A. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.


 Maude Payne
 6 years ago
 Views:
Transcription
1 A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c How to find it: Try nd find limits by trditionl methods (plugging in). If you get 0 0 or!!, pply L Hopitl s rule, which sys tht lim x! c ( ) ( ) = lim x! c f x g x ( ) ( ) f " x g " x x! c. L Hopitl s rule cn be pplied whenever plugging in cretes n indeterminte form: 0 0,!!,0"!,! #!,!,0 0, nd! 0. A limit involving 0! " or " # " is found by creting quotient out of tht expression. A limit involving exponents (!,0 0, or! 0 ) involves tking nturl log of the expression to move the exponent down. e x + cos x " x ". Find lim x! 0 x 4 " x 3 A.! 30 B.! 4 C.! 6 D. 0 E. nonexistent. Find lim ( x ")ln( x ") x! + A. 0 B.! C. D.  E. nonexistent Demystifying the BC Clculus MC Exm
2 3. Find lim x! x e t " # dt x 3 " 4x A. 0 B. e4 C. e3 8 D.!e 6 E. nonexistent 4. A prticle moves in the xyplne so tht the position of the prticle t ny time t is given by x( t) = cost nd y( t) = sin 4t. Find lim dx. t! 0 dy A. 3 B. 6 C. 8 D. 4 E. 0 t. Find the! 5. A popultion of bcteri is growing nd t ny time t, the popultion is given by " t mximum limit of the popultion. A. 500 B e C. 500e D. + ln500 E. 500e # $ Demystifying the BC Clculus MC Exm
3 B. Integrtion by Prts Wht you re finding: When you ttempt to integrte n expression, you try ll the rules you hve been given to tht point  typiclly power, substitution, nd the like. But if these don t work, integrtion by prts my do the trick. Integrtion by prts is usully used when you re need to find the integrl of product. How to find it: Integrtion by prts sttes tht " u dv = uv! " v du + C. To perform integrtion by prts, set up: u = v =. You need to fill in the u nd the dv from the originl problem. Determine du = dv = du nd v, then substitute into the formul. You re replcing one integrtion problem with nother tht might more esily be done with simple methods. The trick is to determine the u nd the dv. Functions tht cn be powered down re typiclly the u nd functions tht hve repetitive derivtives (exponentil nd trig) re typiclly the dv. 6.! x cos4x dx = A. x sin4x + cos4x + C B. x sin4x + 8 cos4x + C C. 8x sin4x + 3cos4x + C D. 8x sin4x! 3cos4x + C E. x sin4x! cos4x + C 7.! x e x dx = A. e x x! 8x +6 ( ) + C B. e x ( x! 8x + ) + C " x C. e x! x! % " x $ ' + C D. e x # &! x + %! x 3 $ $ ' + C E. e x # & + C # 4 & " 3 % Demystifying the BC Clculus MC Exm
4 8. Let R be the region bounded by the grph of y = x ln x, the xxis nd the line x = e, s shown by the figure to the right. Find the re of R. A. e + C. e B. e! D. e E. e + 9. The shded region between the grph of y = tn! x nd the xxis for 0 x s shown in the figure is the bse of solid whose crosssections perpendiculr to the xxis re squres. Find the volume of the solid. A.! + ln " B.! + e " ln 4 C.! " ln D.! 4 " ln E.! " ln 0. The function f is twicedifferentible nd its derivtives re continuous. The tble below gives the vlue of f, f! nd f!! for x = 0 nd x =. Find the vlue of " x f!! ( x ) dx. x ( ) f!( x) f!! ( x) f x "5 " 4 A. 0 B. 8 C. 6 D. 4 E Demystifying the BC Clculus MC Exm
5 B. Integrtion using Prtil Frctions Wht you re finding: When you ttempt to integrte frction, typiclly you let u be the expression in the denomintor nd hope tht du will be in the numertor. When this doesn t hppen, the technique of prtil dx frctions my work. One form of this type of problem is! where x + mx + n fctors into two x + mx + n nonrepeting binomils. dx How to find it: Use the Heviside method. Fctor your denomintor to get! ( x + ) ( x + b). You need to write ( x + ) ( x + b) s x + +. To find the numertor of the x + expression, cover up the x + in x + b expression, nd plug in x =!. To find the numertor of the x + b expression, cover up the ( x + ) ( x + b) x + b in expression, nd plug in x =!b. From there, ech expression cn be integrted. x + ( )( x + b).! 4x + x + 4x + 3 dx = A. ln x + 4 x C B. ln x + 4x C C. 5ln x + 3! ln x + + C D. ln x +! 5ln x C E. ln x C x +. Use the substitution u = cos x to find " sin x cos x cos x! dx. ( ) cos x! A. lncos x! + C B.!lncos x! + C C.!ln + C cos x! cos x D. ln + C E. ln cos x cos x cos x! + C Demystifying the BC Clculus MC Exm
6 3. x " 3 x! dx = A. x x + ln x + + ln x x!! + C B. + C C. x! ln x! + C D. x + ( ln x! + ln x +) + C E. ln x C x + 4. Region R is defined s the region between the grph of 9 y =, x = nd the xxis s shown in the x + x! figure to the right. Find the re of region R. A. ln B. + 3ln4 C. 3ln 4 D. 6ln E. infinite Demystifying the BC Clculus MC Exm
