AP Calculus BC Chapter 6 - AP Exam Problems solutions
|
|
- Posy McLaughlin
- 5 years ago
- Views:
Transcription
1 P Calculus BC Chapter 6 - P Eam Problems solutions. E Epand the integrand. ( + ) d = ( + ) d = C.. f ( ) = sin+ C, f( ) = +cos+c+k. Option is the only one with this form. sec d = d ( tan ) =tan + C 4. Since f is linear, its second derivative is zero. The integral gives the area of a rectangle with zero height and width ( b a). This area is zero. 5. D d= = 4 = D = ( ) 6 = ( +k) d d k d = 0 + ( ) k = 4k k = 4 7. C = d d = = 8. e e E d d = ln e 0 e = = = e 9 9. D ( ) d = ) ( d) = ( ) ( 8 ) = ( = ( ) E d d lim d lim 0 = L 0 + L = L 0 + = which does not eist. L
2 . ( + )( d = d = ( )d = ( ) ) = + +. C k 0 8 = k = k k = 7, so k =. k k d= ( k (k ) ) )= 7 0 = ( + = only when k =. 4. The value of this integral is. Option is also and none of the others have a value of. Visualizing the graphs of y = sin and y = cos is a useful approach to the problem. 5. D The graph is a V with verte at =. The integral gives the sum of the areas of the two triangles that the V forms with the horizontal ais for from 0 to. These triangles have areas of / and respectively. 6. C The graph is a V with verte at =. The integral gives the sum of the areas of the two triangles that the V forms with the horizontal ais for from to 4. These triangles have areas of and 0.5 respectively. 7. By the Fundamental Theorem of Calculus c f ( d ) = f( ) = f(c) f(0) 0 0 c
3 8. B e e f( )d d d= + lne= = B ( ) d + ) d = ( ) d ( ( ) d = ( ) d = = = ln ln ln ln 0. E Q ( ) = p() degree of Q is n+. Differentiating the epression in (i) gives f ( ) = a + b Let =. Then from (ii), a+ b= f () =6 a+ b= f ( ) = 8 Solving these two equations gives a = and b = 6. Therefore f ( ) = 6 and hence f ( ) = 4 +C Using (iii) gives 8 = (4 +C) d 4 = ( + C) = (6 8 + C) ( + C ) = 8 + C Hence C = 0 and f( ) =
4 . D ( d ) +5) ( + 5) ( d= 6 = ( + 5) +C = ( + 5) + C 6 6. D ( ) θ = + sin θ ( cos θdθ ) =( + sin ) ( ) Let u = +. Then / d = C u / du = u +C = D 9 ( + ) d = ( d)= ( + ) = ( )= ( + ) C cos() d = cos()(d) = sin( ) + C 7. D = = ( cos cos0)= sin( ) d cos() C sin( + )d = )(d) = cos( sin( + + ) + C 9. B )= ( cos ) = y ( +C; Let =, 0 cos + C C= 0. y(0) = ( cos0) = 0. a du u =sin u + C, a> 0 a sin ( 0) = 4 d = sin 0 =sin 0
5 . du sin u d du = d =sin +C a u a 5 5. B f ( ) = e +e = e ( + ); f ( ) < 0 for < < 0. E + e e d = e e ( e d ). This is of the form u edu, u= e, so e + e e e d= +C 4. Use the technique of antiderivatives by parts u = dv= e d du = d v = e e e d= e e +C 4 5. e d (4 d) = 0 4 e = e = ( e ) 6. B sin( ) tan( ) d ln cos( cos( ) d = ) +C = 7. B d d = = + ( ) ln = ( ln0 ln 5 ) + + = ln 8. E du d d ln d ln = = u (ln ) + C.This is udu with u = ln, so the value is
6 9. C F(9) F() 9 (ln t) = dt =5.87 using a calculator. Since F() = 0, F(9) = t 40. B Or solve the differential equation with an initial condition by finding an antiderivative for (ln ). This is of the form udu where u = ln. Hence F( )= 4 (ln) + C and since 4 F() = 0, C = 0. Therefore (ln 9) 4 F(9) = = ( + ) d + = d = ln = (ln 8 ln ) E Since F is an antiderivative of f, f()d= F( ) = F(6) F() u =, du = d; when =, u = and when = 4, u = u u d du = u = u 4 du ( ) B Separate the variables. y dy =d; =+ C; y =. Substitute the point (, ) y + C to find the value of C. Then = C =, so y + C =. When =, y =. 44. C This is the differential equation for eponential growth. t y = y(0)e = e t ; = e t ; t = ln t = ln = ln 45. dy = y d dy = d ln y = + K ; y = Ce and y(0) = 8 so, y= 8e y 46. C dy y t an = sec d ln y = tan + k y= Ce. y(0) = 5 y= 5e tan 47. C,ln y + Ce dy d y = = C,y =. Only C is of this form.
