Name: Math 10550, Final Exam: December 15, 2007
|
|
- Norah Fowler
- 6 years ago
- Views:
Transcription
1 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 the rest of your test. Whe told to stop, had i just this oe page. The Hoor Code is i effect for this examiatio, icludig keepig your aswer sheet uder cover. PLEASE MARK YOUR ANSWERS WITH AN X, ot a circle!. (b) (c). (b) (c) 3. (b) (c) 4. (b) (c) 5. (b) (c) 6. (b) (c) 7. (b) (c) 8. (b) (c) 9. (b) (c). (b) (c). (b) (c). (b) (c) 3. (b) (c) 4. (b) (c) 5. (b) (c) 6. (b) (c) 7. (b) (c) 8. (b) (c) 9. (b) (c). (b) (c). (b) (c). (b) (c) 3. (b) (c) 4. (b) (c) 5. (b) (c)
2 Multiple Choice Name:.(6 pts.) Fid the limit lim x x +. x (b) (c) The limit does ot exist. 3 Solutio: x + lim x x ( x +) = lim ( + x +) x x ( + x +) (x +) = lim x x( + x +) x = lim x = lim x = + x( + x +) + x + =.(6 pts.) The fuctio f(x) = x + x 6 x 4 has a removable discotiuity at x =. We ca remove this discotiuity by defiig f() to be 3 (b) (c) 3 5 4
3 Solutio: Sice x + x 6 lim x x 4 (x +3) (x = lim ) x (x +) (x ) = 5 4, Thus defiig f() = 5 4 yields so f(x) is cotiuous at. lim f(x) =f(), x 3
4 3.(6 pts.) If the dr dθ = r = si θ +cosθ, (c) cos θ +cos θ si θ ( + cos θ) (b) +cosθ +cosθ cos θ ( + cos θ) cos θ +cos θ si θ ( + cos θ) Solutio: dr dθ = ( + cos θ)cosθ si θ( si θ) ( + cos θ) = cos θ +cos θ + si θ ( + cos θ) = +cosθ ( + cos θ) = +cosθ 4.(6 pts.) If the f (8) = f(x) = + +x, 4 (b) (c) 8 9 Solutio: Sice f (x) = ( + +x) ( + x) 4
5 pluggig i x = 8 gives f (8) = ( + +8) ( + 8) = ( + 9) = 4 3 (9) = 4. 5
6 5.(6 pts.) The secod derivative of is y =(x +)(x )(x +) 4x (b) x +x (c) x 4x 3 4x x + Solutio: Sice y =(x +)(x )(x +)=(x )(x +)=x 4, differetiatig yields y =4x 3 ad so y =x. 6.(6 pts.) A body travels alog a straight lie accordig to the law s = t 4 4t 3 +t, t. At what positio, after the motio gets started, does the body first come to rest? s = 3 (b) s =36 (c) s = s = s =4 Solutio: We first seek t> such that v(t) =, where v(t) =s (t) = 4t 3 t +4t. Sice factorig gives v(t) = 4t(t +3t ) = 4t(t )(t +5), we see that t = is the first time whe the body is at rest. The positio at t = is s() = =3. 6
7 7.(6 pts.) The equatio of the taget lie to the curve at x = is y = x 3 +6x +x +6 y = x (b) y = x (c) y = x + y = x + y = x Solutio: Sice pluggig i x = gives y =3x +x +, y () = 3 ( ) + ( ) + =. Whe x =, the y-coordiate o the give curve is Therefore the equatio of the taget is y =( ) 3 +6 ( ) + ( ) + 6 =. y = (x +) = y = x. 8.(6 pts.) Use the implicit differetiatio to fid the equatio of the taget lie to the curve 5x +9y =+xy + y at the poit (, ). y = 4 3 x + (b) y = 5 6 x (c) y = 3 x + y = 5 6 x + y = 3 x Solutio: Implicitly differetiatig both sides of the give equatio yields (5x +9y) / (5 + 9y )=y +xyy + y, 7
8 which after pluggig i x = ad y = simplifies to 6 (5 + 9y )=+y. Solvig for y gives y =. Therefore the equatio of the taget lie is 3 y = 3 x +. 8
