AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

Size: px
Start display at page:

Download "AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)"

Transcription

1 SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the fuctio give by g ftdt. (a) Fid g (, ) g ( ), ad g ( ). (b) Fid the average rate of chage of g o the iterval. (c) For how may values c, where < c <, is g () c equal to the average rate foud i part (b)? Eplai your reasoig. (d) Fid the -coordiate of each poit of iflectio of the graph of g o the iterval < < 7. Justify your aswer. (a) g() f( t) dt ( 4 + ) g () f() 4 g() f() 4 : : g() : g() : g() (b) g() g() () ftdt 7 ()(4) + (4 + ) : : g() g() f( t) dt : aswer (c) There are two values of c. We eed 7 g( c) f( c) The graph of f itersects the lie places betwee ad. 7 y at two : aswer of : : reaso Note: / if aswer is by MVT (d) ad 5 because g f chages from icreasig to decreasig at, ad from decreasig to icreasig at 5. : ad 5 oly : : justificatio (igore discussio at 4) Copyright by College Etrace Eamiatio Board. All rights reserved. Available at apcetral.collegeboard.com. 6

2 SCORING GUIDELINES (Form B) Questio 6 Let g be the piecewise-liear fuctio defied o [, 4 ] whose graph is give above, ad let f g 4 cos. (a) Fid f d. Show the computatios that lead to your aswer. (b) Fid all -values i the ope iterval (, 4 ) for which f has a critical poit. (c) Let h gt dt. Fid h. 4 4 (a) ( cos ) 4 f d g d 6 si 6 : { : atiderivative : aswer (b) f g si + si for < < + + si for < < 4 f does ot eist at. For < <, f. For < < 4, f whe. ( ) d : cos d 4 : : g : : f has critical poits at ad. (c) h g h ( ) g( ) : h : : aswer The College Board. Visit the College Board o the Web:

3 9 SCORING GUIDELINES (Form B) Questio A cotiuous fuctio f is defied o the closed iterval 4 6. The graph of f cosists of a lie segmet ad a curve that is taget to the -ais at, as show i the figure above. O the iterval < < 6, the fuctio f is twice differetiable, with f >. (a) Is f differetiable at? Use the defiitio of the derivative with oe-sided limits to justify your aswer. (b) For how may values of a, 4 a 6, equal to? Give a reaso for your aswer. (c) Is there a value of a, 4 a 6, < is the average rate of chage of f o the iterval [ a,6] < for which the Mea Value Theorem, applied to the iterval [ a ] guaratees a value c, a < c < 6, at which f ( c)? Justify your aswer. (d) The fuctio g is defied by g f( t) dt for 4 6. O what itervals cotaied i [ 4, 6] is the graph of g cocave up? Eplai your reasoig. f( h) f (a) lim h h f( h) f lim < + h h Sice the oe-sided limits do ot agree, f is ot differetiable at. f( 6) f( a) (b) whe f( a) f( 6. ) There are 6 a two values of a for which this is true. (c) Yes, a. The fuctio f is differetiable o the iterval < < 6 ad cotiuous o 6. f( 6) f Also,. 6 6 By the Mea Value Theorem, there is a value c, < c < 6, such that f ( c). (d) g f, g f g > whe f > This is true for 4 < < ad < < 6.,6, : sets up differece quotiet at : { : aswer with justificatio : epressio for average rate of chage : { : aswer with reaso : aswers yes ad idetifies a : { : justificatio : g f : : cosiders g > : aswer 9 The College Board. All rights reserved. Visit the College Board o the Web:

4 8 SCORING GUIDELINES (Form B) Questio 4 The fuctios f ad g are give by f 4 + t dt ad g f( si ). (a) Fid f ad g. (b) Write a equatio for the lie taget to the graph of y g at. (c) Write, but do ot evaluate, a itegral epressio that represets the maimum value of g o the iterval. Justify your aswer. (a) f 4 + g f ( si ) cos 4 : : : f g 4 + ( si ) cos (b) g( ), g ( ) 6 Taget lie: y 6( ) : g or g : : taget lie equatio (c) For < <, g oly at. g g( ) + > g 4 t dt, is The maimum value of g o [ ] 4 + t dt. : : sets g : justifies maimum at : itegral epressio for g 8 The College Board. All rights reserved. Visit the College Board o the Web:

5 5 SCORING GUIDELINES (Form B) Questio 4 The graph of the fuctio f above cosists of three lie segmets. (a) Let g be the fuctio give by g f( t) dt. 4 For each of g(, ) g (, ) ad g (, ) fid the value or state that it does ot eist. (b) For the fuctio g defied i part (a), fid the -coordiate of each poit of iflectio of the graph of g o the ope iterval 4 < <. Eplai your reasoig. (c) Let h be the fuctio give by h f( tdt ). Fid all values of i the closed iterval 4 for which h. (d) For the fuctio h defied i part (c), fid all itervals o which h is decreasig. Eplai your reasoig. 5 (a) g( ) f( t) dt ( 5) 4 g ( ) f( ) g ( ) does ot eist because f is ot differetiable at. : : g( ) : g ( ) : g ( ) (b) g f chages from icreasig to decreasig at. : (oly) : : reaso (c),, : correct values each missig or etra value (d) h is decreasig o [, ] h f < whe f > : iterval : : reaso Copyright 5 by College Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for AP studets ad parets). 5

6 Let be the lie taget to the graph of y where >, as show above. (a) Fid d i terms of. AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Questio 6 at the poit (, ), (b) Let T be the triagular regio bouded by, the -ais, ad the lie. Show that the area of T is. (c) Let S be the regio bouded by the graph of y, the lie, ad the -ais. Epress the area of S i terms of ad determie the value of that maimizes the area of S. (a) + d : + + : atiderivative of : aswer (b) Let b be the legth of the base of triagle T. b is the slope of lie, which is Area( T) b() : : slope of lie is : base of T is : shows area is (c) Area( S) d Area( T) + d Area( S ) + d ( + ) 4 : : area of S i terms of : derivative : sets derivative equal to : solves for ( + ) ( + ) + Copyright 4 by College Etrace Eamiatio Board. All rights reserved. Visit apcetral.collegeboard.com (for AP professioals) ad (for AP studets ad parets). 7

7 AP CALCULUS BC SCORING GUIDELINES (Form B) Questio 4 6DACHFDB@EBBAHAJE>ABK?JEBJDA?IA@ EJAHLГ! # EIIDMEJDABECKHA>LA6DACHFDB BDIDHEJJCAJEAJN$AJ CN # BJ@J BH Г! > N > # $ N.E@C$ C $ > MDJEJAHLIEIC@A?HAIECKIJEBOOKHIMAH? MDJEJAHLIEIJDACHFDBC??LA@MKIJEBOOKHIMAH BJ@J KIECIENIK>EJAHLIBACJD J! Г! $ C$ # # C$ $! C C $ $ B$! C $ C$ B$ > # IE?A Г! C N BN # KIJEBE?JE? 6DACHFDBCEI??LA@M$# IE?A EJAHL C B EI@A?HAIECJDEIEJAHL Г! Г JHFAE@AJD@ Copyright by College Etrace Eamiatio Board. All rights reserved. Advaced Placemet Program ad AP are registered trademarks of the College Etrace Eamiatio Board. 5

AP Calculus BC 2007 Scoring Guidelines Form B

AP Calculus BC 2007 Scoring Guidelines Form B AP Calculus BC 7 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

AP Calculus BC 2005 Scoring Guidelines

AP Calculus BC 2005 Scoring Guidelines AP Calculus BC 5 Scorig Guidelies The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad

More information

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

AP Calculus BC 2011 Scoring Guidelines Form B

AP Calculus BC 2011 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The College Board The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad opportuity. Fouded i 9, the College

More information

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES The figure above shows the graph of the piecewise-linear function f. For 4, the function g is defined by g( ) = f ( t) (a) Does g have a relative minimum, a relative maimum, or neither at =? Justify your

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

MAT136H1F - Calculus I (B) Long Quiz 1. T0101 (M3) Time: 20 minutes. The quiz consists of four questions. Each question is worth 2 points. Good Luck!

MAT136H1F - Calculus I (B) Long Quiz 1. T0101 (M3) Time: 20 minutes. The quiz consists of four questions. Each question is worth 2 points. Good Luck! MAT36HF - Calculus I (B) Log Quiz. T (M3) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each

More information

1988 AP Calculus BC: Section I

1988 AP Calculus BC: Section I 988 AP Calculus BC: Sectio I 9 Miutes No Calculator Notes: () I this eamiatio, l deotes the atural logarithm of (that is, logarithm to the base e). () Uless otherwise specified, the domai of a fuctio f

More information

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4 . If f = e ta -, the f = e e p e e p e p+ 4 f = e ta -, so f = e ta - + e, so + f = e p + e = e p + e or f = e p + 4. The slope of the lie taget to the curve - + = at the poit, - is - 5 Differetiate -

More information

Taylor Series (BC Only)

Taylor Series (BC Only) Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor

More information

MATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.)

MATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.) MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral

More information

Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 105: Review for Final Exam, Part II - SOLUTIONS Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum

More information

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i) Math PracTest Be sure to review Lab (ad all labs) There are lots of good questios o it a) State the Mea Value Theorem ad draw a graph that illustrates b) Name a importat theorem where the Mea Value Theorem

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

Objective Mathematics

Objective Mathematics 6. If si () + cos () =, the is equal to :. If <

More information

(A) 0 (B) (C) (D) (E) 2.703

(A) 0 (B) (C) (D) (E) 2.703 Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released

More information

Student s Printed Name:

Student s Printed Name: Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide

More information

PRACTICE FINAL/STUDY GUIDE SOLUTIONS

PRACTICE FINAL/STUDY GUIDE SOLUTIONS Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)

More information

Continuous Functions

Continuous Functions Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio

More information

AP CALCULUS BC 2015 SCORING GUIDELINES

AP CALCULUS BC 2015 SCORING GUIDELINES 05 SCORING GUIDELINES Question 5 Consider the function f =, where k is a nonzero constant. The derivative of f is given by k f = k ( k). (a) Let k =, so that f =. Write an equation for the line tangent

More information

CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.

CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve

More information

AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 6. Scoring Guideline.

AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 6. Scoring Guideline. 207 AP Calculus BC Sample Studet Resposes ad Scorig Commetary Iside: RR Free Respose Questio 6 RR Scorig Guidelie RR Studet Samples RR Scorig Commetary 207 The College Board. College Board, Advaced Placemet

More information

Honors Calculus Homework 13 Solutions, due 12/8/5

Honors Calculus Homework 13 Solutions, due 12/8/5 Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y

More information

Maximum and Minimum Values

Maximum and Minimum Values Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f

More information

f t dt. Write the third-degree Taylor polynomial for G

f t dt. Write the third-degree Taylor polynomial for G AP Calculus BC Homework - Chapter 8B Taylor, Maclauri, ad Power Series # Taylor & Maclauri Polyomials Critical Thikig Joural: (CTJ: 5 pts.) Discuss the followig questios i a paragraph: What does it mea

More information

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(

More information

Mathematics Extension 1

Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

AP CALCULUS AB 2001 SCORING GUIDELINES. Question 3

AP CALCULUS AB 2001 SCORING GUIDELINES. Question 3 AP CALCULUS AB SCORING GUIDELINES Question A car is traveling on a straight road with velocity ft/sec at time t =. For > t > 8 seconds, the car s acceleration at (), in ft/sec, is the piecewise linear

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

Calculus I Practice Test Problems for Chapter 5 Page 1 of 9

Calculus I Practice Test Problems for Chapter 5 Page 1 of 9 Calculus I Practice Test Problems for Chapter 5 Page of 9 This is a set of practice test problems for Chapter 5. This is i o way a iclusive set of problems there ca be other types of problems o the actual

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Diploma Programme. Mathematics HL guide. First examinations 2014

Diploma Programme. Mathematics HL guide. First examinations 2014 Diploma Programme First eamiatios 014 33 Topic 6 Core: Calculus The aim of this topic is to itroduce studets to the basic cocepts ad techiques of differetial ad itegral calculus ad their applicatio. 6.1

More information

7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.

7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3. Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud

More information

GULF MATHEMATICS OLYMPIAD 2014 CLASS : XII

GULF MATHEMATICS OLYMPIAD 2014 CLASS : XII GULF MATHEMATICS OLYMPIAD 04 CLASS : XII Date of Eamiatio: Maimum Marks : 50 Time : 0:30 a.m. to :30 p.m. Duratio: Hours Istructios to cadidates. This questio paper cosists of 50 questios. All questios

More information

AP Exam Practice Questions for Chapter 9

AP Exam Practice Questions for Chapter 9 AP Eam Practice Questios for Chater 9 AP Eam Practice Questios for Chater 9. Evaluate each series. I: Because ad 0, the series diverges. e II: si Because r e si.99 >, the series diverges. 5 III:! ( ) +

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This

More information

Name: Math 10550, Final Exam: December 15, 2007

Name: Math 10550, Final Exam: December 15, 2007 Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder

More information

Calculus 2 Test File Spring Test #1

Calculus 2 Test File Spring Test #1 Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad

More information

AP CALCULUS BC 2014 SCORING GUIDELINES

AP CALCULUS BC 2014 SCORING GUIDELINES AP CALCULUS BC 04 SCORING GUIDELINES Questio 6 + The Taylor series for a fuctio f about x = is give by ( ) ( x ) ad coverges to f( x ) for = x < R, where R is the radius of covergece of the Taylor series.

