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

Size: px
Start display at page:

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

Transcription

1 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, Oak Ridge, Tennessee Thomas Dick Oregon State University Joe Milliet St. Mark's School of Texas, Dallas, Texas Skylight Publishing Andover, Massachusetts

2 Copyright 5- by Skylight Publishing Chapter. Annotated Solutions to Past Free-Response Questions This material is provided to you as a supplement to the book Be Prepared for the AP Calculus Exam (ISBN ). You are not authorized to publish or distribute it in any form without our permission. However, you may print out one copy of this chapter for personal use and for face-to-face teaching for each copy of the Be Prepared book that you own or receive from your school. Skylight Publishing 9 Bartlet Street, Suite 7 Andover, MA 8 web: sales@skylit.com support@skylit.com

3 AB AP Calculus Free-Response Solutions and Notes Question AB- 6 (a) f () t dt (b) f ( 8) g( 8) = ft /hr., for t < 6 ht = 5 t 6, for 6 t< , for 7 t 9 (c) () ( ) ( t ) 9 (d) f () t dt h( 9) ( ) = Students should expect that Part A of the free-response section of the exam (the calculator part) will continue to involve questions with a real-world context.

4 4 FREE-RESPONSE SOLUTIONS ~ AB Question AB- (a) The rate at which entries were being deposited (in hundreds of entries per hour) is, E( 7) E( 5) 8 approximately, = = = (b) The approximation is This represents the average number of hundreds of entries in the box over the 8 hours (c) E(8) P() t dt 8 = hundred entries. (d) The question asks for the maximum value of P(t). P ( t) = at t = 9.85 and t =.86. The maximum value of P(t) must occur at one of these, or at the endpoints, t = 8 or t =. P(9.85) = 5.89, P(.86) =.9, P(8) =, and P() = 8. The maximum is at t =.. No need to evaluate. For the curious, the value is ( 85.5 ) = The candidate test is the simplest way to find the maximum.

5 FREE-RESPONSE SOLUTIONS ~ AB 5 Question AB- (a) The number of people who arrived at the ride between t = and t = is r() t dt = + =. (b) For t < people arrive at the rate greater than 8 people/hour and are removed from the line at the constant rate of 8 people/hour. The rate of arrival is greater than the rate of removal, so the number of people in the line is increasing. (c) Let P() t be the number of people in the line at time t. Then P () t = r() t 8. P () t > for < t <, and P ( t) < for < t < 8. Therefore, P() t reaches its maximum at t =. The line is longest at t = ; at that time there are 7 + r() t dt 8 = = 5 people in the line. t (d) r( u) ( ) du =. Question AB-4 (a) Area = / 4 6 xdx = 6x x = =. ( 7 ) 9 (b) Volume = ( x ) π dx. 6 6 (c) Volume = ( ) 4 y x x dy = dy 6.. This is the second consecutive year that an area-volume problem has appeared in Part B, the closed calculator part, of the free response. Concern over calculator programs that can unfairly assist students in the solution of area-volume problems is one possible reason for this.. Not 9 6 ( x ) dx. The sections are perpendicular to the y-axis, not the x-axis.

6 6 FREE-RESPONSE SOLUTIONS ~ AB Question AB-5 g x = + g t dt, so g( ) = 5+ g ( t) dt 5 π = + + and g = 5+ g t dt = 5 π. x (a) ( ) 5 ( ) ( ) ( ) (b) x = x =, and vice-versa at these points. x =, because g ( x) (c) At a critical point, ( ) ( ) ( ) changes from increasing to decreasing or h x = g x x= g x = x. The line y = x intersects the semicircle where x > and x + y = x = 4 h x changes sign from positive to negative, so h(x) has a relative maximum there. x =. At that point, ( ) The line y = x also intersects the graph of y = g ( x) at x =. h ( x) does not change sign at x =, so there is neither a minimum nor a maximum there.. Use geometry to calculate the areas, not the Fundamental Theorem!. The integral from to is negative.

7 FREE-RESPONSE SOLUTIONS ~ AB 7 Question AB-6 (a) At the point ( ) dy,, 8 dx =. An equation of the tangent line is y 8 ( x ) d y =. (b) The approximation is y = + 8.=.8. Since > on the open interval dx (,.), the graph of f is concave up there, and the tangent line lies below the graph of y = f( x) for x. f.. (c) dy = x dx y. Thus the approximation is less than ( ) y = +. Substituting ( ) x C, we get / 5 5 C =. Solving for y, x 8 y = y = x or 4 Since y () = >, the particular solution we are looking for is / 5 y = x =. 4 5 x 4 8 = + C 5 y = x 4 /.

