2008 AP Calculus AB Multiple Choice Exam


 Theodore Manning
 4 years ago
 Views:
Transcription
1 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active
2 AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active
3 AP Calculus 008 Multiple Choice. ( )( ) ( )( + ) lim is (A) (B) (C) (D) (E) noneistent. d = (A) ln + C (B) ln + C (C) + C (D) + C (E) + C
4 AP Calculus 008 Multiple Choice. If ( ) ( )( ) f, = + then '( ) (A) 6( + ) (B) 6( )( + ) (C) ( + ) ( + ) (D) ( + ) ( ) (E) ( )( + ) f = ( ) 4. sin ( ) cos( ) + d = + + C (A) cos( ) sin ( ) + + C (B) cos( ) sin ( ) (C) cos( ) sin ( ) + + C (D) cos( ) sin ( ) + C (E) cos( ) sin ( ) + + C
5 AP Calculus 008 Multiple Choice lim is (A) (B) 0 (C) (D) 5 + (E) noneistent 4 if f ( ) = if = 6. Let f be the function defined above. Which of the following statements about f are true? I. f has a limit at =. II. f is continuous at =. III. f is differentiable at =. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III
6 AP Calculus 008 Multiple Choice 7. A particle moves along the ais with velocity given by v( t) = t + 6t for time t 0. If the particle is at position = at time t = 0, what is the position of the particle at t =? (A) 4 (B) 6 (C) 9 (D) (E) 8. If f ( ) ( ) π = cos, then f ' = 9 (A) (B) (C) (D) (E)
7 AP Calculus 008 Multiple Choice 0 9. The graph of the piecewise linear function f is shown in the figure above. If g = f t dt, which of the following values is greatest? ( ) ( ) Graph of f (A) g ( ) (B) g ( ) (C) g ( 0) (D) g ( ) (E) g ( )
8 AP Calculus 008 Multiple Choice y Graph of f 0. The graph of function f is shown above for 0. f the following, which has the least value? length (A) f ( ) d (B) Left Riemann sum approimation of f ( ) d with 4 subintervals of equal length (C) Right Riemann sum approimation of f ( ) d with 4 subintervals of equal length (D) Midpoint Riemann sum approimation of ( ) f d with 4 subintervals of equal (E) Trapezoidal sum approimation of f ( ) d with 4 subintervals of equal length
9 AP Calculus 008 Multiple Choice y Graph of f. The graph of a function f is shown above. Which of the following could be the graph of f ', the derivative of f? (A) y (B) y (C) y (D) y (E) y
10 AP Calculus 008 Multiple Choice ( / ). If f ( ) = e, then '( ) f = (A) ( / ) e ln (B) ( / ) e (C) ( / ) e (D) ( / ) e (E) e ( / ) d d. If f ( ) = +, then ( f ( ln ) ) = (A) ln + (B) ln + (C) ln + (D) ln + (E) +
11 AP Calculus 008 Multiple Choice 0 f "( ) The polynomial function f has selected values of its second derivative f " given in the table above. Which of the following statements must be true? (A) f is increasing on the interval ( 0, ). (B) f is decreasing on the interval ( 0, ). (C) f has a local maimum at =. (D) The graph of f has a point of inflection at =. (E) The graph of f changes concavity in the interval ( 0, ). 5. d = 4 (A) ( ) C (B) ( ) 4 + C (C) (D) ln 4 ln 4 + C + C (E) arctan + C
12 AP Calculus 008 Multiple Choice 6. If ( y) sin =, then dy d = (A) cos( y) (B) (C) (D) (E) cos cos cos ( y) ( y) ( y) y cos y cos ( y) ( y) ( cos( y) ) y Graph of f 7. The graph of the function f shown above has horizontal tangents at = and = 5. Let g be the function defined by g ( ) ( ) have a point of inflection? = f t dt. For what values of does the graph of g 0 (A) only (B) 4 only (C) and 5 only (D), 4, and 5 (E) 0, 4, and 6
13 AP Calculus 008 Multiple Choice 8. In the y plane, the line + y = k, where k is a constant, is tangent to the graph of y = + +. What is the value of k? (A) (B) (C) (D) 0 (E) 9. What are all horizontal asymptotes of the graph of 5 + y = in the y plane? (A) y = only (B) y = 0 only (C) y = 5 only (D) y = and y = 0 (E) y = and y = 5
14 AP Calculus 008 Multiple Choice 0. Let f be a function with a second derivative given by f ( ) ( )( ) the coordinates of the points of inflection of the graph of f? (A) 0 only (B) only (C) 0 and 6 only (D) and 6 only (E) 0,, and 6 " = 6. What are ( t ) t. A particle moves along a straight line. The graph of the particle s position ( t ) at time t is shown above for 0 < t < 6. The graph has horizontal tangents at t = and t = 5 and a point of inflection at t =. For what values of t is the velocity of the particle increasing? (A) 0 < t < (B) < t < 5 (C) < t < 6 (D) < t < 5 only (E) < t < and 5 < t < 6
