(2)(5 x) dx = 10x x x 3 dx = 1 4 x (x+1)( x1/2 ) dx = = 1 2 { } 1 2 { } = m3 4.
|
|
- Lynette May
- 5 years ago
- Views:
Transcription
1 Odd Answers: Chapter Five Contemporary Calculus PROBLE ANSWERS Chapter Five Section.. volume (.. ) + (.. ) + (.. ) 7. v v (π...) + (π.. ) + (π.. ) 9π. volume (9..) + (..) + (..). 7. (ase)(height)(thickness) ()( x) dx x x ( 9) () in 8. (ase)(height)(thickness) (x)( x ) dx x dx x 8 in. in 9. (ase)(height)(thickness) (x+)( x/ ) dx x / + x / dx { x/ + x/ } { + } { + } 8 m. π(radius) (thickness) π( ( x ) ) dx π 8x + x dx π (x x + x ) π { } π { } π m. Solids A and B have equal volumes. Why?. (ase)(height)(thickness) ( x )( x ) dx x dx ( x ) 9 cuic units.. (side)(side)(thickness) ( x )( x ) dx x dx x ( ) ( ) cuic units.. π(radius) (thickness) π( x ) dx π ( x ( ) x/ + x ) 8 π π(radius) (thickness) π( x ) dx π (9 x + x) dx π ( 9x ( ) x/ + x ) π( + 8) π() π 7.7 cuic units. π π π 8. π(radius) (thickness) π( sin(x) ) dx π ( sin(x) + sin (x) ) dx π{ x + cos(x) + x sin(x) } π π { π + ( ) + π } π{ + + } π π. cuic units.
2 Odd Answers: Chapter Five Contemporary Calculus 9. (ase)(height)(thickness) ( x )( x ) dx x dx ( x ) 8 () 8 () cuic units. π/.. cos(x) dx cuic units.. π(radius) (thickness) π( x ) dx π( ) dx π x dx π( x x) π( ) π( ) π. cuic units.. π 8.8 cuic units.. π. π 7. Rr π 9. on your own. (a) H L () H L (c) ratio. (a) BL () Volume BL (c) ratio Section ( ) ( ) 7.97 feet ( ) + ( 7) + ( ) + ( 9).8 feet (/) + + (/) + + (/).. (a) (,) to (,): + () y ' : L + ( ) dx dx x (same as in part (a) ) 7. (a) (,) to (, ): + () x ', y ' : L + ( ) dt dt t 9. y ' x x / : L + ( y ' ) dx + ( x ) dx + x dx. y ' x ( + x) / () / () /.787 x : + (y ') + (x + x ) x + + x ( x + x )
3 Odd Answers: Chapter Five Contemporary Calculus L + ( y ' ) dx x + x dx x + x dx x x ( () () ) ( (). y ' x x : + (y ') + (x 8 + x 8 ) x x 8 ( x + x ) L + ( y ' ) dx x + x dx x x \ () ).. y ' x: L ( () + ( y ' ) dx () ) ( () + ( x ) dx () ).88 + x dx.79 (using calculator). y ' x : L + ( y ' ) dx + 9x dx.8 (using calculator) 7. y ' x / : L ( y ' ) dx + x dx 8.8 (using calculator) e 8. y ' x : L e + ( y ' ) dx + x dx. (using calculator) π/ 9. y ' cos(x): (a) L π/ () L π/ + cos (x) dx.8 (using calculator) + cos (x) dx.8 (using calculator). x '. sin(t), y '. cos(t) π L (. sin(t) ) + (. cos(t) ) dt π 9. sin (t) +. cos (t) dt. (using calculator). x '. sin(t), y '. cos(t) π L. sin (t) +. cos (t) dt.8 (using calculator). x ' t. sin(t) + cos(t), y ' t. cos(t) + sin(t). Then π ( x ' ) + ( y ' ) t + (check it) so L + t dt. (using calculator)