8 8 FREE-RESPONSE SOLUTIONS ~ AB

9 BC AP Calculus Free-Response Solutions and Notes Question BC- See AB Question. Question BC- See AB Question. 9

10 FREE-RESPONSE SOLUTIONS ~ BC Question BC- (a) dx dx t 4 dt dt t= dy dt t = =. ( te ) + = 8. t= = =. The speed at t = is t= 4 t te dt.588. t (b) Total distance traveled = ( 4) + ( ) dy (c) The tangent to the path is horizontal when =. Solving gives.8 dt t. At dx that time, dt >, so the particle is moving to the right. (d) (i) x() t = 5 t 4t+ 8= 5 t = or t =. dy dy dx (ii) The slope is dx = dt dt =. At t =, dy / e = = dx e dy At t =, dx = =.. t (iii) ( ) ( ) ( t y = y + te ) dt = + + ( te ) dt 4. e. Alternatively, we could calculate y().. The integral can be evaluated precisely using integration by parts:., t t t t t ( te ) dt te dt te e dt = = = ( t ) e =. e

11 FREE-RESPONSE SOLUTIONS ~ BC Question BC-4 See AB Question 4. Question BC-5 (a) dy The step in Euler s method is. At (, ), the slope y dx = =, so y new = + =. At,, the slope is =, so 5 y new = + = =. 4 4 (b) Since f is continuous, f ( x) f ( ) ( ) ( ) lim = =, so we can use l Hôpital s Rule: x f x f x lim = lim =. x x x x (c) dy = dx y ln y x C = +. Using the initial condition, we get C = ln y = x y = e x. We are told that y = f( x) < for x x this solution, so y = y y e = y = e.

12 FREE-RESPONSE SOLUTIONS ~ BC Question BC-6 4 n x x n x cos = ! 4!! 4 n cos x x x n x f( x) = = x! 4! 6! n! (a) x ( ) ( n ) (b) f ( ) =. ( ) ( ) f () = f () >. Therefore, by the Second Derivative Test,! 4! f has a relative minimum at x =. (c) Integrating the first three terms of the series for f, we get 5 x x P5 ( x) = x+ + C. Since g() =, C =! 4! 5 6! 5 x x P5 ( x) = x+.! 4! 5 6! (d) g () = ! 5 6! + and P () = +. The alternating series 4! estimate error g() P () does not exceed the absolute value of the first omitted term in the series, which is. Therefore, g() P () < <. 56! 56! 6!

13 AB (Form B) AP Calculus Free-Response Solutions and Notes Question AB- (Form B) ( ) (a) Area = 6 4ln( x) dx (( ) ) (b) Volume = π 8 4ln( x) dx 68.8 ( ) (c) 6 4ln( x) dx Question AB- (Form B) (a) The graph of g has a horizontal tangent where g ( x) (b) The graph of g is concave down where g ( x) subinterval,.9 < x <... (c) (.) ( ) ( ) g g g x dx (.) = = : x =.6 and x =.59. <. This happens on one. The slope at x =. is g.47. An equation of the tangent line is ( x ) y.546 =.47.. > for.5 < x <, the graph of g is concave up there, so the tangent line at x =. lies under the graph of g. (d) Since g ( x) < on (.9,.), we could say the graph of g is concave down on the closed interval [.9,.].. Since g ( x)

14 4 FREE-RESPONSE SOLUTIONS ~ AB (FORM B) Question AB- (Form B) (a) The midpoint Riemann sum is 4 ( ) = 66 cubic feet. (b) The amount of water leaked = R () t dt cubic feet. (c) Volume = = cubic feet. (d) Let V () t be the volume of water in the pool at time t. Then ( 8) ( 8) ( 8) V = P R = 4.4 cubic feet per hour. dh dh V t = =, so V () t = 44π =. Therefore, dt dt 44π feet per hour. 44π V π r h 44π h dh dt t= 8. Or leave the answer as a fraction with π.. Don t forget the units. ( )