7 48. (a) dy d = + y : answer dy d = y = 4 = + 4 = 4 8 = (b) y 4 = ( ) f(.) 4 (. ) { : equation of tangent line : uses equation to approimate f(.) (c) f(.) = 4. y dy = ( + ) d y dy = ( + ) d y = + + C 4 = + + C 4 = C y = y = is branch with point (, 4) f() = : separates variables : antiderivative of dy term : antiderivative of d term 5 : uses y = 4 when = to pick one function out of a family of functions : solves for y 0/ if solving a linear equation in y 0/ if no constant of integration Note: ma 0/5 if no separation of variables Note: ma /5 [ ] if substitutes value(s) for, y, or dy/d before antidifferentiation (d) f(.) = : answer, from student s solution to the given differential equation in (c)
8 49. (a) ln y dy = d y (ln y) = + C or (ln y) = + C or ln y or =± C y = e ± C (b) (ln e ) C = 4 (ln y) = 0 + C = 4 ln y =± 4 But =0, y = e ln y = 4 y = e 4 (c) y If =, then y = and ln y = 0. This causes ln y to be undefined.
9 50. Å Y E E Y E DY X DX Y X # X # Y LN X # LN Y LN X # ÅÅ# E E ÅSEPRTESÅVRIBLES ÅNTIDERIVTIVEÅOFÅDYÅTERM ÅNTIDERIVTIVEÅOFÅDX ÅTERM ÅCONSTNTÅOFÅINTEGRTION Å ÅUSESÅINITILÅCONDITIONÅF ÅSOLVESÅFORÅY Å.OTEÅÅIFÅY ÅISÅNOTÅÅLOGRITHMICÅFUNCTIONÅOFÅX.OTEÅMXÅÅ;=ÅIFÅNOÅCONSTNTÅOF INTEGRTION.OTEÅÅIFÅNOÅSEPRTIONÅOFÅVRIBLES B $OMINÅX E X E X E E NGEÅd Y d ÅX E ÅDOMIN Å ÅÅÅÅÅÅ.OTEÅÅIFÅÅISÅNOTÅINÅTHEÅDOMIN ÅRNGE.OTEÅÅIFÅYÅISÅNOTÅÅLOGRITHMICÅFUNCTIONÅOFÅX
10 Г O ГO=! O!Г!Г IBD=I=?=EEK=JJDEIFEJ H *?=KIBEI?JEKKIBH N #JDH JDBJBN!=@ HECDJBN!6DHBHBD=I=?= EEK=JN! > O@O O! ГN@N! N Г N + & & Г& + +& O $ N ГN $ O Г $ N ГN $ N!!?=EEK KIJEBE?=JE IF=H=JIL=HE=>I =JE@HEL=JELB@OJH =JE@HEL=JELB@N JH $?IJ=JBEJCH=JE KIIEEJE=?@EJEC$ Г" ILIBHO J=N!$EB?IJ=J BEJCH=JE J$EBIF=H=JEBL=HE=>I
11 5. (a) dy d dy d dy y (6 Г ) Гy d = y (6 Г) Гy, 4 4 0Г Г 8 : dy : d < Г > product rule or chain rule error : value at, 4 (b) dy (6 )d Г y Г 6 Г C y Г 4 8 Г 9 C 9 C C Г 6 : : separates variables : antiderivative of dy term : antiderivative of d term : constant of integration : uses initial condition f () 4 : solves for y y Г6 Note: ma /6 [ ] if no constant of integration Note: 0/6 if no separation of variables
12 5. Г f ()d f= ( )d lim f = ()d lim f( ) b@ = lim fb) ( Г f() 0 Г 4 Г4 b@ b b@ b : : use of FTC : answer from limiting process (b) f(.5) N f() f =() (0.5) = 4 Г ()(4)(0.5) Г f() NГ f= (.5)(0.5) NГГ (.5)( Г)(0.5).5 : : Euler's method equations or equivalent table : Euler approimation to f () (not eligible without first point) (c) dy Г d y ln y Г k y Ce Г 4 Ce Г ; C 4e y 4e e Г : separates variables : antiderivatives 5 : : constant of integration : uses initial condition f () 4 : solves for y Note: ma /5 [ ] if no constant of integration Note: 0/5 if no separation of variables
13 5. E 4 4 dt = ln t =ln 4 ln = ln 4 t 54. C Use the technique of antiderivatives by parts: u = dv= e d du =d v = e e e d= ( e e ) = e E Use the technique of antiderivatives by parts: Let u = and dv = cos d. ( ) = cos d = sin sin d ( s in +cos ) 0 = E Use the technique of antiderivatives by parts: Let u = and dv = cos d. ( ) = cos d = sin sin d ( s in +cos ) 0 = E Use the technique of antiderivatives by parts: u = and dv= sec d sec d= tan tan d= tan +ln cos + C 58. B Use the technique of antiderivative by parts, which is no longer in the B Course Description. The formula is udv= uv vdu. Let u = f( ) and dv = d. This leads to f ( ) d = f ( ) f ( )d.