More information

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

More information

f x x c x c x c... x c...

f x x c x c x c... x c... CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate

More information

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questios. For the quadratic fuctio show below, the coordiates of its verte are () 0, (), 7 (3) 6, (4) 3, 6. A quadratic

More information

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

More information

SCORE. Exam 2. MA 114 Exam 2 Fall 2017

SCORE. Exam 2. MA 114 Exam 2 Fall 2017 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator

More information

Math 122 Test 3 - Review 1

Math 122 Test 3 - Review 1 I. Sequeces ad Series Math Test 3 - Review A) Sequeces Fid the limit of the followig sequeces:. a = +. a = l 3. a = π 4 4. a = ta( ) 5. a = + 6. a = + 3 B) Geometric ad Telescopig Series For the followig

More information

} is said to be a Cauchy sequence provided the following condition is true.

} is said to be a Cauchy sequence provided the following condition is true. Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r

More information

Math 21B-B - Homework Set 2

Math 21B-B - Homework Set 2 Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio

More information

MATH CALCULUS II Objectives and Notes for Test 4

MATH CALCULUS II Objectives and Notes for Test 4 MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius

More information

SCORE. Exam 2. MA 114 Exam 2 Fall 2016

SCORE. Exam 2. MA 114 Exam 2 Fall 2016 MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use

More information

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent. 06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.

The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE. NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES

More information

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b) Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

More information

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart. MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a

More information

Calculus 2 Test File Fall 2013

Calculus 2 Test File Fall 2013 Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

Series Solutions (BC only)

Series Solutions (BC only) Studet Study Sessio Solutios (BC oly) We have itetioally icluded more material tha ca be covered i most Studet Study Sessios to accout for groups that are able to aswer the questios at a faster rate. Use

More information

MIDTERM 2 CALCULUS 2. Monday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM

MIDTERM 2 CALCULUS 2. Monday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM MIDTERM 2 CALCULUS 2 MATH 23 FALL 218 Moday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

More information

18.01 Calculus Jason Starr Fall 2005

18.01 Calculus Jason Starr Fall 2005 Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative

More information

MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of

MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial

More information

6.) Find the y-coordinate of the centroid (use your calculator for any integrations) of the region bounded by y = cos x, y = 0, x = - /2 and x = /2.

6.) Find the y-coordinate of the centroid (use your calculator for any integrations) of the region bounded by y = cos x, y = 0, x = - /2 and x = /2. Calculus Test File Sprig 06 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = +. Produce a solid by revolvig the regio aroud

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

JEE ADVANCED 2013 PAPER 1 MATHEMATICS

JEE ADVANCED 2013 PAPER 1 MATHEMATICS Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot

More information

NATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2

NATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2 NATIONAL JUNIOR COLLEGE SENIOR HIGH PROMOTIONAL EXAMINATIONS Higher MATHEMATICS 9758 9 September 06 hours Additioal Materials: Aswer Paper List of Formulae (MF6) Cover Sheet READ THESE INSTRUCTIONS FIRST

More information

TEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2

TEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2 TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS 9740 9 Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write your

More information

Solutions to quizzes Math Spring 2007

Solutions to quizzes Math Spring 2007 to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x

More information

1 Approximating Integrals using Taylor Polynomials

1 Approximating Integrals using Taylor Polynomials Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................

More information

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Name: Date: Part I Questios UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS. For the quadratic fuctio show, the coordiates. of its verte are 0, (3) 6, (), 7 (4) 3, 6. A quadratic fuctio

More information

Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Example 2. Find the upper bound for the remainder for the approximation from Example 1. Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

More information

Indian Institute of Information Technology, Allahabad. End Semester Examination - Tentative Marking Scheme

Indian Institute of Information Technology, Allahabad. End Semester Examination - Tentative Marking Scheme Idia Istitute of Iformatio Techology, Allahabad Ed Semester Examiatio - Tetative Markig Scheme Course Name: Mathematics-I Course Code: SMAT3C MM: 75 Program: B.Tech st year (IT+ECE) ate of Exam:..7 ( st

More information

Taylor Polynomials and Taylor Series

Taylor Polynomials and Taylor Series Taylor Polyomials ad Taylor Series Cotets Taylor Polyomials... Eample.... Eample.... 4 Eample.3... 5 Eercises... 6 Eercise Solutios... 8 Taylor s Iequality... Eample.... Eample.... Eercises... 3 Eercise

More information

Error for power series (Day 2) YOU MAY USE YOUR CALCULATOR TO COMPUTE FRACTIONS AND OTHER SIMPLE OPERATIONS