4 Odd Answers: Chapter Five Contemporary Calculus. same x ' and y ' as in #. L + t dt. (using calculator). x ' R. ( cos(t)) and y ' R. sin(t): ( x ' ) + ( y ' ) R. ( cos(t) ) π L π R. ( cos(t)) dt R cos(t) dt 8R (Actually, 8R ). mile,8 feet at π feet per revolution so mile 8 ft π ft/rev 8.8 revolutions. From #, total distance ( 8.8 rev.)( 8 feet/rev) 7.7 feet (.7 miles).. (Why?) 7. y ' x: total length + x dx.88 (using calculator) T (a) Find T so + x dx (.88) 8.9. By experimenting on a calculator, T.77. A () Find A so + x dx (.88). and B B so + x dx (.88).. By experimenting on a calculator, A. and B.. 8., 9. on your own.. (a) A: SA x π()() π B: SA x π()() 8π () A: SA y π()() π B: SA y π()() π. (a) A: SA x π()() 8π B: SA x π()( + ) π () A: SA y π()() π B: SA y π(.)( ) π. (a) aout y : A: SA π()() π B: SA π()( ) π () aout x : A: SA π()() π B: SA π(8.)( ) 8π. The largest surface area occurs when the midpoint is farthest from the line of rotation, the x axis, when θ 9 o. Then SA π(dist. to x axis)(length) π()() π. 7. y ' x : SA y πx + ( y ' ) dx πx + ( x ) dx πx + x dx.7 (using calculator)
5 Odd Answers: Chapter Five Contemporary Calculus 9. y ' x: SA x. y ' x : SA x. y ' x: SA y πy + ( y ' ) dx πy + (y ') dx πx + (y ') dx π( x ) + x dx. (using calculator) π( x ) + 9x dx. (using calculator) π x + x dx.77 (using calculator) or use a u sustitution with u + x : SA y. on your own Section. u7 u π u / 8 du.77.. Fig.. Weight (volume)(density) (.. y)(. ), distance raised 7 y i, so Work (7 y)( 7 y i ). 7 Total work (7 y)(7) dy 7( 7y ) 7 y 8,7 ft ls.. Slice at height y: weight (.. y)() y l, distance raised y ft. (a) Total work ( y)() dy ( y ) y,8 ft ls. () The top cuic feet of water corresponds to dimensions of ft. long, ft wide, and Weight of slice y, distance raised y (where y goes from. ft. to ft.) Total work ( y)() dy ( y ) y,888 ft ls... (c) With HP pump: it takes 8 ()() minutes. min 8.7 seconds to empty the tank. With / HP pump: 8.7 / 7. seconds. Both pumps do the same amount of work ut in different times and thus have different power.. Slice at height y: weight (π. r. y)() (π.. y)() π y l, distance raised y ft. (a) Total work ( y)(π) dy π( y ) y π(8),9 ft ls. () weight of slice π y l, distance raised ( + ) y 8 y ft. Total work (8 y)(π) dy π( 8y ) y π 7. ft ls. (c) First find the work needed to empty the top feet; using integration of part (a) with new limits.. ft. high.
6 Odd Answers: Chapter Five Contemporary Calculus work ( y)(π) dy π( y y ) π(.) 88 ft ls. Knowing that HP pump works at a rate of, ft ls/minute, then it takes 88.7 minutes. seconds. 7. Fig.. Use similar triangles: slice at height y has weight (volume)(density) (7.. y. y)( 8 ) 8y y and distance raised y. Total work ( y)(8y) dy 8( y ) y, ft ls. 8. Fig.. Each slice (perpendicular to the y axis) is a disk with volume π(radius) (thickness) π ( y ) y so weight (volume)(density) ( π ( y ) y )( ) π y y. (a) Distance to top 8 y so 8 Total work (8 y)( π y ) dy π 8 8y y dy π( 8 y y ) 8 π. 8,. π,7 ft ls. () The limits of integration are now to 8: 8 work (8 y)( π y ) dy π 8 8y y dy π( 8 y y ) π,8 ft ls. H 9. Find H so π 8y y dy { total work done in 8(a) } {,. π },7.π. H π 8y y dy π{ 8 y ) H y π H { 8 H }. Some "exploring" with a calculator shows that if H.9 then π H { 8 H },7.π, the value we want. One person should remove the top feet of grain and the other should remove the ottom.9 feet.