14 59. B Use the technique of antiderivatives by parts: = dv =sin d u f ( ) du f ( ) = d v = cos f ( )sin d = f( )cos + f ( )sin d= f ( )cos + f ( )cos d and we are given that cos d f ( ) = f ( ) = 60. Use partial fractions to rewrite ( ) ( + ) as + = d ( ln ln + ) + C = ln ( + d = + C + + ) ( ) 6. D Use partial fractions: d = d = (ln ln + ) = ln ln 5 ln+ ln 4 = ln 8 ( ) ( ) 6. Use partial fractions. = = 6+ 8 ( 4)( ) 4 4 d = ( ln 4 ln ) + C = ln + C C ll slopes along the diagonal y = appear to be 0. This is consistent only with option (C). For each of the others you can see why they do not work. Option () does not work because all slopes at points with the same coordinate would have to be equal. Option (B) does not work because all slopes would have to be positive. Option (D) does not work because all slopes in the third quadrant would have to be positive. Option (E) does not work because there would only be slopes for y > 0.
15 64. Å ÅZEROÅSLOPEÅTÅÅPOINTSÅWITH ÅÅÅÅÅÅY ÅNDÅX Å Å ÅNEGTIVEÅSLOPEÅTÅ ÅNDÅ ÅÅÅÅÅÅPOSITIVEÅSLOPEÅTÅ ÅNDÅ ÅÅÅÅÅÅSTEEPERÅSLOPEÅTÅY ÅTHNÅY Å Å B 4HEÅGRPHÅDOESÅNOTÅHVEÅSLOPEÅÅWHEREÅY nåorån 4HEÅSLOPEÅFIELDÅSHOWNÅSUGGESTSÅTHTÅSOLUTIONS REÅSYMPTOTICÅTOÅY ÅFROMÅBELOWÅBUTÅTHE GRPHÅDOESÅNOTÅEXHIBITÅTHISÅBEHVIOR Å RESON C Y DY X DX Y X # # ÅÅÅ # Y X ÅÅÅY X ÅSEPRTESÅVRIBLES ÅNTIDERIVTIVES ÅCONSTNTÅOFÅINTEGRTION Å ÅUSESÅINITILÅCONDITIONÅF ÅSOLVESÅFORÅY ÅÅÅÅÅÅÅIFÅYÅISÅLINER.OTE MXÅÅ;=ÅIFÅNOÅCONSTNTÅOF INTEGRTION.OTE ÅIFÅNOÅSEPRTIONÅOFÅVRIBLES D ÅRNGEÅISÅ b Y Å NSWER ÅIFÅnÅNOTÅINÅRNGE Copyright 000 by College Entrance Eamination Board and Educational Testing Service. ll rights reserved.
16 65.