9 9.(6 pts.) A cylider is carved out of ice ad the left i the su to melt. If the radius decreases at a rate of 3 iches per hour ad the height decreases at a rate of 6 iches per hour, how fast is the surface area of the cylider decreasig whe the cylider is at height 5 feet ad radius oe foot? (Hit: iches i a foot.) Aswer: The total surface area decreases at a rate of 3 4 ft /hr (b) 9 ft /hr ft /hr Solutio: The surface are is give by 5 4 ft /hr (c) A =r +rh. Differetiatig with respect to t the gives da dt =4rdr dt +(hdr dt + r dh dt ). Pluggig i the give values h =5,r =, dr = dt 4, ad dh dt = yields da dt =4 4 +(5 4 + )=9. 5 ft /hr.(6 pts.) Use liear approximatio to estimate (b) (c) Solutio: With f(x) =x, ad hece f (x) = x 3, the liear approximatio of f at x =4. is f(4.) = f(4) + f (4)(4. 4) = 79. =
10 .(6 pts.) The maximum ad miimum values of f(x) = x x +, o the iterval [,] are M =,m= (b) M =,m= (c) M =,m= 3 5 M = 5,m= m = is a miimum; there is o maximum. Solutio: The critical poits are where f (x) = x + x(x) (x +) = x ( + x ) equals zero ad the edpoits x =,x =. Sice f (x) = if ad oly if x = ±, we take x = as our third critical poits. Sice f() =, f() =,adf() =,wesee 5 that M = ad m =..(6 pts.) Determie the umber of solutios of the equatio x 3 5x += i the iterval [, ]. The umber of solutios is (b) (c) 3 4 Solutio: Set f(x) =x 3 5x +, so that f (x) =3x 5. Sice f( ) = 3 ad f() =, the itermediate value theorem guaratees that f has at least oe root i [, ]. Because x < 5forx [, ], it follows that 3x < 5 ad hece f (x) =3x 5 < forx [, ]. Thus f is strictly decreasig o [, ] ad hece caot have more tha oe zero o [, ]. Therefore f has exactly oe root o [, ].
11 3.(6 pts.) Cosider the fuctio f(x) = x +3 x. Oe of the followig statemets is true. Which oe? The lie y = x + is a slat asymptote of f, ad the lie x = is a vertical asymptote of f. (b) f has o horizotal or slat asymptotes, ad the lie x = is a vertical asymptote. (c) The lie y = is a horizotal asymptote of f, ad the lie x = is a vertical asymptote of f. The lie y = x + is a slat asymptote of f, ad the lie f has o vertical asymptotes. The lie y = x is a slat asymptote of f ad the lie x = is a vertical asymptote of f. Solutio: Sice (as log divisio easily verifies) x +3 x = x ++ 4 x, the slat asymptote is y = x +. Thus there is o horizotal asymptote. Because the deomiator is udefied at x = ad x is ot a factor of the umerator, x = is a vertical asymptote. 4.(6 pts.) Cosider the fuctio f(x) = x +3 x. Oe of the followig statemets is true. Which oe? f is icreasig o the iterval (, 3). (b) f has a local miimum at x =. (c) f is decreasig o the itervals (, ) ad (, 3). f is icreasig o the itervals (, ) ad (, 3). f has a local miimum at x =. Solutio: From the previous problem, we kow f(x) =x ++ 4 x,
12 so that The for x = f (x) = f (x) > > Name: 4 (x ). 4 (x ) (x ) > 4 (x 3)(x +)> x< orx>3. Thus f is icreasig o (, ) ad (3, ) ad decreasig everywhere else (i.e. o (, ) ad (, 3)). Clearly x = is ot a local miimum sice f has a vertical asymptote there. Although f ( ) =, this is actually because of a local maximum. Ideed f 8 (x) = (x ), 3 so f ( ) <.