Error for power series (Day 2) YOU MAY USE YOUR CALCULATOR TO COMPUTE FRACTIONS AND OTHER SIMPLE OPERATIONS AP Calculus BC CHAPTE B WOKSHEET INFINITE SEQUENCES AND SEIES Name Seat # Date Error or power series (Day ) YOU MAY USE YOU CALCULATO TO COMPUTE FACTIONS AND OTHE SIMPLE OPEATIONS a) Approimate si usig

More information

MTH Assignment 1 : Real Numbers, Sequences

MTH Assignment 1 : Real Numbers, Sequences MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a

More information

+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We

More information

MEI Conference 2009 Stretching students: A2 Core

MEI Conference 2009 Stretching students: A2 Core MEI Coferece 009 Stretchig studets: A Core Preseter: Berard Murph berard.murph@mei.org.uk Workshop G How ca ou prove that these si right-agled triagles fit together eactl to make a 3-4-5 triagle? What

More information

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.

COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n. COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where

More information

AP CALCULUS AB/CALCULUS BC 2017 SCORING GUIDELINES

AP CALCULUS AB/CALCULUS BC 2017 SCORING GUIDELINES AP CALCULUS AB/CALCULUS BC 07 SCORING GUIDELINES Question 4 H ( 0) = ( 9 7) = 6 4 H ( 0) = 9 An equation for the tangent line is y = 9 6 t. : slope : : tangent line : approimation The internal temperature

More information

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist. Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si

More information

Area Approximation and Accumulation

Area Approximation and Accumulation Area Approximatio ad Accumulatio Studet should be able to: Recogize that a defiite itegral gives a accumulatio or total Always give meaig to the itegral i CONTEXT to the problem Give the uits of measuremet

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P1 FEBRUARY/MARCH 011 MARKS: 150 TIME: 3 hours This questio paper cosists of 8 pages, 3 diagram sheets ad 1 iformatio sheet. Please tur over Mathematics/P1

More information

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =

More information

MATHEMATICS (Three hours and a quarter)

MATHEMATICS (Three hours and a quarter) MATHEMATICS (Three hours ad a quarter) (The first fiftee miutes of the eamiatio are for readig the paper oly. Cadidates must NOT start writig durig this time.) Aswer Questio from Sectio A ad questios from

More information

For use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel)

For use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel) For use oly i Badmito School November 0 C Note C Notes (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i Badmito School November 0 C Note Copyright www.pgmaths.co.uk

More information

MA1200 Exercise for Chapter 7 Techniques of Differentiation Solutions. First Principle 1. a) To simplify the calculation, note. Then. lim h.

MA1200 Exercise for Chapter 7 Techniques of Differentiation Solutions. First Principle 1. a) To simplify the calculation, note. Then. lim h. MA00 Eercise for Chapter 7 Techiques of Differetiatio Solutios First Priciple a) To simplify the calculatio, ote The b) ( h) lim h 0 h lim h 0 h[ ( h) ( h) h ( h) ] f '( ) Product/Quotiet/Chai Rules a)

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

Math 116 Second Midterm November 13, 2017

Math 116 Second Midterm November 13, 2017 Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

B U Department of Mathematics Math 101 Calculus I

B U Department of Mathematics Math 101 Calculus I B U Departmet of Mathematics Math Calculus I Sprig 5 Fial Exam Calculus archive is a property of Boğaziçi Uiversity Mathematics Departmet. The purpose of this archive is to orgaise ad cetralise the distributio

More information

Math 142, Final Exam. 5/2/11.

Math 142, Final Exam. 5/2/11. Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute

1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

JEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018)

JEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018) JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART- : JEE(Advaced) 08/Paper- SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse

More information

Math 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals

Math 1314 Lesson 16 Area and Riemann Sums and Lesson 17 Riemann Sums Using GeoGebra; Definite Integrals Math 1314 Lesso 16 Area ad Riema Sums ad Lesso 17 Riema Sums Usig GeoGebra; Defiite Itegrals The secod questio studied i calculus is the area questio. If a regio coforms to a kow formula from geometry,

More information

HOMEWORK #10 SOLUTIONS

HOMEWORK #10 SOLUTIONS Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous

More information

Power Series: A power series about the center, x = 0, is a function of x of the form

Power Series: A power series about the center, x = 0, is a function of x of the form You are familiar with polyomial fuctios, polyomial that has ifiitely may terms. 2 p ( ) a0 a a 2 a. A power series is just a Power Series: A power series about the ceter, = 0, is a fuctio of of the form

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information