17 Г O ГO=! O!Г!Г IBD=I=?=EEK=JJDEIFEJ H *?=KIBEI?JEKKIBH N #JDH JDBJBN!=@ HECDJBN!6DHBHBD=I=?= EEK=JN! > O@O O! ГN@N! N Г N + & & Г& + +& O $ N ГN $ O Г $ N ГN $ N!!?=EEK KIJEBE?=JE IF=H=JIL=HE=>I =JE@HEL=JELB@OJH =JE@HEL=JELB@N JH $?IJ=JBEJCH=JE KIIEEJE=?@EJEC$ Г" ILIBHO J=N!$EB?IJ=J BEJCH=JE J$EBIF=H=JEBL=HE=>I
18 67. (a) : : zero slope at each point (, y) where = 0 or y = positive slope at each point (, y) where 0 and y > : negative slope at each point (, y) where 0 and y < (b) Slopes are positive at points (, y) where 0 and y >. : description (c) dy = d y ln y = + C y = 0 y = Ke, K = ±e = Ke = K y = + e e C e C 6 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y 0 if y is not eponential Note: ma 6 [ ] if no constant of integration Note: 0 6 if no separation of variables
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 informationAP CALCULUS AB 2004 SCORING GUIDELINES (Form B)
004 SORING GUIDELINES (Form B) dy 4 onsider the differential equation ( y. ) d = On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use
More informationAP CALCULUS AB 2017 SCORING GUIDELINES
AP CALCULUS AB 07 SCORING GUIDELINES Consider the differential equation AP CALCULUS AB 06 SCORING GUIDELINES y =. Question 4 On the aes provided, sketch a slope field for the given differential equation
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More information1969 AP Calculus BC: Section I
969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric
More informationAP 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 information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationAP CALCULUS BC 2016 SCORING GUIDELINES
Consider the differential equation (a) Find in terms of x an. AP CALCULUS BC 06 SCORING GUIDELINES x y. = Question 4 (b) Let y = f ( x) be the particular solution to the given differential equation whose
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 9 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More informationFind the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis
Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais
More informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationAP CALCULUS BC 2007 SCORING GUIDELINES
AP CALCULUS BC 2007 SCORING GUIDELINES Question 4 Let f be the function defined for x > 0, with f( e ) = 2 and f, the first derivative of f, given by f ( x) = x 2 ln x. (a) Write an equation for the line
More information1998 AP Calculus AB: Section I, Part A
55 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.. What is the -coordinate of the point
More informationsin x (B) sin x 1 (C) sin x + 1
ANSWER KEY Packet # AP Calculus AB Eam Multiple Choice Questions Answers are on the last page. NO CALCULATOR MAY BE USED IN THIS PART OF THE EXAMINATION. On the AP Eam, you will have minutes to answer
More informationAP Calculus BC : The Fundamental Theorem of Calculus
AP Calculus BC 415 5.3: The Fundamental Theorem of Calculus Tuesday, November 5, 008 Homework Answers 6. (a) approimately 0.5 (b) approimately 1 (c) approimately 1.75 38. 4 40. 5 50. 17 Introduction In
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More informationdu u C sec( u) tan u du secu C e du e C a u a a Trigonometric Functions: Basic Integration du ln u u Helpful to Know:
Integration Techniques for AB Eam Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at
More informationNote: 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 information1998 AP Calculus AB: Section I, Part A
998 AP Calculus AB: 55 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.. What is the -coordinate
More informationCLEP 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 informationdx. Ans: y = tan x + x2 + 5x + C
Chapter 7 Differential Equations and Mathematical Modeling If you know one value of a function, and the rate of change (derivative) of the function, then yu can figure out many things about the function.
More informationAmherst College, DEPARTMENT OF MATHEMATICS Math 11, Final Examination, May 14, Answer Key. x 1 x 1 = 8. x 7 = lim. 5(x + 4) x x(x + 4) = lim
Amherst College, DEPARTMENT OF MATHEMATICS Math, Final Eamination, May 4, Answer Key. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value,
More informationMath 181, Exam 2, Fall 2014 Problem 1 Solution. sin 3 (x) cos(x) dx.
Math 8, Eam 2, Fall 24 Problem Solution. Integrals, Part I (Trigonometric integrals: 6 points). Evaluate the integral: sin 3 () cos() d. Solution: We begin by rewriting sin 3 () as Then, after using the
More informationMath 1000 Final Exam Review Solutions. (x + 3)(x 2) = lim. = lim x 2 = 3 2 = 5. (x + 1) 1 x( x ) = lim. = lim. f f(1 + h) f(1) (1) = lim
Math Final Eam Review Solutions { + 3 if < Consider f() Find the following limits: (a) lim f() + + (b) lim f() + 3 3 (c) lim f() does not eist Find each of the following limits: + 6 (a) lim 3 + 3 (b) lim
More information90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
More informationIntegration. 5.1 Antiderivatives and Indefinite Integration. Suppose that f(x) = 5x 4. Can we find a function F (x) whose derivative is f(x)?