7 C. Improper Integrls Wht you re finding: An improper integrl is in the form be in the form continuous. b! " f ( x ) dx or " f ( x) dx or " f ( x) dx. It lso cn! f ( x ) dx where there is t lest one vlue c such tht c b for which f x How to find it: Improper integrls re just limit problems in disguise: b # f ( x ) dx = lim!" into two pieces: b $!" b with re nd volume problems.! #!! #! ( ) is not " f ( x ) dx = lim f ( x) b#! " dx or # f ( x ) dx. In the cse where there is discontinuity t x = c, the improper integrl is split! f ( x ) dx = lim! f ( x) dx + lim! f ( x) dx. Improper integrls usully go hndinhnd k" c # k" c + 5. Which of the following re convergent? k b k b I.! " dx II. x! dx III. x 0! " dx x 3 A. I only B. II only C. III only D. II nd III only E. I, II nd III " 6. # xe!4x dx = 0 A.! 6 B. 6 C. 6 D. 6 E. infinite Demystifying the BC Clculus MC Exm
8 7. The region bounded by the grph of y = 4, the line x = 4 nd the x  xis is rotted bout the xxis. x Find the volume of the solid. A. π B. π C. 4π D. 6π E. infinite 8.! " x ( x +) dx = A.! 4 B.! C. π D. π E. infinite 4 9. To the right is grph of f ( x) =. Find the vlue of 3 " f ( x ) x! dx. ( )! A. 0 B. 3 C.! 3 D. 6 E. Divergent Demystifying the BC Clculus MC Exm
9 D. Euler s Method Wht you re finding: Euler s Method provides numericl procedure to pproximte the solution of differentil eqution with given initil vlue. How to find it: ) Strt with given initil point (x, y) on the grph of the function nd given!x = dx. ) Clculte the slope using the DEQ t the point. 3) Clculte the vlue of dy using the fct tht dy! dy dx "x. 4) Find the new vlues of y nd x: y new = y old + dy nd x new = x old +!x 5) Repet the process t step ). There re clcultor progrms vilble to perform Euler s Method. Typiclly, Euler Method problems occur in the nonclcultor section where only one or two steps of the method need to be performed. 0. Let y = f ( x) be the solution to the differentil eqution dy f 3 size of 0.5? dx = x + y with the initil condition tht ( ) =!. Wht is the pproximtion for f ( 4) if Euler s Method is used, strting t x = 3 with step A..5 B. 3.5 C. 4.5 D. 5.5 E..5. Let y = f ( x) be solution to the differentil eqution dy dx = y x with initil condition f 0 k constnt, k! 0. If Euler s method with 3 steps of equl size strting t x = 0 gives the pproximtion f 3 ( )! 0, find the vlue of k. ( ) = k, A.! B. C. D.  E.! Demystifying the BC Clculus MC Exm
10 . Consider the differentil eqution dy dx = y x the exct vlue of f 8 with initil condition f ( ) =!4. Find the difference between ( ) nd n Euler pproximtion of f ( 8) using step of 0.5. A. 0 B. C. D. 5 E (Clc) Consider the differentil eqution dy dx "!% "!% between the exct vlue of f $ ' nd n Euler pproximtion of f $ ' using two equl steps. # & # & = cos x with initil condition f ( 0) = 0. Find the difference A. 0 B C D E Demystifying the BC Clculus MC Exm
11 E. Logistic Curves Wht you re finding: Logistic curves occur when quntity is growing t rte proportionl to itself nd the room vilble for growth. This room vilble is clled the crrying cpcity. This constntly incresing curve hs distinctive Sshpe where the initil stge of growth is exponentil, then slows, nd eventully the growth essentilly stops. How to find it: Logistic growth is signled by the differentil eqution dp dt = kp ( P! t ). While this DEQ C cn be solved into P( t) =, students re not responsible for tht eqution. They need to know how to!ckt + de determine the time when the logistic growth is the fstest. This is ccomplished by d P = 0. Also students dt need to know tht the curve hs horizontl symptote mening limp t t!" ( ) = C ( the crrying cpcity). 4. A popultion of students hving contrcted the flu in school yer is modeled by function P tht stisfies the logistic differentil eqution with dp dt = P " 600! P % $ '. If P( 0) =00, find lim # 800& P( t). t!" A. 400 B. 800 C.,600 D.,400 E. 4, A popultion is modeled by function G tht stisfies the logistic differentil eqution dg dt = G " e! G % $ '. If G 0 # 4e& A. 4 B. e C. e D. 4e E. 4e ( ) =, for wht vlue of G is the popultion growing the fstest? 6. Consider the differentil eqution dy dx = ky ( L! y ). Let y = f ( x) be the prticulr solution to the differentil eqution with f ( 0) =. If x! 0, find the rnge of f ( x). A. 0,L ( ) B. ( 0,) C. ( L,] D. [,L) E. [,kl) Demystifying the BC Clculus MC Exm