7 Odd Answers: Chapter Five Contemporary Calculus 7. Fig.. Slice at height y: weight (volume)(density) ( π r y)(.) ( π ( y ) y)(.) π y y. (a) Distance raised y so work ( y)( π y ) dy π { y y } π ( 8 ). ft ls. () The only change is the distance raised: distance y. work ( y)( π y ) dy π { y y } π ( 8 ),879.8 ft ls.. Fig.. Slice at height y: weight (volume)(density) ( π r y)(.) ( π ( y ) y)(.). π ( y ) y. distance y. work ( y)(.π )( y ) dy.π y y + y dy.π{ y 8y y + } y.π( 89. ) 7, ft ls.. { work for (a) } > { work for () } > { work for (c) }. For container (c), most of the water is raised only a small distance, ut for container (a), most of the water is raised a larger distance. 7. Work area under the curve of force vs. position. (a) work ()() + ()() + ()() + ()() 8 ft ls. () work ()() + ()() + ()() ft ls. 9. F kx so x kx and k. (a) work x dx x 7 in oz. () work x dx x 8 in oz.. Work { area under graph of force vs. length }. Use the graph in Fig. to approximate the area. (a) From x to x, area ( cm) { avg. height of aout g } g cm.
8 Odd Answers: Chapter Five Contemporary Calculus 8 () From x 8 to x, area ( 8 cm) { g } + ( ) { g) 97. g cm.. F kx so k. and k /. + x dx 9 x.8 in ls. work R+h R P. Work f(x) dx a R x dx 9. { +h } +h () () x dx 9. { x +h } (a) h : work 9. { } 7, mile pounds. () h : work 9. { }, mile pounds. (c) h : work 9. { }, mile pounds. 7. Assume you weigh P pounds. R+h Work f(x) dx R P a R x dx +h () P x dx.. 7 P { x } +h +h }.. 7 P { (a) h : work.. 7 P { } mile pounds. () h : work.. 7 P { } mile pounds. (c) h : work.. 7 P { + } mile pounds. What happens when h is really large? 9. f(x) kx and we know that f. when x so. work a f(x) dx a (.)x dx { x } { a a }. k and k.. (a) from x to x : work { }. ft l. () from x to x : work { } 9 ft l. (c) from x to x.: work {. } 9 ft l. t. Work ta f(t) ( dx/dt ) + ( dy/dt ) dt π π t sin (t) + cos (t) dt t dt t π π.
9 Odd Answers: Chapter Five Contemporary Calculus 9 t. Work ta f(t) ( dx/dt ) + ( dy/dt ) dt t t + dt. (Use the sustitution u t + to find an antiderivative, and then evaluate.). "Unroll" the region to get a triangle with ase π(radius) π() π and height f(π) π. The "area" of the triangle is (ase)(height) (π)(π) π. (This is the same as prolem.) Section.. (a) + +. o ()() + ()() + ()() 8. x o 8. () new x o ()() + ()() + ()() + (8)(? ) (? ) 8 so? 8.. (c) x o 8 + (? )() +? so 7 + (? ) 8 + (? ) and (? ) so?.. (a) + +. x y ()() + ()() + ()() 8. x y () new x new y ()() + ()() + ()(). 8 + (? ) + + (?) + so? 8.. so?.8.. See Fig. 9. A: mass ()()(density). Center of mass (., ) B: mass ()()(density). Center of mass (,.) Total mass +. y ()(.) + ()(). x ()() + ()(.) 7. y x.7 and y x 7.. Notice that the center of mass is not "in" the region. 7. See Fig.. A: mass ()(). Center of mass (.,.) B: mass ()(). Center of mass (,.) C: mass ()(). Center of mass (.,.) Total mass ++. y ()(.)+()()+()(.). x ()(.)+()(.)+()(.) 7. y x.8. y x See Fig.. A: mass (8)(). Center of mass (, )
10 Odd Answers: Chapter Five Contemporary Calculus B: mass.π( ).8. Center of mass (, 8+.() ) (, 8.8) C: mass.π( ).. Center of mass (+.(), ) (.7, ) Total mass.. y ()()+(.8)()+(.)(.7) 9.8. x ()() + (.8)(8.8) + (.)() y x y x Fig.. x x. f(x) dx f(x) dx x. x dx x dx x x 9. y ( f(x) ) dx f(x) dx x dx x dx x x... Fig.. Because of symmetry aout the y axis, x. y x a x. { f (x) g (x) } dx { ( x ) } dx { x x } { ( ) ( + )}. a { x } dx { x x } { (8 8 ) ( 8+8 ) }.7. y x..7.