5 Integration 5. Antiderivatives and Indefinite Integration Suppose that f() = 5 4. Can we find a function F () whose derivative is f()? Definition. A function F is an antiderivative of f on an interval
More informationAP CALCULUS AB 2003 SCORING GUIDELINES
CORING GUIDELINE Question Let R be the shaded region bounded by the graphs of y the vertical line =, as shown in the figure above. = and y = e and (b) Find the volume of the solid generated when R is revolved
More informationOBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
More informationFinal Examination F.5 Mathematics M2 Suggested Answers
Final Eamination F.5 Mathematics M Suggested Answers. The (r + )-th term C 9 r ( ) 9 r r 9 C r r 7 7r For the 8 term, set 7 7r 8 r 5 Coefficient of 8 C 9 5 5. d 8 ( ) set d if > slightly, d we have
More informationReview of elements of Calculus (functions in one variable)
Review of elements of Calculus (functions in one variable) Mainly adapted from the lectures of prof Greg Kelly Hanford High School, Richland Washington http://online.math.uh.edu/houstonact/ https://sites.google.com/site/gkellymath/home/calculuspowerpoints
More information6.1 Antiderivatives and Slope Fields Calculus
6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationAP CALCULUS AB 2017 SCORING GUIDELINES
AP CALCULUS AB 07 SCORING GUIDELINES 06 SCORING GUIDELINES Question 6 f ( ) f ( ) g( ) g ( ) 6 8 0 8 7 6 6 5 The functions f and g have continuous second derivatives. The table above gives values of the
More information( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx
Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin
More informationMath 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
More informationLecture 4: Integrals and applications
Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18
More informationLecture 5: Integrals and Applications
Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The
More informationAP Calculus Prep Session Handout. Integral Defined Functions
AP Calculus Prep Session Handout A continuous, differentiable function can be epressed as a definite integral if it is difficult or impossible to determine the antiderivative of a function using known
More informationCurriculum 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 informationCalculus 2 - Examination
Calculus - Eamination Concepts that you need to know: Two methods for showing that a function is : a) Showing the function is monotonic. b) Assuming that f( ) = f( ) and showing =. Horizontal Line Test:
More informationSection 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44
Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.
More informationAP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationPart I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
More informationAP 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 informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationSection: I. u 4 du. (9x + 1) + C, 3
EXAM 3 MAT 168 Calculus II Fall 18 Name: Section: I All answers must include either supporting work or an eplanation of your reasoning. MPORTANT: These elements are considered main part of the answer and
More informationdx dx x sec tan d 1 4 tan 2 2 csc d 2 ln 2 x 2 5x 6 C 2 ln 2 ln x ln x 3 x 2 C Now, suppose you had observed that x 3
CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Partial Fractions Understand the concept of a partial fraction decomposition. Use partial fraction decomposition with
More informationMath 112 Section 10 Lecture notes, 1/7/04
Math 11 Section 10 Lecture notes, 1/7/04 Section 7. Integration by parts To integrate the product of two functions, integration by parts is used when simpler methods such as substitution or simplifying
More informationMidterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
More informationPartial Fractions. dx dx x sec tan d 1 4 tan 2. 2 csc d. csc cot C. 2x 5. 2 ln. 2 x 2 5x 6 C. 2 ln. 2 ln x
460_080.qd //04 :08 PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Partial Fractions Understand the concept of a partial fraction decomposition. Use partial
More informationAP 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 informationM151B Practice Problems for Exam 1
M151B Practice Problems for Eam 1 Calculators will not be allowed on the eam. Unjustified answers will not receive credit. 1. Compute each of the following its: 1a. 1b. 1c. 1d. 1e. 1 3 4. 3. sin 7 0. +
More informationFundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
More informationMATH 101 Midterm Examination Spring 2009
MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.