15 AP Calculus 008 Multiple Choice. A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have heard the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant? (A) dp kp dt = dp dt = (B) kp ( N p) dp dt = (C) kp ( p N ) dp dt = (D) kt ( N t) dp dt = (E) kt ( t N )
16 AP Calculus 008 Multiple Choice. Which of the following is the solution to the differential equation condition y ( ) =? dy = with the initial d y (A) (B) 9 / y = e + 9 / y = e + (C) y = (D) (E) y = 4 y = 4 4. The function f is twice differentiable with f ( ) =, f '( ) = 4, and "( ). value of the approimation of f (.9) using the line tangent to the graph of f at =? (A) 0.4 (B) 0.6 (C) 0.7 (D). (E).4 f = What is the
17 AP Calculus 008 Multiple Choice f ( ) = c + d for c for > 5. Let f be the function defined above, where c and d are constants. If f is differentiable at =, what is the value of c + d? (A) 4 (B) (C) 0 (D) (E) 4 6. What is the slope of the line tangent to the curve y arctan ( 4) =? 4 = at the point at which (A) (B) (C) 0 (D) (E)
18 AP Calculus 008 Multiple Choice 5 y Shown above is a slope field for which of the following differential equations? (A) dy y d = (B) dy y y d = (C) dy y y d = + (D) dy y d = + dy d = + (E) ( )
19 AP Calculus 008 Multiple Choice 8. Let f be a differentiable function such that f ( ) = 5, f ( 6) =, ( ) f '( 6) =. The function g is differentiable and g ( ) f ( ) g '( )? (A) (B) 8 f ' = 8, and = for all. What is the value of (C) 6 (D) (E) The value of g '( ) cannot be determined from the information given.
20 AP Calculus 008 Multiple Choice y 4 5 Graph of f ' 76. The graph of f ', the derivative f, is shown above for 5. n what intervals is f increasing? (A) [, ] only (B) [, ] (C) [, 5] only (D) [0,.5] and [, 5] (E) [, ], [, ], and [4, 5]
21 AP Calculus 008 Multiple Choice y 4 Graph of f 77. The figure above shows the graph of a function f with domain 0 4. Which of the following statements are true? I. lim f ( ) II. lim f ( ) + eists. eists. III. lim f ( ) eists. (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 78. The first derivative of the function f is defined by f '( ) sin ( ) what interval(s) is f increasing? (A).445 (B).69 (C) (D) and.875 (E) 0 and.69 = for 0. n
22 AP Calculus 008 Multiple Choice If f ( ) d = 7 and f ( ) d = 4, 5 what is the value of ( ) 5 5 (A) (B) (C) 0 (D) (E) f d? 80. The derivative of the function f is given by f ( ) ( ) inflection does the graph of f have on the open interval (, )? ' = cos. How many points of (A) ne (B) Two (C) Three (D) Four (E) Five
23 AP Calculus 008 Multiple Choice 8. If G ( ) is an antiderivative for f ( ) and G ( ) = 7, then ( 4) (A) f '( 4) (B) 7 + f '( 4) 4 (C) f ( t ) dt ( ) 4 (D) + ( ) 7 f t dt 4 (E) ( ) + 7 f t dt G = 8. A particle moves along a straight line with velocity given by v( t) 7 (.0) t t 0. What is the acceleration of the particle at time t =? (A) 0.94 (B) (C) (D) (E) = at time
24 AP Calculus 008 Multiple Choice 8. What is the area enclosed by the curves = and y = + 5? y (A) (B).8 (C) 4.58 (D). (E) y 4 5 Graph of f ' 84. The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent lines at =, =, and =. At which of the following values of does f have a relative maimum? (A) only (B) only (C) 4 only (D) and only (E),, and 4
25 AP Calculus 008 Multiple Choice 4 f ( ) f '( ) The table above gives values of a function f and its derivative at selected values of. If is continuous on the interval [ 4, ], what is the value of f ( ) 4 ' d? (A) 4.5 (B).5 (C) 0 (D).5 (E) 4.5 f '
26 AP Calculus 008 Multiple Choice t 0 4 v( t ) The table gives selected values of the velocity, v( t ), of a particle moving along the ais. At time t = 0, the particle is at the origin. Which of the following could be the graph of the position, ( t ), of the particle for 0 t 4? (A) ( t ) (B) ( t ) t t (C) ( t ) (D) ( t ) t t (E) ( t ) t
27 AP Calculus 008 Multiple Choice 87. An object traveling in a straight line has position ( t ) at time t. If the initial position is ( 0) = and the velocity of the object is v( t) t time t =? = +, what is the position of the object at (A) 0.4 (B).54 (C) 4.5 (D) 6.5 (E) The radius of a sphere is decreasing at a rate of centimeters per second. At the instant when the radius of the sphere is centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area S of a sphere with radius r is S = 4π r ) (A) 08π (B) 7π (C) 48π (D) 4π (E) 6π
