Calculus AB 2012 Scoring Guidelines AP Calculus AB Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

More information

2008 AP Calculus AB Multiple Choice Exam

2008 AP Calculus AB Multiple Choice Exam 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Teacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image.

Teacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image. Teacher Page Geometr / Da # 10 oordinate Geometr (5 min.) 9-.G.3.1 9-.G.3.2 9-.G.3.3 9-.G.3. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric

More information

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period: AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be

More information

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

D) A block pulled with a constant force will have a constant acceleration in the same direction as the force.

D) A block pulled with a constant force will have a constant acceleration in the same direction as the force. Phsics 100A Homework 4 Chapter 5 Newton s First Law A)If a car is moving to the left with constant velocit then the net force applied to the car is zero. B) An object cannot remain at rest unless the net

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

More information

2014 2015 Geometry B Exam Review

2014 2015 Geometry B Exam Review Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

7.3 Parabolas. 7.3 Parabolas 505

7.3 Parabolas. 7.3 Parabolas 505 7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

AP CALCULUS AB 2008 SCORING GUIDELINES

AP CALCULUS AB 2008 SCORING GUIDELINES AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line

More information

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010 MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic

More information

SECTION 9-1 Conic Sections; Parabola

SECTION 9-1 Conic Sections; Parabola 66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships

More information

Student Performance Q&A:

Student Performance Q&A: Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

More information

4 More Applications of Definite Integrals: Volumes, arclength and other matters

4 More Applications of Definite Integrals: Volumes, arclength and other matters 4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the

More information

Triple Integrals in Cylindrical or Spherical Coordinates

Triple Integrals in Cylindrical or Spherical Coordinates Triple Integrals in Clindrical or Spherical Coordinates. Find the volume of the solid ball 2 + 2 + 2. Solution. Let be the ball. We know b #a of the worksheet Triple Integrals that the volume of is given

More information

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc. Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

More information

AP Calculus AB. Practice Exam. Advanced Placement Program

AP Calculus AB. Practice Exam. Advanced Placement Program Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this

More information

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

More information

Section 6.4: Work. We illustrate with an example.

Section 6.4: Work. We illustrate with an example. Section 6.4: Work 1. Work Performed by a Constant Force Riemann sums are useful in many aspects of mathematics and the physical sciences than just geometry. To illustrate one of its major uses in physics,

More information

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

More information

Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.

Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and

More information

AP Calculus AB 2007 Scoring Guidelines Form B

AP Calculus AB 2007 Scoring Guidelines Form B AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exams in Single Variable Calculus (MA 165, 166)

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exams in Single Variable Calculus (MA 165, 166) June PURDUE UNIVERSITY Stud Guide for the Credit Eams in Single Variable Calculus (MA 65, 66) Eam and Eam cover respectivel the material in Purdue s courses MA 65 (MA 6) and MA 66 (MA 6). These are two

More information

( ) where W is work, f(x) is force as a function of distance, and x is distance.

( ) where W is work, f(x) is force as a function of distance, and x is distance. Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

AP Calculus AB 2004 Free-Response Questions

AP Calculus AB 2004 Free-Response Questions AP Calculus AB 2004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be

More information

LINEAR FUNCTIONS OF 2 VARIABLES

LINEAR FUNCTIONS OF 2 VARIABLES CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

More information

AP Physics - Chapter 8 Practice Test

AP Physics - Chapter 8 Practice Test AP Physics - Chapter 8 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A single conservative force F x = (6.0x 12) N (x is in m) acts on

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

HSC Mathematics - Extension 1. Workshop E4

HSC Mathematics - Extension 1. Workshop E4 HSC Mathematics - Extension 1 Workshop E4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong

More information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

More information

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1. Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit

More information

AP Calculus BC 2003 Free-Response Questions

AP Calculus BC 2003 Free-Response Questions AP Calculus BC 2003 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11.

SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xy-plane. It is sketched in Figure 11. SECTION 9.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is

More information

AP Calculus AB 2009 Free-Response Questions

AP Calculus AB 2009 Free-Response Questions AP Calculus AB 2009 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

More information

Calculus with Analytic Geometry I Exam 10 Take Home part

Calculus with Analytic Geometry I Exam 10 Take Home part Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the

More information

9.5 CALCULUS AND POLAR COORDINATES

9.5 CALCULUS AND POLAR COORDINATES smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet

More information

AP Calculus AB 2009 Scoring Guidelines

AP Calculus AB 2009 Scoring Guidelines AP Calculus AB 9 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 19,

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

Section 2-3 Quadratic Functions

Section 2-3 Quadratic Functions 118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the

More information

Fluid Pressure and Fluid Force

Fluid Pressure and Fluid Force 0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9. .(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

More information

LESSON SUMMARY. Measuring Shapes

LESSON SUMMARY. Measuring Shapes LESSON SUMMARY CXC CSEC MATHEMATICS UNIT SIX: Measurement Lesson 11 Measuring Shapes Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1 (Some helpful exercises and page numbers are given

More information

UNIVERSITY OF WISCONSIN SYSTEM

UNIVERSITY OF WISCONSIN SYSTEM Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Eam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice eam contributors: Benita Albert Oak Ridge High School,

More information

Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math 122 Spring 12 Instructor: Jeff Lang Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

More information

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

More information

Math Placement Test Practice Problems

Math Placement Test Practice Problems Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

More information

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

More information

Classical Physics I. PHY131 Lecture 7 Friction Forces and Newton s Laws. Lecture 7 1

Classical Physics I. PHY131 Lecture 7 Friction Forces and Newton s Laws. Lecture 7 1 Classical Phsics I PHY131 Lecture 7 Friction Forces and Newton s Laws Lecture 7 1 Newton s Laws: 1 & 2: F Net = ma Recap LHS: All the forces acting ON the object of mass m RHS: the resulting acceleration,

More information

AP Calculus AB 2003 Scoring Guidelines

AP Calculus AB 2003 Scoring Guidelines AP Calculus AB Scoring Guidelines The materials included in these files are intended for use y AP teachers for course and exam preparation; permission for any other use must e sought from the Advanced

More information

Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 2015 Solutions for Class Activity #2 Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

More information

Parametric Surfaces. Solution. There are several ways to parameterize this. Here are a few.

Parametric Surfaces. Solution. There are several ways to parameterize this. Here are a few. Parametric Surfaces 1. (a) Parameterie the elliptic paraboloid = 2 + 2 + 1. Sketch the grid curves defined b our parameteriation. Solution. There are several was to parameterie this. Here are a few. i.

More information

Geometry Notes PERIMETER AND AREA

Geometry Notes PERIMETER AND AREA Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

More information

Unit 04: Fundamentals of Solid Geometry - Shapes and Volumes

Unit 04: Fundamentals of Solid Geometry - Shapes and Volumes Unit 04: Fundamentals of Solid Geometry - Shapes and Volumes Introduction. Skills you will learn: a. Classify simple 3-dimensional geometrical figures. b. Calculate surface areas of simple 3-dimensional

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

Week #15 - Word Problems & Differential Equations Section 8.1

Week #15 - Word Problems & Differential Equations Section 8.1 Week #15 - Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by

More information

Calculus AB 2014 Scoring Guidelines

Calculus AB 2014 Scoring Guidelines P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official

More information

Discussion Session 1

Discussion Session 1 Physics 102 Fall 2016 NAME: Discussion Session 1 Math Review and Temperature The goal of Physics is to explain the Universe in terms of equations, and so the ideas of mathematics are central to your success

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

More information

AP Calculus AB 2006 Free-Response Questions

AP Calculus AB 2006 Free-Response Questions AP Calculus AB 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM

M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM 68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof

More information

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

Student Performance Q&A:

Student Performance Q&A: Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief

More information