13 5.(6 pts.) Cosider the fuctio 9x6 x f(x) = x 3 +. Oe of the followig statemets is true. Which oe? (b) y = 3 is a horizotal asymptote of f, ad y = 3 is ot a horizotal asymptote. f has o horizotal asymptotes. (c) y = ad y = 3 are both horizotal asymptotes of f. y = ±3 are both horizotal asymptotes of f. y = is a horizotal asymptote of f. Solutio: Sice the problem oly asks about horizotal asymptotes, we compute limits as x ± : 9x6 x lim f(x) = lim x 6 x x x 3 + = lim x = =3, 9 + ad similarly (sice x 3 = x 6 whe x<) lim f(x) = x lim x = lim x = = 3, 9 + so y = ±3 are both horizotal asymptotes of f. 9 x 5 + x 3 9x6 x x x 5 + x 3 x 3 x 6 x 3 3
14 6.(6 pts.) The fuctio f(x) =(x +) 4 4x +5x is cocave dow o which of the followig itervals? (, ) (b) (, ) (c) (, ) (, ) (, ) ad Solutio: Sice f (x) =4(x +) 3 48x +5=8(x +) 3 48x +5 f (x) =4(x +) 48 = 48((x +) a b )=48(x)(x +)=9x(x +), fidig where f (x) < amouts to solvig x(x +) <. The curve x(x + ) is a upward-opeig parabola with roots at x = ad x =, ad whece is egative whe <x<. Therefore f is cocave dow o (, ). 4
15 7.(6 pts.) A ope box is to be made from a square of side oe by cuttig four idetical squares ear the vertices. The box with the largest volume has a height of 6 (b) Solutio: If the height of the box is h (which is also the side legth of the cutout square), the the volume is give by Thus (c) V = h( h) = h 4h +4h 3. V = 8h +h =(4h )(3h ), so that V = whe h = or h =. 6 I order to make a box, h must be i the iterval (, /). Because V is a upwardopeig parabola, it must switch from positive to egative at h = ad be egative util 6 h =,soh = is gives a maximum o (, /) (6 pts.) Whe applyig Newto s method to approximate a root of the equatio x 3 x + =, with iitial guess x =, the value of x is:.5 (b).5 (c) 3 Solutio: With f(x) =x 3 x +, we have Thus f (x) =3x. x = x f(x ) f (x ) = f() f () = =. 5
16 9.(6 pts.) Which of the followig is a Riema sum correspodig to the itegral 3 x 4 dx? ( + i )4 (b) i= ( + i )4 (c) i= ( i )4 i= i= ( +i )4 ( i )4 i= Solutio: With f(x) =x 4 ad x = 3 =, the Riema sum i this case is f( + i x) x = ( + i )4 = ( + i )4. i= i= i=.(6 pts.) A fuctio f(x) defied o the iterval [, ] has a atiderivative F (x). Assume that F ( ) = 8 ad F () = 7. Which oe of the statemets below is true? (b) (c) f(x)dx =. F (x) is a icreasig fuctio. f(x) ca be a odd fuctio. f(x)dx =. f(x)dx =. Solutio: If f is ot assumed cotiuous, the f might ot be itegrable, so that oe of the choices are correct. Thus we add the hypothesis that f is cotiuous. By the fudametal theorem of calculus, f(x) dx = F () F ( ) = 7 8=. Note that f(x) caot be a odd fuctio sice if it were, the cotrary to the calculatio above. f(x) dx =, 6
17 .(6 pts.) Calculate the itegral 3 si x dx. (b) (c) Solutio: Sice si x< oly o (, 3/), 3/ / si x dx = / si xdx+ 3/ = cos x / +cosx 3/ =. si xdx.(6 pts.) The volume of the solid obtaied by rotatig the regio give by x + y =, x ad y, about the lie y = is (b) (c) ( x )dx [ x + x ]dx x[ x + x ]dx x x dx ( + x ) dx 7
18 Solutio: The outer radius is x + ad the ier is, so V = = = ((outer radius) (ier radius) ) dx (( x +) ) dx [ x x ] dx. 8
19 3.(6 pts.) Fid the volume of the solid obtaied by rotatig about the y-axis the regio betwee y = x ad y = x 4. 6 (b) (c) Solutio: The curves itersect whe x = ad x = (ad x =, but sice the solid is obtaied by rotatig aroud the y-axis, this itersectio poit is irrelevat). Thus V = x[x x 4 ] dx x 4 = 4 x6 6 = 4 6 = (6 pts.) Fid the average of f(x) = si (x) cos(x) over [, ]. 3 (b) Solutio: The average is give by / / 3 si (x) cos(x) dx = u du = 3. (c) u du 3 9