11 Odd Answers: Chapter Five Contemporary Calculus. Fig.. Because of symmetry aout the y axis, x. y x. x a { f (x) } dx ( x ) dx { x 8 x + x } { ( + ) ( + )} 7.7 a { x } dx { x x } { (8 8 ) ( 8+8 ) }.7. x y Fig.. f(x) g(x) dx 9 x dx x x. y " x. { f(x) g(x) } dx x. (9 x ) dx x x x { f (x) g (x) } dx { (9 x) () } dx { 8 8x + x 9 } dx a { 7x 9x + x } { 8+9) 7. y x 8... y x Fig.. a 9 f(x) dx x dx x/ y 9 x. { f(x) } dx x. ( x ) dx 9 x/ x a 9 f (x) dx ( x ) dx x y x y x Fig.. a f(x) dx e e x dx e. x e x (e e) ( )
12 Odd Answers: Chapter Five Contemporary Calculus y x. { f(x) } dx x. ( e e x ) dx e. x e x (x ) ( e ) (+).9 x a { f (x) g (x) } dx e e x dx { e. x ex } {e e } { }.97. y x.9.9. y x (a) h o ()() + ( x )( x )( ) + ( x )( ) + x + x. () Find minimum of h in part (a): calculate d h d t, set d h d t and solve for x to find critical points. d h x( + x) ( + d t x ) ( + x). If d h d t then x( + x) ( + x so x ± + 9. inches.. Empty glass: weight oz., y 8 cm. Full glass (liquid only): weight oz., y cm, volume ( cm)( cm)(8 cm) 8 cm. density of liquid weight volume oz. 8 cm oz. 8 cm Partially full glass (liquid only): x height of liquid, y x cm, weight ( cm)( cm)(x cm)( 8 oz. cm ) x oz h height of cm of glass containing x cm of liquid 7. and 9. On your own. ( oz)( cm) + (x oz)( x cm) oz + x oz + x + x cm.. Center of gravity is feet aove the ground and is raised to a height of feet: distance c.g. raised 8 feet. work (force)(distance) ()(8), ft l.. C.g. is raised + feet. Work ()(), ft l.
13 Odd Answers: Chapter Five Contemporary Calculus. (a) Volume aout x axis A. π. y. π. π ft. Surface area aout x axis P. π. y 8. π. π ft. () Volume aout y axis A. π. x. π. π ft. Surface area aout y axis P. π. x 8. π. 8π ft. (c) Volume aout line A. π. (dist. of c.g. from line). π. π ft. Surface area aout line P. π. (dist of c.g. from line) 8. π. π ft. 7. (a) Volume aout x axis A. π. y (π ). π. π ft. Surface area aout x axis P. π. y π(). π. π ft. () Volume aout y axis A. π. x π. π. π ft. Surface area aout y axis P. π. x π. π. π ft. (c) Volume aout line A. π. (dist. of c.g. from line) π. π. π ft. Surface area aout line P. π. (dist of c.g. from line) π. π. π ft. 9. Each rectangle has area 8 ft, perimeter ft., and centroid ft. from the line of rotation. Volume of each π(radius)(area) π( ft)(8 ft ) 8π ft.8 ft. Surface area of each π(radius)(perimeter) π( ft)( ft) 7π ft. ft. Section. Liquid Pressure. A: d. x. () dx d x dx d x d. Fig.. B: d. x. (x ) dx d x x dx. Fig.. C: d ( x x ) d. d. x. ( x) dx. d D: d. x. () dx d.
14 Odd Answers: Chapter Five Contemporary Calculus. Rectangular end: d. x. () dx d x dx d x d. Triangular end (Fig. ): d. x. ( )( x) dx d x x dx d { x x ) d. The total force is unchanged if the length is douled.. Fig.. d. ( y). ( y ) dy d y / y / dy d { ( y/ ) y/ } d { ( (8) ) () } d.. All three have the same total force against their sides. 7. The one with the largest perimeter: not (a), proaly () 9. Fig.. (a) 8 () Kinetic Energy d. x. ( π) dx πd x d. x. ( π) dx πd x 8 πd (),πd. πd (),7πd.. (a) g. v rev/sec ( π(cm) )/sec 9π cm/sec. KE v ( g) ( 9π cm/s ) 8, π ergs 799,8 ergs. () g. v rev/sec ( π(cm) )/sec π cm/sec. KE v ( g) ( π cm/s ), π ergs,, ergs.. Fig.. KE i ( x )(. πx ) π x x π x dx 8π x KE.7. π ergs ergs.. Fig. 7. ( g)/( cm). g/cm. KE i (. x )( πx ).π x x KE.π x dx.π x Total KE {,7 π ergs } 8,98 ergs.,7 π ergs,9 ergs.