More informationName Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y
10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the
More information1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve
MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the
More informationReview sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston
Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar
More informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More informationSolutions to Math 41 Final Exam December 9, 2013
Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain
More information1. 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 informationAP 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 informationHOMEWORK 3 MA1132: ADVANCED CALCULUS, HILARY 2017
HOMEWORK MA112: ADVANCED CALCULUS, HILARY 2017 (1) A particle moves along a curve in R with position function given by r(t) = (e t, t 2 + 1, t). Find the velocity v(t), the acceleration a(t), the speed
More informationCalculus 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 informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationReview: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review: A Cross Section of the Midterm Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it eists. 4 + ) lim - - ) A) - B) -
More informationMethods of Integration
Methods of Integration Professor D. Olles January 8, 04 Substitution The derivative of a composition of functions can be found using the chain rule form d dx [f (g(x))] f (g(x)) g (x) Rewriting the derivative
More information1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x)
APPM 5 Final Eam (5 pts) Fall. The following problems are not related: (a) (5 pts, 5 pts ea.) Find the following limits or show that they do not eist: (i) lim e (ii) lim arcsin() (b) (5 pts) Find and classify
More information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationAP Calculus BC 2003 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationExploring Substitution
I. Introduction Exploring Substitution Math Fall 08 Lab We use the Fundamental Theorem of Calculus, Part to evaluate a definite integral. If f is continuous on [a, b] b and F is any antiderivative of f
More informationSOLUTIONS TO THE FINAL - PART 1 MATH 150 FALL 2016 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS
SOLUTIONS TO THE FINAL - PART MATH 5 FALL 6 KUNIYUKI PART : 5 POINTS, PART : 5 POINTS, TOTAL: 5 POINTS No notes, books, or calculators allowed. 5 points: 45 problems, pts. each. You do not have to algebraically
More information18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then
8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =
More informationIntegration Techniques for the BC exam
Integration Techniques for the B eam For the B eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation
More information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
More informationNovember 13, 2018 MAT186 Week 8 Justin Ko
1 Mean Value Theorem Theorem 1 (Mean Value Theorem). Let f be a continuous on [a, b] and differentiable on (a, b). There eists a c (a, b) such that f f(b) f(a) (c) =. b a Eample 1: The Mean Value Theorem
More informationUBC-SFU-UVic-UNBC Calculus Exam Solutions 7 June 2007
This eamination has 15 pages including this cover. UBC-SFU-UVic-UNBC Calculus Eam Solutions 7 June 007 Name: School: Signature: Candidate Number: Rules and Instructions 1. Show all your work! Full marks
More informationAP Calculus AB Sample Exam Questions Course and Exam Description Effective Fall 2016
P alculus Sample Eam Questions ourse and Eam escription Effective Fall 6 Section I, Part ( graphing calculator may not be used) Multiple hoice Questions.. 3.. 5. lim f( g( )) f(lim g( )) f() 3 7 sin lim
More informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationReview Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.
Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:
More informationWithout fully opening the exam, check that you have pages 1 through 10.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 10. Show all your work on the standard
More informationBE 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 informationProperties of Derivatives
6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationThe Fundamental Theorem of Calculus Part 3
The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative
More informationMath 102 Spring 2008: Solutions: HW #3 Instructor: Fei Xu
Math Spring 8: Solutions: HW #3 Instructor: Fei Xu. section 7., #8 Evaluate + 3 d. + We ll solve using partial fractions. If we assume 3 A + B + C, clearing denominators gives us A A + B B + C +. Then
More informationArkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam
0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker
More informationAP Calculus BC. Practice Exam. Advanced Placement Program
Advanced Placement Program AP Calculus BC Practice Eam The questions contained in this AP Calculus BC Practice Eam are written to the content specifications of AP Eams for this subject. Taking this practice
More informationCalifornia Subject Examinations for Teachers
California Subject aminations for Teachers TST GUID MTHMTICS SUBTST III Sample Questions and Responses and Scoring Information Copyright 04 Pearson ducation, Inc. or its affiliate(s). ll rights reserved.
More informationSection 4.4 The Fundamental Theorem of Calculus
Section 4.4 The Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '() = f() throughout this interval
More informationMath Calculus II Homework # Due Date Solutions
Math 35 - Calculus II Homework # - 007.08.3 Due Date - 007.09.07 Solutions Part : Problems from sections 7.3 and 7.4. Section 7.3: 9. + d We will use the substitution cot(θ, d csc (θ. This gives + + cot
More informationWest Essex Regional School District. AP Calculus AB. Summer Packet
West Esse Regional School District AP Calculus AB Summer Packet 05-06 Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral
More informationWorkbook for Calculus I
Workbook for Calculus I By Hüseyin Yüce New York 2007 1 Functions 1.1 Four Ways to Represent a Function 1. Find the domain and range of the function f(x) = 1 + x + 1 and sketch its graph. y 3 2 1-3 -2-1
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationCalculus II Practice Test Problems for Chapter 7 Page 1 of 6
Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on
More informationDepartment of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO
Department of Mathematical Sciences Math 6 Calculus Spring 6 Eam V DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO NAME (Printed): INSTRUCTOR: SECTION NO.: When instructed, turn over this cover page
More informationChapter 5: Integrals
Chapter 5: Integrals Section 5.5 The Substitution Rule (u-substitution) Sec. 5.5: The Substitution Rule We know how to find the derivative of any combination of functions Sum rule Difference rule Constant
More information