12 F. Arc Length Wht you re finding: Given function on n intervl [, b], the rc length is defined s the totl length of the function from x = to x = b. For this section, we will only concentrte on curves tht re defined in function form. Functions defined prmetriclly, in polr or in vectorvlued forms hve their own formuls. How to find it: The rc length of continuous function f x b [ ] ( ) over n intervl [, b] is given by L = " + f! ( x ) dx. Most problems involving rc length need clcultors becuse of the difficulty of integrting the expression. 7. (Clc) An nt wlks round the first qudrnt region R bounded by the yxis, the line y = x nd the curve f ( x) = 6! 4x 3 s shown in the figure to the right. Find the distnce the nt wlked. A B. 0.3 C..485 D..88 E If the length of curve from x = to x = 8 is given by! + 8x 4 dx, nd the curve psses through the point (, 4), which of the following could be the eqution for the curve? 8 A. y =3! 9x B. y = 4! 3x 3 C. y = 7 + 3x 3 D. y =!! 3x 3 E. y = 9x! Demystifying the BC Clculus MC Exm
13 9. (Clc) The yellow bird in the populr gme Angry Birds flies long the pth y = 4 + 3x! x when x 0. When x = 4 (the point on the figure to the right), the plyer touches the screen nd the bird leves the pth nd trvels long the line tngent to the pth t tht point. If the bird crshes into the xxis, find the totl distnce the bird flies. A B..34 C..000 D E (Clc) The grphs of i) y = x, ii) y = x nd iii) y = x! ll pss through the points (0,0) nd (,). Find the difference in rc length from the lrgest rc length to the shortest rc length of these functions on the intervl [0,]. A B. 008 C D E Find the rc length of the grph of x = ( 3 y + ) 3 for 0! y! 3. A. B. 0 C. 6 D. 3 3 E Demystifying the BC Clculus MC Exm
14 G. Prmetric Equtions Wht you re finding: Prmetric equtions re continuous functions of t in the form x = f ( t) nd y = g( t). Tken together, the prmetric equtions crete grph where the points x nd y re independent of ech other nd both dependent on the prmeter t (which is usully time). Prmetric curves when grphed do not hve to be functions. Typiclly, it is necessry to tke derivtives of prmetrics. Since the study of vectors prllels the study of prmetrics, in this section we will only nlyze the very few problems tht re not ssocited with motion in the plne. How to find it: If smooth curve C is given by the prmetric equtions x = slope of C t the point (x, y) is given by dy dy dx = dt,dx dx dt! 0. dt f ( t) nd y = g( t), then the d! dy $ The nd derivtive of the curve is given by d y dx = d # &! dy $ dt " dx % # & =. dx " dx % dx dt t = b! dx# The rc length is given by L = + dy! # % dt. The curve must be smooth nd my not intersect itself. " dt $ " dt $ t = 3. Wht is the re under the curve described by the prmetric equtions x = cost nd y = 3sin t for 0! t! "? A. 4 B. 8 C. 4 D. E A position of prticle moving in the xyplne is given by x = t 3! 6t + 9t + nd y = t 3! 9t!. For wht vlues of t is the prticle t rest? A. 0 only B. only C. 3 only D. 0 nd only E. 0, nd Demystifying the BC Clculus MC Exm
15 34. A curve C is defined by the prmetric equtions x = t! t! 4 nd y = t 3! 7t!. Which of the following is the eqution of the line tngent to the grph of C t the point (, 4)? A. y = 6! x B. x! 4y +4 = 0 C. 5x! 3y + = 0 D. y = 4x! 4 E. No tngent line t (, 4) 35. Describe the behvior of curve C defined by the prmetric equtions x = + t nd y = t 3 + t! t! t t =. A. Incresing, concve up B. Decresing, concve up C. Incresing, concve down D. Decresing, concve down E. Incresing, no concvity 36. Find the expression which represents the length L of the pth described by the prmetric equtions x = sin ( t) nd y = cos( 3t ) for 0! t! ". " A. L = # sint cost! 3sin3t dt B. L = " 4sin 4t + 9sin 9t dt 0! C. L = " 6sin 4t cos 4t + 9sin 9t dt D. L = " 4 sin t cos t + 9sin 3t dt 0! " E. L = 6sin t cos t + 9sin 3t dt 0! 0! Demystifying the BC Clculus MC Exm
16 H. VectorVlued Functions Wht you re finding: While concepts like unit vectors, dot products, nd ngles between vectors re importnt for multivrible clculus, vectors in BC clculus re little more thn prmetric equtions in disguise. How to find it: Typiclly, you will be given sitution where n object is moving in the plne. You could be given either its position vector x( t) nd y( t), its velocity vector x!( t) nd y!( t) or its ccelertion vector x!! ( t) nd y!! ( t) nd use the bsic derivtive or integrl reltionships tht hve been tught in AB clculus to find the other vectors. The one formul tht students should know is tht the speed of the object is defined s the bsolute vlue of the velocity: v t ( ) = x!( t) [ ] + [ y!( t) ]. The speed is sclr, not vector. 37. A prticle moves on plne curve such tht t ny time t > 0, its xcoordinte is t! t + t 3 while its ycoordinte is! t ( ). Find the mgnitude of the prticle s ccelertion t t =. A. 4 B. C. D. 3 E The position of n object moving in the xyplne with position function r( t) = + sint,t + cost, t 0. Wht is the mximum speed ttined by the object? A. B. C. D. 4 E. 39. A xyplne hs both its x nd ycoordintes mesured in inches. An nt is wlking long this plne with its position vector s t 3,3t!, t mesured in minutes. Wht is the verge speed of the nt mesured in inches per minute from t = 0 to t = 3 minutes? A. B. 4 3 C. 3! D. 3 E Demystifying the BC Clculus MC Exm