15 Odd Answers: Chapter Five Contemporary Calculus 7. (a) m i π r x π x x, v i rev/sec ( π radians )/sec πx radians/sec. KE i m i ( v i ) ( π x x )( πx ) π x x KE π x dx π ( ) x 9 π ( ) 7 π () m i π x x, v i rev/sec ( π radians )/sec πx radians/sec. KE i m i ( v i ) ( π x x )( πx ) 7 π x x KE 7 π x dx 7 π ( ) x 8 π ( ), π 9. density g/cm. (a) Fig. 8. m i (area)( g/cm ) ( x)( g/cm ) x. v i rev/sec ( π x radians )/sec π x radians/sec. KE i m i ( v i ) ( x )( πx ) π x x KE π x dx 8 π x 8 π (7), π Total KE {, π ), π () Fig. 9. m i (area)( g/cm ) ( x)( g/cm ) 8 x. v i π x radians/sec. KE i m i ( v i ) ( 8 x )( πx ) π x x KE π x dx 8 π x 8 π (), π Total KE {, π ), π Volumes using tues: volume a π(radius)(height)(thickness). Fig.. a π(radius)(height)(thickness) π( x )( x ) dx (put u x, then du x dx) u π u / $ # " du ' & ) π ( u % ( ) u/ u u $ "# ' $ & )() " "# ' & )() # % ( % (.
16 Odd Answers: Chapter Five Contemporary Calculus. Fig.. a π(radius)(height)(thickness) π π( - x )( x x ) dx + " π( - x )( x x) dx "x + x " 8x dx + π x " x + 8xdx π{ } + π{ 7 }.. Fig.. a π(radius)(height)(thickness) π( - x )( x ) dx π x " 8 + x dx π{ ln(x) " 8 x + x } π{ - + ln() } Fig.. a π(radius)(height)(thickness) π( x )( x ln(x) ) dx 8.8 (Using calculator.) 9. Fig.. π(radius)(height)(thickness) a / π( x )( 9 x x) dx finish on your own. Voting. a votes for cand. A, votes for cand. A, c votes for cand. C.. a votes for cand. B, votes for cand. C, c votes for cand. C.. Fig. 9: B wins.. Fig. : A wins.. (a) Fig. a: A wins. () Fig. : C wins. (c) Fig. c: B wins. Notice that in candidate elections, A loses to B and A loses to C. In the candidate election, A wins.. (a) B wins. () B wins (c) A wins 7. (a) A wins. () A wins (c) looks like C wins 8. (a) B wins. () A wins (c) looks like A wins 9. on your own
Exam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationCalculus II - Fall 2013
Calculus II - Fall Midterm Exam II, November, In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.. Find the area between
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationVirginia Tech Math 1226 : Past CTE problems
Virginia Tech Math 16 : Past CTE problems 1. It requires 1 in-pounds of work to stretch a spring from its natural length of 1 in to a length of 1 in. How much additional work (in inch-pounds) is done in
More informationTurn off all cell phones, pagers, radios, mp3 players, and other similar devices.
Math 25 B and C Midterm 2 Palmieri, Autumn 26 Your Name Your Signature Student ID # TA s Name and quiz section (circle): Cady Cruz Jacobs BA CB BB BC CA CC Turn off all cell phones, pagers, radios, mp3
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More informationPractice Exam 1 Solutions
Practice Exam 1 Solutions 1a. Let S be the region bounded by y = x 3, y = 1, and x. Find the area of S. What is the volume of the solid obtained by rotating S about the line y = 1? Area A = Volume 1 1
More informationAB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2
AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6
More information5.2 LENGTHS OF CURVES & AREAS OF SURFACES OF REVOLUTION
5.2 Arc Length & Surface Area Contemporary Calculus 1 5.2 LENGTHS OF CURVES & AREAS OF SURFACES OF REVOLUTION This section introduces two additional geometric applications of integration: finding the length
More informationCalculus I Sample Final exam
Calculus I Sample Final exam Solutions [] Compute the following integrals: a) b) 4 x ln x) Substituting u = ln x, 4 x ln x) = ln 4 ln u du = u ln 4 ln = ln ln 4 Taking common denominator, using properties
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationPractice 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 informationMATH 162. Midterm Exam 1 - Solutions February 22, 2007
MATH 62 Midterm Exam - Solutions February 22, 27. (8 points) Evaluate the following integrals: (a) x sin(x 4 + 7) dx Solution: Let u = x 4 + 7, then du = 4x dx and x sin(x 4 + 7) dx = 4 sin(u) du = 4 [
More informationPrelim 1 Solutions V2 Math 1120
Feb., Prelim Solutions V Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Problem ) ( Points) Calculate the following: x a)
More informationc) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0
Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationMath 122 Fall Handout 15: Review Problems for the Cumulative Final Exam