15 FREE-RESPONSE SOLUTIONS ~ AB (FORM B) 5 Question AB-4 (Form B) (a) The squirrel changes direction at t = 9 and at t = 5, because its velocity changes sign at each of these points in time. (b) The area of the trapezoid extending from t = to t = 9 is = 4, so the squirrel moves 4 units towards B during that time. The area of the trapezoid from t = 9 to t = 5 is = 5, so the squirrel moves back 5 units towards A during that time. The area of the trapezoid from t = 5 to t = 8 is + = 5, so the squirrel moves 5 units towards B during that time. Therefore, the distances from A are 4 at t = 9, 9 at t = 5, and 5 at t = 8. The maximum is 4 at t = 9. (c) Based on the calculations in Part (b), the total distance traveled by the squirrel is = 5. (d) a() t = = ; v( t) = ( t 9) = t+ 9 ; 7 x() t = v() t dt = 5t + 9t+ C. From Part (b), x (9) = C = 4 C = 65 x( t) = 5t + 9t 65.. The graph of velocity on the interval 7< t < is a straight line. Its slope is equal to the acceleration. The equation of the line is written in the point-slope form for the v t = t 7. point (9, ). We could use the point (7, ) instead: ( ) ( ). Be careful not to confuse v(9) = on the graph, which applies to the equation for the velocity, with x(9) = 4, which we use to find the equation for the distance.

16 6 FREE-RESPONSE SOLUTIONS ~ AB (FORM B) Question AB-5 (Form B) (a) dy (b) dx = when x + =, that is, y = x and y. y ydy x dx, so (c) = ( + ) y x x = y x = + x + C. y () = C = y x x = Since the initial condition has y <, we choose y x x = We are asked to sketch the solution on < x <. Without this restriction, the domain of this solution would be (, ). A particular solution to a differential equation is always defined on a connected interval. This differential equation is not defined when y =, so no solution can cross or touch the x-axis.

17 FREE-RESPONSE SOLUTIONS ~ AB (FORM B) 7 Question AB-6 (Form B) (a) Particle R moves to the right when r ( t) = t t+ 9> ( t )( t ) t < and < t 6. > π π (b) Particle P moves to the right when p () t = sin t > 4 π π < 4 t < π 4< t 6. π sin 4 t < R P t The particles travel in opposite directions for < t < and < t < 4. (c) The acceleration is p ( ) π π π π = cos = = The velocity 6 π π π at t = is p ( ) = sin = <. Since the velocity and acceleration 4 4 have opposite signs, the particle is slowing down. (d) The distance between the particles is p ( t) r( t) interval t is () () p t r t dt.. A chart is not required but might be helpful. and the average distance on the

18 8 FREE-RESPONSE SOLUTIONS ~ AB (FORM B)

19 Question BC- (Form B) See AB Question. BC (Form B) AP Calculus Free-Response Solutions and Notes Question BC- (Form B) dx (a) The tangent line is vertical when and t.5. (b) Let the position of the particle be ( ( ), ( )) dt = and dy dt. This occurs at t.45 x t y t. Then () = ( ) + 4cos( ) sin( t ) () = ( ) + + sin( ) x x t e dt y y t dt dy dy dx = dt dt dx t = t = + = and ( x ) y 4.6 = =. At t =,.86. An equation for the tangent line is (c) The speed is dx dy + = dt dt t= 4.5. (d) The acceleration vector is the derivative of the velocity vector. At t =, this is 8.45,.6. ( ). Enter the given functions dx dt of the solution. and dy dt into your calculator and save them for the rest 9

20 FREE-RESPONSE SOLUTIONS ~ BC (FORM B) Question BC- (Form B) See AB Question. Question BC-4 (Form B) See AB Question 4. Question BC-5 (Form B) (a) g ( x) ( + x ) x x x ( + 4x ) ( + 4x ) = =. For = when x >, g ( x) x =. For < x <, g ( x) > and g is increasing; for x >, g ( x) < and g is decreasing. The absolute maximum value of g therefore occurs at x =. It is g = =. g has no minimum on (, ) (b) The area is given by the improper integral 4x dx = x + 4x b lim ln x ln ( + 4x ) = lim ln b ln ( + 4b ) + ln 5 = b b b lim ln + ln 5 = lim ln + ln 5 = ln + ln 5. b 4 b + b + 4 b. Leave it at this to save time and avoid mistakes.