28 AP Calculus 008 Multiple Choice 89. The function f is continuous for and f ( ) f ( ) = = 0. If there is no c, where < c <, for which f '( c ) = 0, which of the following statements must be true? (A) For k, < < ( ) (B) For k, f ' k > 0. < < ( ) (C) For k, f ' k < 0. < < f '( ) (D) For k, < < '( ) k eists. (E) For some k, where k, f k eists, but f ' is not continuous. < < f '( ) k does not eist. 90. The function f is continuous on the closed interval [, 4] and twice differentiable on the open interval (, 4). If f '( ) = and ( ) following could be a table of values for f? (A) (B) (C) f " < 0 on the open interval (, 4), which of the f ( ) f ( ) f ( ) (D) (E) f ( ) f ( )
29 AP Calculus 008 Multiple Choice cos 9. What is the average value of y = + + on the closed interval [, ]? (A) (B) (C) 0.8 (D) 0.44 (E) miles River City 7 miles 9. A city located beside a river has a rectangular boundary as shown in the figure above. The population density of the city at any point along a strip miles from the river s edge is f persons per square mile. Which of the following epressions gives the population of ( ) the city? 4 (A) f ( ) d 0 4 (B) ( ) 7 f d 0 4 (C) ( ) 8 f d 7 (D) f ( ) d (E) ( ) 4 f d 0
AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 13 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 informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
More informationAP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationMark 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 informationAP 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 informationAP Calculus AB 2010 FreeResponse Questions Form B
AP Calculus AB 2010 FreeResponse Questions Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity.
More informationStudent Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the 2008 freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief
More informationAP 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 notforprofit membership association whose mission is to connect students to
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationCalculus 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 informationAP Calculus BC 2006 FreeResponse Questions
AP Calculus BC 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus AB 2011 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the
More informationAP Calculus BC 2013 FreeResponse Questions
AP Calculus BC 013 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus AB 2001 Scoring Guidelines
P Calculus Scing Guidelines The materials included in these files are intended f noncommercial use by P teachers f course and eam preparation; permission f any other use must be sought from the dvanced
More informationAP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationAP Calculus AB 2005 FreeResponse Questions
AP Calculus AB 25 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationAP Calculus AB 2013 FreeResponse Questions
AP Calculus AB 2013 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded
More informationAP Calculus BC 2001 FreeResponse Questions
AP Calculus BC 001 FreeResponse Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
More informationMark 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 informationAP 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 notforprofit membership association whose mission is to connect students to college
More informationAP Calculus AB 2011 FreeResponse Questions
AP Calculus AB 11 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus AB 2012 FreeResponse Questions
AP Calculus AB 1 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationAP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 4
AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 4 The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function 1 3 3 2 f. In the figure
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationAP Calculus AB 2009 FreeResponse Questions
AP Calculus AB 2009 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More informationAP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in
More informationAP CALCULUS AB 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 2007 SCORING GUIDELINES (Form B) Question 4 Let f be a function defined on the closed interval 5 x 5 with f ( 1) = 3. The graph of f, the derivative of f, consists of two semicircles and
More informationAP Calculus BC Exam. The Calculus BC Exam. At a Glance. Section I. SECTION I: MultipleChoice Questions. Instructions. About Guessing.