Math 122 Fall 2008 Handout 15: Review Problems for the Cumulative Final Exam The topics that will be covered on Final Exam are as follows. Integration formulas. U-substitution. Integration by parts. Integration
More informationSpring 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 informationFinal Exam SOLUTIONS MAT 131 Fall 2011
1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as
More informationFor the intersections: cos x = 0 or sin x = 1 2
Chapter 6 Set-up examples The purpose of this document is to demonstrate the work that will be required if you are asked to set-up integrals on an exam and/or quiz.. Areas () Set up, do not evaluate, any
More informationMath 190 (Calculus II) Final Review
Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the
More informationMULTIVARIABLE CALCULUS
MULTIVARIABLE CALCULUS Summer Assignment Welcome to Multivariable Calculus, Multivariable Calculus is a course commonly taken by second and third year college students. The general concept is to take the
More informationAP Calculus Free-Response Questions 1969-present AB
AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationQuestions from Larson Chapter 4 Topics. 5. Evaluate
Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of
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 informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More informationMATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
More informationProblem Out of Score Problem Out of Score Total 45
Midterm Exam #1 Math 11, Section 5 January 3, 15 Duration: 5 minutes Name: Student Number: Do not open this test until instructed to do so! This exam should have 8 pages, including this cover sheet. No
More informationPractice Questions From Calculus II. 0. State the following calculus rules (these are many of the key rules from Test 1 topics).
Math 132. Practice Questions From Calculus II I. Topics Covered in Test I 0. State the following calculus rules (these are many of the key rules from Test 1 topics). (Trapezoidal Rule) b a f(x) dx (Fundamental
More informationMATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed
MATH 11 FINAL EXAM FALL 01 Part I, No Calculators Allowed 1. Evaluate the limit: lim x x x + x 1. (a) 0 (b) 0.5 0.5 1 Does not exist. Which of the following is the derivative of g(x) = x cos(3x + 1)? (a)
More informationFree Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
Free Response Questions 1969-010 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions
More informationMath 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator
Math Test - Review Use differentials to approximate the following. Compare your answer to that of a calculator.. 99.. 8. 6. Consider the graph of the equation f(x) = x x a. Find f (x) and f (x). b. Find
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 informationSection 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10
Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More information(b) x = (d) x = (b) x = e (d) x = e4 2 ln(3) 2 x x. is. (b) 2 x, x 0. (d) x 2, x 0
1. Solve the equation 3 4x+5 = 6 for x. ln(6)/ ln(3) 5 (a) x = 4 ln(3) ln(6)/ ln(3) 5 (c) x = 4 ln(3)/ ln(6) 5 (e) x = 4. Solve the equation e x 1 = 1 for x. (b) x = (d) x = ln(5)/ ln(3) 6 4 ln(6) 5/ ln(3)
More informationMath 113 (Calculus II) Final Exam KEY
Math (Calculus II) Final Exam KEY Short Answer. Fill in the blank with the appropriate answer.. (0 points) a. Let y = f (x) for x [a, b]. Give the formula for the length of the curve formed by the b graph
More informationFinal Examination Solutions
Math. 5, Sections 5 53 (Fulling) 7 December Final Examination Solutions Test Forms A and B were the same except for the order of the multiple-choice responses. This key is based on Form A. Name: Section:
More informationSolutions to Exam 2, Math 10560
Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If
More informationMath 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
More informationDepartment of Mathematical 1 Limits. 1.1 Basic Factoring Example. x 1 x 2 1. lim
Contents 1 Limits 2 1.1 Basic Factoring Example...................................... 2 1.2 One-Sided Limit........................................... 3 1.3 Squeeze Theorem..........................................
More informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
More informationVolumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
More informationMA FINAL EXAM Form A December 16, You must use a #2 pencil on the mark sense sheet (answer sheet).