21 FREE-RESPONSE SOLUTIONS ~ BC (FORM B) Question BC-6 (Form B) (a) By the ratio test, the series converges when gives x < or < x <. At ( x) ( x) n+ lim n <. Evaluating the limit n n n x =, the series is n ( ) ( ) = n n. n= n= This is the harmonic series, which diverges. At x =, the series is. This n= n is an alternating series with terms decreasing by absolute value and approaching zero. Therefore, it converges by the Alternating Series Test. The interval of convergence is < x. n ( ) n (b) y f ( x) n n ( ) n( x) n ( ) n ( x) = =, so xy = and n= n n= n n n n n ( ) n ( x) ( ) ( x) xy y =. Factoring gives n= n n n n ( ) ( x) ( n ) n xy y = = ( x) n= n n= 4x first term 4x and the common ratio x, so it converges to + x that is, for < x <.. This is a geometric series with the n when x <,

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

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

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

More information

Mark Howell Gonzaga High School, Washington, D.C. Benita Albert Oak Ridge High School, Oak Ridge, Tennessee

Mark Howell Gonzaga High School, Washington, D.C. Benita Albert Oak Ridge High School, Oak Ridge, Tennessee Be Prepared for the Third Editio Calculus Exam * AP ad Advaced Placemet Program are registered trademarks of the College Etrace Examiatio Board, which was ot ivolved i the productio of ad does ot edorse

More information

AP Calculus BC Fall Final Part IIa

AP Calculus BC Fall Final Part IIa AP Calculus BC 18-19 Fall Final Part IIa Calculator Required Name: 1. At time t = 0, there are 120 gallons of oil in a tank. During the time interval 0 t 10 hours, oil flows into the tank at a rate of

More information

Students! (1) with calculator. (2) No calculator

Students! (1) with calculator. (2) No calculator Students! (1) with calculator Let R be the region bounded by the graphs of y = sin(π x) and y = x 3 4x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line y = splits the region

More information

Answer Key for AP Calculus AB Practice Exam, Section I

Answer Key for AP Calculus AB Practice Exam, Section I Answer Key for AP Calculus AB Practice Exam, Section I Multiple-Choice Questions Question # Key B B 3 A 4 E C 6 D 7 E 8 C 9 E A A C 3 D 4 A A 6 B 7 A 8 B 9 C D E B 3 A 4 A E 6 A 7 A 8 A 76 E 77 A 78 D

More information

AP Calculus AB 2015 Free-Response Questions

AP Calculus AB 2015 Free-Response Questions AP Calculus AB 015 Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online

More information

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental

More information

AP Calculus BC Fall Final Part IA. Calculator NOT Allowed. Name:

AP Calculus BC Fall Final Part IA. Calculator NOT Allowed. Name: AP Calculus BC 18-19 Fall Final Part IA Calculator NOT Allowed Name: 3π cos + h 1. lim cos 3π h 0 = h 1 (a) 1 (b) (c) 0 (d) -1 (e) DNE dy. At which of the five points on the graph in the figure below are

More information

Section 2 Practice Tests

Section 2 Practice Tests Section Practice Tests This section gives 18 sample problems that mix and match concepts similar to problems on the free-response section of the AP exam. While any of these examples would be a good review

More information

Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value

Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value AP Calculus Unit 6 Basic Integration & Applications Day 5 Notes: The Fundamental Theorem of Calculus, Particle Motion, and Average Value b (1) v( t) dt p( b) p( a), where v(t) represents the velocity and

More information

BC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true?

BC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true? BC Exam 1 - Part I 8 questions No Calculator Allowed - Solutions 6x 5 8x 3 1. Find lim x 0 9x 3 6x 5 A. 3 B. 8 9 C. 4 3 D. 8 3 E. nonexistent ( ) f ( 4) f x. Let f be a function such that lim x 4 x 4 I.

More information

AP Calculus AB. Free-Response Questions

AP Calculus AB. Free-Response Questions 2018 AP Calculus AB Free-Response Questions College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online

More information

AP Calculus. Analyzing a Function Based on its Derivatives

AP Calculus. Analyzing a Function Based on its Derivatives AP Calculus Analyzing a Function Based on its Derivatives Student Handout 016 017 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

More information

Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION. tangent line, a+h. a+h

Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION. tangent line, a+h. a+h Sample Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION B B A B tangent line,, a f '(a) = lim h 0 f(a + h) f(a) h a+h a b b f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x ] n a n B B

More information

Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION. tangent line, a+h. a+h

Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION. tangent line, a+h. a+h Sample Questions PREPARING FOR THE AP (BC) CALCULUS EXAMINATION B B A B tangent line,, a f '(a) = lim h 0 f(a + h) f(a) h a+h a b b f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x ] n a n B B

More information

AP Calculus Exam Format and Calculator Tips:

AP Calculus Exam Format and Calculator Tips: AP Calculus Exam Format and Calculator Tips: Exam Format: The exam is 3 hours and 15 minutes long and has two sections multiple choice and free response. A graphing calculator is required for parts of

More information

Spring 2015 Sample Final Exam

Spring 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 information

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time 55 minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems,

More information

AP Calculus BC Spring Final Part IA. Calculator NOT Allowed. Name:

AP Calculus BC Spring Final Part IA. Calculator NOT Allowed. Name: AP Calculus BC 6-7 Spring Final Part IA Calculator NOT Allowed Name: . Find the derivative if the function if f ( x) = x 5 8 2x a) f b) f c) f d) f ( ) ( x) = x4 40 x 8 2x ( ) ( x) = x4 40 +x 8 2x ( )

More information

NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:

NO 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 information

AP Calculus BC Summer Assignment (June)

AP Calculus BC Summer Assignment (June) AP Calculus BC Summer Assignment (June) Solve each problem on a separate sheet of paper as if they are open ended AP problems. This means you must include all justifications necessary as on the AP AB exam.

More information

Graphical Relationships Among f, f,

Graphical Relationships Among f, f, Graphical Relationships Among f, f, and f The relationship between the graph of a function and its first and second derivatives frequently appears on the AP exams. It will appear on both multiple choice

More information

LSU AP Calculus Practice Test Day

LSU AP Calculus Practice Test Day LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3

More information

The Princeton Review AP Calculus BC Practice Test 2

The Princeton Review AP Calculus BC Practice Test 2 0 The Princeton Review AP Calculus BC Practice Test CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each

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?

= 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 information

AP Calculus AB 2006 Free-Response Questions Form B

AP Calculus AB 2006 Free-Response Questions Form B AP Calculus AB 006 Free-Response Questions 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

More information

CALCULUS AB SECTION II, Part A

CALCULUS AB SECTION II, Part A CALCULUS AB SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. pt 1. The rate at which raw sewage enters a treatment tank

More information

AP Calculus AB 1998 Free-Response Questions

AP Calculus AB 1998 Free-Response Questions AP Calculus AB 1998 Free-Response Questions These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced

More information

K. Function Analysis. ). This is commonly called the first derivative test. f ( x) is concave down for values of k such that f " ( k) < 0.

K. Function Analysis. ). This is commonly called the first derivative test. f ( x) is concave down for values of k such that f  ( k) < 0. K. Function Analysis What you are finding: You have a function f ( ). You want to find intervals where f ( ) is increasing and decreasing, concave up and concave down. You also want to find values of where

More information

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher

Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Sudoku Puzzle A.P. Exam (Part B) Questions are from the 1997 and 1998 A.P. Exams A Puzzle by David Pleacher Solve the 4 multiple-choice problems below. A graphing calculator is required for some questions

More information

Final Value = Starting Value + Accumulated Change. Final Position = Initial Position + Displacement

Final Value = Starting Value + Accumulated Change. Final Position = Initial Position + Displacement Accumulation, Particle Motion Big Ideas Fundamental Theorem of Calculus and Accumulation AP Calculus Course Description Goals page 6 Students should understand the meaning of the definite integral both

More information

AP Calculus AB 2nd Semester Homework List

AP Calculus AB 2nd Semester Homework List AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this

More information

MATH 1242 FINAL EXAM Spring,

MATH 1242 FINAL EXAM Spring, MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t

More information

Unit #6 Basic Integration and Applications Homework Packet

Unit #6 Basic Integration and Applications Homework Packet Unit #6 Basic Integration and Applications Homework Packet For problems, find the indefinite integrals below.. x 3 3. x 3x 3. x x 3x 4. 3 / x x 5. x 6. 3x x3 x 3 x w w 7. y 3 y dy 8. dw Daily Lessons and

More information

t (hours) H t ( ) (percent)

t (hours) H t ( ) (percent) 56. Average Value of a Function (Calculator allowed) Relative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. A storm is passing through an area and the

More information

Answer 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 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 information

Practice Final Exam Solutions

Practice 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 information

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the

A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the attached packet of problems, and turn it in on Monday, August

More information

I. Horizontal and Vertical Tangent Lines

I. Horizontal and Vertical Tangent Lines How to find them: You need to work with f " x Horizontal tangent lines: set f " x Vertical tangent lines: find values of x where f " x I. Horizontal and Vertical Tangent Lines ( ), the derivative of function

More information

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. AP Calculus AB Exam SECTION I: Multiple Choice 016 DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing

More information

AP Calculus AB 2008 Free-Response Questions Form B

AP Calculus AB 2008 Free-Response Questions Form B AP Calculus AB 2008 Free-Response Questions 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

More information

Euler s Method and Logistic Growth (BC Only)

Euler s Method and Logistic Growth (BC Only) Euler s Method Students should be able to: Approximate numerical solutions of differential equations using Euler s method without a calculator. Recognize the method as a recursion formula extension of

More information

Chapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry

Chapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) 7 8 4 7 d b) 9 4 7 d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate

More information

Math 106 Answers to Exam 3a Fall 2015

Math 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 information

AP Calculus BC. Free-Response Questions

AP Calculus BC. Free-Response Questions 017 AP Calculus BC Free-Response Questions 017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. 017 AP CALCULUS

More information

MA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM. Name (Print last name first):... Instructor:... Section:... PART I

MA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM. Name (Print last name first):... Instructor:... Section:... PART I CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM Name (Print last name first):............................................. Student ID Number:...........................

More information

2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION 8 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems. After eamining the form

More information

. CALCULUS AB. Name: Class: Date:

. CALCULUS AB. Name: Class: Date: Class: _ Date: _. CALCULUS AB SECTION I, Part A Time- 55 Minutes Number of questions -8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using

More information

Limits 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. 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) During what time intervals on [0, 4] is the particle traveling to the left?

(a) During what time intervals on [0, 4] is the particle traveling to the left? Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t /

More information

AP Calculus. Particle Motion. Student Handout

AP Calculus. Particle Motion. Student Handout AP Calculus Particle Motion Student Handout 016-017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss Copyright 016

More information

2018 FREE RESPONSE QUESTION TENTATIVE SOLUTIONS

2018 FREE RESPONSE QUESTION TENTATIVE SOLUTIONS 8 FREE RESPONSE QUESTION TENTATIVE SOLUTIONS L. MARIZZA A BAILEY Problem. People enter a line for an escalator at a rate modeled by the function, r given by { 44( t r(t) = ) ( t )7 ) t t > where r(t) is

More information

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c)

L. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c) L. Function Analysis What you are finding: You have a function f ( x). You want to find intervals where f ( x) is increasing and decreasing, concave up and concave down. You also want to find values of

More information

AP Calculus AB Free-Response Scoring Guidelines

AP Calculus AB Free-Response Scoring Guidelines Question pt The rate at which raw sewage enters a treatment tank is given by Et 85 75cos 9 gallons per hour for t 4 hours. Treated sewage is removed from the tank at the constant rate of 645 gallons per

More information

Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2)

Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2) Motion with Integrals Worksheet 4: What you need to know about Motion along the x-axis (Part 2) 1. Speed is the absolute value of. 2. If the velocity and acceleration have the sign (either both positive

More information

Math 116 Second Midterm November 14, 2012

Math 116 Second Midterm November 14, 2012 Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that

More information

Justifications on the AP Calculus Exam

Justifications on the AP Calculus Exam Justifications on the AP Calculus Exam Students are expected to demonstrate their knowledge of calculus concepts in 4 ways. 1. Numerically (Tables/Data) 2. Graphically 3. Analytically (Algebraic equations)

More information

AP Calculus BC. Functions, Graphs, and Limits

AP Calculus BC. Functions, Graphs, and Limits AP Calculus BC The Calculus courses are the Advanced Placement topical outlines and prepare students for a successful performance on both the Advanced Placement Calculus exam and their college calculus

More information

AP Calculus BC 2015 Free-Response Questions

AP Calculus BC 2015 Free-Response Questions AP Calculus BC 05 Free-Response Questions 05 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central

More information

AP Calculus AB 2001 Free-Response Questions

AP Calculus AB 2001 Free-Response Questions AP Calculus AB 001 Free-Response 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 information

AP Calculus AB/BC ilearnmath.net

AP Calculus AB/BC ilearnmath.net CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or

More information

x f(x)

x f(x) CALCULATOR SECTION. For y y 8 find d point (, ) on the curve. A. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) e, A(t) is measured in tons of silver and

More information

x+1 e 2t dt. h(x) := Find the equation of the tangent line to y = h(x) at x = 0.

x+1 e 2t dt. h(x) := Find the equation of the tangent line to y = h(x) at x = 0. Math Sample final problems Here are some problems that appeared on past Math exams. Note that you will be given a table of Z-scores for the standard normal distribution on the test. Don t forget to have