20 5.(6 pts.) A (vertical) cylidrical tak has a height meter ad base radius meter. It is filled full with a liquid with a desity kg/m 3. Fid the work required to empty the tak by pumpig all of the liquid to the top of the tak. 5 kg-m (b) kg-m (c) kg-m kg-m 5 kg-m Solutio: Noe of the give solutios are correct. We cosider the cylider sliced ito slabs of equal height x so that the work doe o the i th slice is W i = F i x i =( V i 9.8)x i =( () x o.8)x i =(98 x)x i, where x i is a poit i the i th slab. The the total work is W = lim W i i= =98 lim =98 =98/ =49. x i x i= xdx
21 Math 55, Fial Exam: December 5, 7 Name: ANSWERS 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 the rest of your test. Whe told to stop, had i just this oe page. The Hoor Code is i effect for this examiatio, icludig keepig your aswer sheet uder cover. PLEASE MARK YOUR ANSWERS WITH AN X, ot a circle!. (b) (c) ( ). (b) (c) ( ) 3. ( ) (c) 4. ( ) (b) (c) 5. (b) ( ) 6. ( ) (b) (c) 7. ( ) (c) 8. (b) ( ) 9. (b) (c) ( ). (b) (c) ( ). ( ) (b) (c). (b) ( ) 3. ( ) (b) (c) 4. (b) ( ) 5. (b) (c) ( ) 6. (b) (c) ( ) 7. ( ) (b) (c) 8. (b) ( ) 9. ( ) (c). (b) (c) ( ). (b) (c) ( ). ( ) (c) 3. ( ) (b) (c) 4. (b) ( ) 5. (b) (c) ( )
MATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationMATH Exam 1 Solutions February 24, 2016
MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [
More informationCALCULUS 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 informationMath 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 informationf(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 informationAP 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 informationMath 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 informationMIDTERM 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 informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
More informationMATH 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(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 informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationPRACTICE 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 informationFor example suppose we divide the interval [0,2] into 5 equal subintervals of length
Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit
More informationMath 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(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 informationMath 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 informationn 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 informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. 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 upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationMath 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 informationCalculus with Analytic Geometry 2
Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,
More informationMath 12 Final Exam, May 11, 2011 ANSWER KEY. 2sinh(2x) = lim. 1 x. lim e. x ln. = e. (x+1)(1) x(1) (x+1) 2. (2secθ) 5 2sec2 θ dθ.
Math Fial Exam, May, ANSWER KEY. [5 Poits] Evaluate each of the followig its. Please justify your aswers. Be clear if the it equals a value, + or, or Does Not Exist. coshx) a) L H x x+l x) sihx) x x L
More informationThe 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 informationHOMEWORK #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 information18.01 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 informationFor example suppose we divide the interval [0,2] into 5 equal subintervals of length
Math 120c Calculus Sec 1: Estimatig with Fiite Sums I Area A Cosider the problem of fidig the area uder the curve o the fuctio y!x 2 + over the domai [0,2] We ca approximate this area by usig a familiar
More informationMidterm Exam #2. Please staple this cover and honor pledge atop your solutions.