MA 600 FINAL EXAM Form A December 6, 05 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME. You must use a # pencil on the mark sense sheet (answer sheet).. If the cover of your question booklet is GREEN,
More informationMath 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
More informationFinal Exam Solutions
Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
More informationMath Fall 08 Final Exam Review
Math 173.7 Fall 08 Final Exam Review 1. Graph the function f(x) = x 2 3x by applying a transformation to the graph of a standard function. 2.a. Express the function F(x) = 3 ln(x + 2) in the form F = f
More informationMA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.
MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.
More informationMath 122 Fall Unit Test 1 Review Problems Set A
Math Fall 8 Unit Test Review Problems Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Solutions to Assignment 7.6. sin. sin
Arkansas Tech University MATH 94: Calculus II Dr. Marcel B. Finan Solutions to Assignment 7.6 Exercise We have [ 5x dx = 5 ] = 4.5 ft lb x Exercise We have ( π cos x dx = [ ( π ] sin π x = J. From x =
More informationMCB4UW Handout 7.6. Comparison of the Disk/Washer and Shell Methods. V f x g x. V f y g y
MCBUW Handout 7.6 Comparison of the Disk/Washer and Shell Methods Method Ais of Formula Notes aout the Revolution Representative Rectangle a Disk Method -ais V f d -ais a V g d Washer Method -ais a V f
More informationNORTHEASTERN UNIVERSITY Department of Mathematics
NORTHEASTERN UNIVERSITY Department of Mathematics MATH 1342 (Calculus 2 for Engineering and Science) Final Exam Spring 2010 Do not write in these boxes: pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 Total (100 points)
More informationMATH 152, Spring 2019 COMMON EXAM I - VERSION A
MATH 15, Spring 19 COMMON EXAM I - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: ROW NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN
More informationMath 125 Final Examination Spring 2015
Math 125 Final Examination Spring 2015 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name This exam is closed book. You may use one 8.5 11 sheet of handwritten notes (both sides
More informationGrade: The remainder of this page has been left blank for your workings. VERSION D. Midterm D: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm D: Page of 2 Indefinite Integrals. 9 marks Each part is worth marks. Please
More information6 APPLICATIONS OF INTEGRATION
6 APPLICATIONS OF INTEGRATION 6. Areas Between Curves. A. A x x y y (y T y B) dx (x R x L ) dy + (). A (9 x ) (x +) dx ( x x ) dx x x x 6 + + 9 (x x ) x dx (x x ) dx x x 6 y (y ) dy (y / y +)dy y/ y +
More informationMath 2413 General Review for Calculus Last Updated 02/23/2016
Math 243 General Review for Calculus Last Updated 02/23/206 Find the average velocity of the function over the given interval.. y = 6x 3-5x 2-8, [-8, ] Find the slope of the curve for the given value of
More informationReview: Exam Material to be covered: 6.1, 6.2, 6.3, 6.5 plus review of u, du substitution.
Review: Exam. Goals for this portion of the course: Be able to compute the area between curves, the volume of solids of revolution, and understand the mean value of a function. We had three basic volumes:
More informationMath 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 informationPre-Calculus Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing.
Pre-Calculus Exam 009 University of Houston Math Contest Name: School: Please read the questions carefully and give a clear indication of your answer on each question. There is no penalty for guessing.
More information9. (1 pt) Chap2/2 3.pg DO NOT USE THE DEFINITION OF DERIVATIVES!! If. find f (x).
math0spring0-6 WeBWorK assignment number 3 is due : 03/04/0 at 0:00pm MST some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are help Don
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationMA 114 Worksheet # 1: Improper Integrals
MA 4 Worksheet # : Improper Integrals. For each of the following, determine if the integral is proper or improper. If it is improper, explain why. Do not evaluate any of the integrals. (c) 2 0 2 2 x x
More informationChapter 6: Applications of Integration
Chapter 6: Applications of Integration Section 6.3 Volumes by Cylindrical Shells Sec. 6.3: Volumes: Cylindrical Shell Method Cylindrical Shell Method dv = 2πrh thickness V = න a b 2πrh thickness Thickness
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 information2. Using the graph of f(x) below, to find the following limits. Write DNE if the limit does not exist:
1. [10 pts.] State each of the following theorems. (a) [2 pts.] The Intermediate Value Theorem (b) [2 pts.] The Mean Value Theorem. (c) [2 pts.] The Mean Value Theorem for Integrals. (d) [4 pts.] Both
More informationExam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.
Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x +
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More information5. Find the intercepts of the following equations. Also determine whether the equations are symmetric with respect to the y-axis or the origin.