More information

AP Calculus BC 2008 Free-Response Questions Form B

AP Calculus BC 2008 Free-Response Questions Form B AP Calculus BC 008 Free-Response Questions 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

More information

AP Calculus Prep Session Handout. Table Problems

AP Calculus Prep Session Handout. Table Problems AP Calculus Prep Session Handout The AP Calculus Exams include multiple choice and free response questions in which the stem of the question includes a table of numerical information from which the students

More information

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)

More information

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part

2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 1. Let R be the region in the first and second quadrants bounded

More information

WeBWorK assignment 1. b. Find the slope of the line passing through the points (10,1) and (0,2). 4.(1 pt) Find the equation of the line passing

WeBWorK assignment 1. b. Find the slope of the line passing through the points (10,1) and (0,2). 4.(1 pt) Find the equation of the line passing WeBWorK assignment Thought of the day: It s not that I m so smart; it s just that I stay with problems longer. Albert Einstein.( pt) a. Find the slope of the line passing through the points (8,4) and (,8).

More information

Calculus AB Topics Limits Continuity, Asymptotes

Calculus AB Topics Limits Continuity, Asymptotes Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3

More information

Name: Instructor: Exam 3 Solutions. Multiple Choice. 3x + 2 x ) 3x 3 + 2x 2 + 5x + 2 3x 3 3x 2x 2 + 2x + 2 2x 2 2 2x.

Name: Instructor: Exam 3 Solutions. Multiple Choice. 3x + 2 x ) 3x 3 + 2x 2 + 5x + 2 3x 3 3x 2x 2 + 2x + 2 2x 2 2 2x. . Exam 3 Solutions Multiple Choice.(6 pts.) Find the equation of the slant asymptote to the function We have so the slant asymptote is y = 3x +. f(x) = 3x3 + x + 5x + x + 3x + x + ) 3x 3 + x + 5x + 3x

More information

Correlation with College Board Advanced Placement Course Descriptions

Correlation with College Board Advanced Placement Course Descriptions Correlation with College Board Advanced Placement Course Descriptions The following tables show which sections of Calculus: Concepts and Applications cover each of the topics listed in the 2004 2005 Course

More information

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. 997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..

More information

BC Exam 2 - Part I 28 questions No Calculator Allowed. C. 1 x n D. e x n E. 0

BC Exam 2 - Part I 28 questions No Calculator Allowed. C. 1 x n D. e x n E. 0 1. If f x ( ) = ln e A. n x x n BC Exam - Part I 8 questions No Calculator Allowed, and n is a constant, then f ( x) = B. x n e C. 1 x n D. e x n E.. Let f be the function defined below. Which of the following

More information

AP Calculus AB Winter Break Packet Happy Holidays!

AP Calculus AB Winter Break Packet Happy Holidays! AP Calculus AB Winter Break Packet 04 Happy Holidays! Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of

More information

AB 1: Find lim. x a.

AB 1: Find lim. x a. AB 1: Find lim x a f ( x) AB 1 Answer: Step 1: Find f ( a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f ( x). Do any cancellations and go back to Step 1. If you

More information

AP Calculus BC 2011 Free-Response Questions Form B

AP Calculus BC 2011 Free-Response Questions Form B AP Calculus BC 11 Free-Response Questions Form B About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded

More information

BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points)

BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points) BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: United and Continuous! ( points) For #- below, find the its, if they eist.(#- are pt each) ) 7 ) 9 9 ) 5 ) 8 For #5-7, eplain why

More information

AP Calculus AB. Scoring Guidelines

AP Calculus AB. Scoring Guidelines 17 AP Calculus AB Scoring Guidelines 17 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the

More information

Calculus BC AP/Dual Fall Semester Review Sheet REVISED 1 Name Date. 3) Explain why f(x) = x 2 7x 8 is a guarantee zero in between [ 3, 0] g) lim x

Calculus BC AP/Dual Fall Semester Review Sheet REVISED 1 Name Date. 3) Explain why f(x) = x 2 7x 8 is a guarantee zero in between [ 3, 0] g) lim x Calculus BC AP/Dual Fall Semester Review Sheet REVISED Name Date Eam Date and Time: Read and answer all questions accordingly. All work and problems must be done on your own paper and work must be shown.