Math 50B Itegral Calculus April, 07 Midterm Exam # Name: Aswer Key David Arold Istructios. (00 poits) This exam is ope otes, ope book. This icludes ay supplemetary texts or olie documets. You are ot allowed
More information7.) 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 informationReview Problems for the Final
Review Problems for the Fial Math - 3 7 These problems are provided to help you study The presece of a problem o this hadout does ot imply that there will be a similar problem o the test Ad the absece
More informationMath 176 Calculus Sec. 5.1: Areas and Distances (Using Finite Sums)
Math 176 Calculus Sec. 5.1: Areas ad Distaces (Usig Fiite Sums) I. Area A. Cosider the problem of fidig the area uder the curve o the f y=-x 2 +5 over the domai [0, 2]. We ca approximate this area by usig
More informationMaximum 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 informationCalculus 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 informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More information2 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 informationUnit 4: Polynomial and Rational Functions
48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad
More informationMATH 129 FINAL EXAM REVIEW PACKET (Revised Spring 2008)
MATH 9 FINAL EXAM REVIEW PACKET (Revised Sprig 8) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial exam, but they will
More informationMATHEMATICS. 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 information1. (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 information6.3 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 informationAP 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 informationMath 116 Final Exam December 19, 2016
Math 6 Fial Exam December 9, 06 UMID: EXAM SOLUTIONS Iitials: Istructor: Sectio:. 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 3 pages icludig
More informationCalculus 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 informationMTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1
MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces
More informationSolutions 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 informationSCORE. 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 informationStudent 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 informationMath 128A: Homework 1 Solutions
Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that
More informationFooling Newton s Method
Foolig Newto s Method You might thik that if the Newto sequece of a fuctio coverges to a umber, that the umber must be a zero of the fuctio. Let s look at the Newto iteratio ad see what might go wrog:
More information1 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 informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More information6.) 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 informationSection 13.3 Area and the Definite Integral
Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate
More informationSCORE. 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 informationMath 116 Final Exam December 12, 2014
Math 6 Fial Exam December 2, 24 Name: EXAM SOLUTIONS Istructor: Sectio:. Do ot ope this exam util you are told to do so. 2. This exam has 4 pages icludig this cover. There are 2 problems. Note that the
More informationCalculus 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 informationChapter 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 informationFINALTERM 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 informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationn n 2 + 4i = lim 2 n lim 1 + 4x 2 dx = 1 2 tan ( 2i 2 x x dx = 1 2 tan 1 2 = 2 n, x i = a + i x = 2i
. ( poits) Fid the limits. (a) (6 poits) lim ( + + + 3 (6 poits) lim h h h 6 微甲 - 班期末考解答和評分標準 +h + + + t3 dt. + 3 +... + 5 ) = lim + i= + i. Solutio: (a) lim i= + i = lim i= + ( i ) = lim x i= + x i =
More informationMath 116 Practice for Exam 3
Math 6 Practice for Exam Geerated October 0, 207 Name: SOLUTIONS Istructor: Sectio Number:. This exam has 7 questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationMath 152 Exam 3, Fall 2005
c IIT Dept. Applied Mathematics, December, 005 PRINT Last ame: KEY First ame: Sigature: Studet ID: Math 5 Exam 3, Fall 005 Istructios. For the multiple choice problems, there is o partial credit. For the
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationMath 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 informationMTH 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 informationHonors 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 informationMAT136H1F - 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 informationMath 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 informationChapter 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 informationMTH 246 TEST 3 April 4, 2014
MTH 26 TEST April, 20 (PLEASE PRINT YOUR NAME!!) Name:. (6 poits each) Evaluate lim! a for the give sequece fa g. (a) a = 2 2 5 2 5 (b) a = 2 7 2. (6 poits) Fid the sum of the telescopig series p p 2.
More informationx 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 informationMark 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 informationCalculus II exam 1 6/18/07 All problems are worth 10 points unless otherwise noted. Show all analytic work.
9.-0 Calculus II exam 6/8/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. (5 poits) Prove that the area eclosed i the circle. f( x) = x +, 0 x. Use the approximate the area
More informationB 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 informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationCALCULUS 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 informationMath 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 informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More informationA) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.
M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationAP 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 informationy = 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 informationIndian 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 informationAlgebra 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 informationMATH 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 informationTHE 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 informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationAP 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 information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationSection 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 informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationSubstitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
More informationInfinite 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 informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More informationMATH 2411 Spring 2011 Practice Exam #1 Tuesday, March 1 st Sections: Sections ; 6.8; Instructions:
MATH 411 Sprig 011 Practice Exam #1 Tuesday, March 1 st Sectios: Sectios 6.1-6.6; 6.8; 7.1-7.4 Name: Score: = 100 Istructios: 1. You will have a total of 1 hour ad 50 miutes to complete this exam.. A No-Graphig
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
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 (, )
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More information