MATHEMATICS 1571 Final Examination Review Problems 1. For the function f defined by f(x) = 2x 2 5x find the following: a) f(a + b) b) f(2x) 2f(x) 2. Find the domain of g if a) g(x) = x 2 3x 4 b) g(x) =
More informationUNIT 3: DERIVATIVES STUDY GUIDE
Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average
More informationVolume: The Disk Method. Using the integral to find volume.
Volume: The Disk Method Using the integral to find volume. If a region in a plane is revolved about a line, the resulting solid is a solid of revolution and the line is called the axis of revolution. y
More informationHalldorson Honors Pre Calculus Name 4.1: Angles and Their Measures
.: Angles and Their Measures. Approximate each angle in terms of decimal degrees to the nearest ten thousandth. a. θ = 5 '5" b. θ = 5 8'. Approximate each angle in terms of degrees, minutes, and seconds
More informationIntegration by Substitution
Integration by Substitution Dr. Philippe B. Laval Kennesaw State University Abstract This handout contains material on a very important integration method called integration by substitution. Substitution
More information2.8 Linear Approximation and Differentials
2.8 Linear Approximation Contemporary Calculus 1 2.8 Linear Approximation and Differentials Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use
More information6.0 INTRODUCTION TO DIFFERENTIAL EQUATIONS
6.0 Introduction to Differential Equations Contemporary Calculus 1 6.0 INTRODUCTION TO DIFFERENTIAL EQUATIONS This chapter is an introduction to differential equations, a major field in applied and theoretical
More informationMATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
More informationLSU 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(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationClassroom Voting Questions: Calculus II
Classroom Voting Questions: Calculus II Section 5.1: How Do We Measure Distance Traveled? 1. True or False The left-sum always underestimates the area under the curve. 2. True or False Averaging the left
More informationMATH 1A - FINAL EXAM DELUXE - SOLUTIONS. x x x x x 2. = lim = 1 =0. 2) Then ln(y) = x 2 ln(x) 3) ln(x)
MATH A - FINAL EXAM DELUXE - SOLUTIONS PEYAM RYAN TABRIZIAN. ( points, 5 points each) Find the following limits (a) lim x x2 + x ( ) x lim x2 + x x2 + x 2 + + x x x x2 + + x x 2 + x 2 x x2 + + x x x2 +
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationAP 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 informationMTH 133 Final Exam Dec 8, 2014
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Problem Score Max Score 1 5 3 2 5 3a 5 3b 5 4 4 5 5a 5 5b 5 6 5 5 7a 5 7b 5 6 8 18 7 8 9 10 11 12 9a
More informationL. 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 informationAP Calculus AB 2017 Free-Response Solutions
AP Calculus AB 217 Free-Response Solutions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of Denver May 18, 217 1 Problem 1 1.1 Part a The approximation with a left-hand
More informationThe Princeton Review AP Calculus BC Practice Test 1
8 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 informationDistance And Velocity
Unit #8 - The Integral Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Distance And Velocity. The graph below shows the velocity, v, of an object (in meters/sec). Estimate
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationHave a Safe and Happy Break
Math 121 Final EF: December 10, 2013 Name Directions: 1 /15 2 /15 3 /15 4 /15 5 /10 6 /10 7 /20 8 /15 9 /15 10 /10 11 /15 12 /20 13 /15 14 /10 Total /200 1. No book, notes, or ouiji boards. You may use
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More informationWeBWorK 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 informationThe Chain Rule. Mathematics 11: Lecture 18. Dan Sloughter. Furman University. October 10, 2007
The Chain Rule Mathematics 11: Lecture 18 Dan Sloughter Furman University October 10, 2007 Dan Sloughter (Furman University) The Chain Rule October 10, 2007 1 / 15 Example Suppose that a pebble is dropped
More informationMath 251, Spring 2005: Exam #2 Preview Problems
Math 5, Spring 005: Exam # Preview Problems. Using the definition of derivative find the derivative of the following functions: a) fx) = e x e h. Use the following lim =, e x+h = e x e h.) h b) fx) = x
More informationSolutions to Final Review Sheet. The Math 5a final exam will be Tuesday, May 1 from 9:15 am 12:15 p.m.
Math 5a Solutions to Final Review Sheet The Math 5a final exam will be Tuesday, May 1 from 9:15 am 1:15 p.m. Location: Gerstenzang 1 The final exam is cumulative (i.e., it will cover all the material we
More information