More information

Calculus with the Graphing Calculator

Calculus with the Graphing Calculator Calculus with the Graphing Calculator Using a graphing calculator on the AP Calculus exam Students are expected to know how to use their graphing calculators on the AP Calculus exams proficiently to accomplish

More information

AP* Calculus Free-response Question Type Analysis and Notes Revised to include the 2013 Exam By Lin McMullin

AP* Calculus Free-response Question Type Analysis and Notes Revised to include the 2013 Exam By Lin McMullin AP* Calculus Free-response Question Type Analysis and Notes Revised to include the 2013 Exam By Lin McMullin General note: AP Questions often test several diverse ideas or concepts in the same question.

More information

AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 2. the

AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 2. the AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 2 Let f be the function defined for x 0 with f ( 0) = 5 and f, the ( x 4) 2 first derivative of f, given by f ( x) = e sin ( x ). The graph of y

More information

Applied Calculus I Practice Final Exam Solution Notes

Applied 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 information

Curriculum Framework Alignment and Rationales for Answers

Curriculum Framework Alignment and Rationales for Answers The multiple-choice section on each eam is designed for broad coverage of the course content. Multiple-choice questions are discrete, as opposed to appearing in question sets, and the questions do not

More information

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B) AP CALCULUS AB 004 SCORING GUIDELINES (Form B) Question 4 The figure above shows the graph of f, the derivative of the function f, on the closed interval 1 x 5. The graph of f has horizontal tangent lines

More information

x f(x)

x f(x) CALCULATOR SECTION. For y + y = 8 find d point (, ) on the curve. A. B. C. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) = e, A(t) is measured in tons of

More information

AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66

AP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66 AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle

More information

7.2 Trapezoidal Approximation

7.2 Trapezoidal Approximation 7. Trapezoidal Approximation NOTES Write your questions here! Riemann Sum f(x) = x + 1 [1,3] f(x) = x + 1 The Definite Integral b f(x)dx a Trapezoidal Approximation for interval [1,3] with n subintervals

More information

Purdue University Study Guide for MA Credit Exam

Purdue University Study Guide for MA Credit Exam Purdue University Study Guide for MA 16010 Credit Exam Students who pass the credit exam will gain credit in MA16010. The credit exam is a two-hour long exam with multiple choice questions. No books or

More information

Math 131 Exam II "Sample Questions"

Math 131 Exam II Sample Questions Math 11 Exam II "Sample Questions" This is a compilation of exam II questions from old exams (written by various instructors) They cover chapters and The solutions can be found at the end of the document

More information

MA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM. Name (Print last name first):... Student ID Number (last four digits):...

MA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM. Name (Print last name first):... Student ID Number (last four digits):... CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM Name (Print last name first):............................................. Student ID Number (last four digits):........................

More information

AP Calculus BC. Course Overview. Course Outline and Pacing Guide

AP Calculus BC. Course Overview. Course Outline and Pacing Guide AP Calculus BC Course Overview AP Calculus BC is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide

More information

Chapter 6: The Definite Integral

Chapter 6: The Definite Integral Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives

More information

1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2.

1. Find A and B so that f x Axe Bx. has a local minimum of 6 when. x 2. . Find A and B so that f Ae B has a local minimum of 6 when.. The graph below is the graph of f, the derivative of f; The domain of the derivative is 5 6. Note there is a cusp when =, a horizontal tangent

More information

CLEP Calculus. Time 60 Minutes 45 Questions. For each question below, choose the best answer from the choices given. 2. If f(x) = 3x, then f (x) =

CLEP Calculus. Time 60 Minutes 45 Questions. For each question below, choose the best answer from the choices given. 2. If f(x) = 3x, then f (x) = CLEP Calculus Time 60 Minutes 5 Questions For each question below, choose the best answer from the choices given. 7. lim 5 + 5 is (A) 7 0 (C) 7 0 (D) 7 (E) Noneistent. If f(), then f () (A) (C) (D) (E)

More information

ANOTHER FIVE QUESTIONS:

ANOTHER FIVE QUESTIONS: No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e ln ln @ @ Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE

More information

AP Calculus BC Scope & Sequence

AP Calculus BC Scope & Sequence AP Calculus BC Scope & Sequence Grading Period Unit Title Learning Targets Throughout the School Year First Grading Period *Apply mathematics to problems in everyday life *Use a problem-solving model that

More information

Math 180, Final Exam, Fall 2007 Problem 1 Solution

Math 180, Final Exam, Fall 2007 Problem 1 Solution Problem Solution. Differentiate with respect to x. Write your answers showing the use of the appropriate techniques. Do not simplify. (a) x 27 x 2/3 (b) (x 2 2x + 2)e x (c) ln(x 2 + 4) (a) Use the Power

More information