The Calculus BC Exam AP Calculus BC Exam SECTION I: MultipleChoice Questions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Grade 50% Writing Instrument Pencil required
More informationMATH 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 informationMATH 121 FINAL EXAM FALL 20102011. December 6, 2010
MATH 11 FINAL EXAM FALL 010011 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 informationAP Calculus AB 2003 Scoring Guidelines Form B
AP Calculus AB Scoring Guidelines Form B 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 informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationAP Calculus AB 2004 FreeResponse Questions
AP Calculus AB 2004 FreeResponse 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 informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationAP Calculus BC 2010 FreeResponse Questions
AP Calculus BC 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationArea Under the Curve. Riemann Sums And the Trapezoidal Rule
Area Under the Curve Riemann Sums And the Trapezoidal Rule Who knew that D=R x T would connect to velocity, and now integration, and the area under a curve? Take a look at the attached applications. Let
More informationAP CALCULUS AB 2006 SCORING GUIDELINES. Question 4
AP CALCULUS AB 2006 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) 0 10 20 30 40 50 60 70 80 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity vt () after being launched upward
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationVisualizing Differential Equations Slope Fields. by Lin McMullin
Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should
More informationAP Calculus AB Syllabus
Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.
More informationAn Introduction to Calculus. Jackie Nicholas
Mathematics Learning Centre An Introduction to Calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Some rules of differentiation and how to use them
More information3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.
BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's
More informationL 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 informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More informationWorksheet 1. What You Need to Know About Motion Along the xaxis (Part 1)
Worksheet 1. What You Need to Know About Motion Along the xaxis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationAP 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 informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationAP Calculus AB 2010 FreeResponse Questions
AP Calculus AB 2010 FreeResponse Questions The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Noncalculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM
X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Noncalculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion
More information100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
More informationGive a formula for the velocity as a function of the displacement given that when s = 1 metre, v = 2 m s 1. (7)
. The acceleration of a bod is gien in terms of the displacement s metres as s a =. s (a) Gie a formula for the elocit as a function of the displacement gien that when s = metre, = m s. (7) (b) Hence find
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationPartial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.
Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this
More informationAlgebra. 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 informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationActive Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122. Carroll College Mathematics Department
Active Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122 Carroll College Mathematics Department Last Update: June 2, 2015 2 To The Student This packet is NOT your
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationProblems 121 could be on the no Derive part. Sections 1.2, 2.2, 2.3, 3.1, 3.3, 3.4, 4.1, 4.2
MTH 120 Practice Test #1 Sections 1.2, 2.2, 2.3, 3.1, 3.3, 3.4, 4.1, 4.2 Use the properties of limits to help decide whether the limit eists. If the limit eists, find its value. 1) lim 5 2) lim 3 225
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationW i f(x i ) x. i=1. f(x i ) x = i=1
Work Force If an object is moving in a straight line with position function s(t), then the force F on the object at time t is the product of the mass of the object times its acceleration. F = m d2 s dt
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationPractice 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 information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationSolutions 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 information7.6 Approximation Errors and Simpson's Rule
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationIntegral Calculus  Exercises
Integral Calculus  Eercises 6. Antidifferentiation. The Indefinite Integral In problems through 7, find the indicated integral.. Solution. = = + C = + C.. e Solution. e =. ( 5 +) Solution. ( 5 +) = e
More informationCourse outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems)
Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems) Functions, domain and range Domain and range of rational and algebraic
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationA Resource for Freestanding Mathematics Qualifications
To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve
More informationExponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014
Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,
More informationNumerical Solution of Differential Equations
Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More information(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
More informationVersion 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence
Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This printout should have 47 questions Multiplechoice questions may continue on the net column or page find all choices before answering
More information6. 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 informationThe Fourth International DERIVETI92/89 Conference Liverpool, U.K., 1215 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVETI9/89 Conference Liverpool, U.K., 5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue NotreDame Ouest Montréal
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationAP Calculus BC 2004 Scoring Guidelines
AP Calculus BC Scoring Guidelines 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 sought from
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationAP 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 informationExercises and Problems in Calculus. John M. Erdman Portland State University. Version August 1, 2013
Exercises and Problems in Calculus John M. Erdman Portland State University Version August 1, 2013 c 2010 John M. Erdman Email address: erdman@pdx.edu Contents Preface ix Part 1. PRELIMINARY MATERIAL
More informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More information