17 40. An object moving in the xyplne hs position function r( t) = the motion of the object. ( t +),t! 6ln t + ( ), t 0. Describe A. Left nd up B. Left nd down C. Right nd up D. Right nd down E. Depend on the vlue of t 4. An object moving long curve in the xyplne hs position x( t),y t dx dy = 8t + nd = sint for t! 0. dt dt ( ( )) t time t with At time t = 0, the object is t position (5, π). Where is the object t t =!? ( ) B. (! +! + 5,! +) C.! + 5,! + A.! +! + 5,! " D.!, % $ ' E.! + 5,! # & ( ) ( ) ( ) t time t with 4. (Clc) An object moving long curve in the xyplne hs position x( t),y ( t) dx dt = t dy + 3t + nd dt = et! for t " 0. At time t = 0, the object is t position (6, 7). Find the position of the object t t =. A. (4.667, 3.053) B. (3.683,.78) C. (.37, 9.78) D. (4.47, 6.053) E. (.573, ) Demystifying the BC Clculus MC Exm
A. Limits  L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. 1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationSample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationPractice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.
Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationlim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then
AP Clculus AB/BC Formul nd Concept Chet Sheet Limit of Continuous Function If f(x) is continuous function for ll rel numers, then lim f(x) = f(c) Limits of Rtionl Functions A. If f(x) is rtionl function
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos(  1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin(  1 ) = π 2 6 2 6 Cn you do similr problems?
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the coordinte of ech criticl vlue of g. Show
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationcritical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)
Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationMAT187H1F Lec0101 Burbulla
Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is
More informationFirst Semester Review Calculus BC
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewiseliner function f, for 4, is shown below.
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationMath Bootcamp 2012 Calculus Refresher
Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationn=0 ( 1)n /(n + 1) converges, but not n=100 1/n2, is at most 1/100.
Mth 07H Topics since the second exm Note: The finl exm will cover everything from the first two topics sheets, s well. Absolute convergence nd lternting series A series n converges bsolutely if n converges.
More informationFINALTERM EXAMINATION 9 (Session  ) Clculus & Anlyticl GeometryI Question No: ( Mrs: )  Plese choose one f ( x) x According to PowerRule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationROB EBY Blinn College Mathematics Department
ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationShorter questions on this concept appear in the multiplechoice sections. As always, look over as many questions of this kind from past exams.
22 TYPE PROBLEMS The AP clculus exms contin fresh crefully thought out often clever questions. This is especilly true for the freeresponse questions. The topics nd style of the questions re similr from
More informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationCalculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..
More informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationAB Calculus Path to a Five Problems
AB Clculus Pth to Five Problems # Topic Completed Definition of Limit OneSided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255  Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationIII. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet
III. AB Review The mteril in this section is review of AB concepts. www.mstermthmentor.com  181  Illegl to post on Internet R1: Bsic Differentition The derivtive of function is formul for the slope of
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More information