# 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.

Size: px
Start display at page:

Download "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."

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 ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by trditionl methods (plugging in). If you get or, pply L Hopitl s rule, which sys tht lim c = lim c f g f g. L Hopitl s rule cn be pplied whenever plugging in cretes n indeterminte form:,,,,,, nd. A limit involving or is found by creting quotient out of tht epression. A limit involving eponents (,, or ) involves tking nturl log of the epression to move the eponent down. e + cos. Find lim A. B. C. 6 D. E. noneistent e + cos + C. lim = = e sin lim = = e cos lim 6 = = e sin lim 6 = 6 = 6. Find lim ( )ln + A. B. C. D. - E. noneistent A. lim ( )ln + lim + = ln ( ) = lim + ( ) = = lim ( +) ( ) + = Demystifying the BC Clculus MC Em

2 . Find lim e t A. B. e C. e 8 D. e 6 E. noneistent C. lim lim e t = e t e = e =. A prticle moves in the y-plne so tht the position of the prticle t ny time t is given by t = cost nd y( t) = sin t. Find lim t. A. - B. -6 C. -8 D. - E. A. 8sin cos 8sin cos = lim = sin t sin 8( sin + cos ) lim = t cos 5. A popultion of bcteri is growing nd t ny time t, the popultion is given by 5 + t mimum limit of the popultion. A. 5 B. 5 + e C. 5e D. + ln5 E. 5e E. y = 5 + t ln y = ln5 + t ln + t t lim ln5 + t ln + ln + t t t = ln5 + lim = ln5 + lim t t + t t ln y = ln5 + y = e ln 5 + = e ln 5 e = 5e t t = ln5 + t. Find the Demystifying the BC Clculus MC Em

3 B. Integrtion by Prts Wht you re finding: When you ttempt to integrte n epression, 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 (eponentil nd trig) re typiclly the dv. 6. cos = A. sin + cos + C B. sin + 8 cos + C C. 8 sin + cos + C D. 8 sin cos + C E. sin cos + C B. u = v = sin du = dv = cos cos = sin cos = sin + 8 cos + C = sin sin 7. e = A. e ( 8 +6) + C B. e ( 8 + ) + C C. e + C D. e + + C E. e + C D. u = v = e du = dv = e e = ( e ) e u = v = e du = dv = e e = ( e ) ( e ) = ( e ) e + e = e + + C e Demystifying the BC Clculus MC Em

4 8. Let R be the region bounded by the grph of y = ln, the -is nd the line = e, s shown by the figure to the right. Find the re of R. A. e + C. e e A. A = ln u = ln v = B. e D. e E. e + du = dv = A = ln e 9. The shded region between the grph of y = tn nd the -is for s shown in the figure is the bse of solid whose cross-sections perpendiculr to the -is re squres. Find the volume of the solid. e = e e ln = e + A. + ln B. + e ln C. ln D. ln E. ln E. V = tn = tn u = tn v = du = + dv = tn = tn + = tn ln( + ) = tn ln = ln. The function f is twice-differentible nd its derivtives re continuous. The tble below gives the vlue of f, f nd f for = nd =. Find the vlue of f ( ). f ( ) f ( ) f 6 5 A. - B. -8 C. 6 D. E. 6 C. f ( ) = f [ ] f ( ) = [ ] = + 8 = 6 5 [ ] f f f f Demystifying the BC Clculus MC Em

5 B. Integrtion using Prtil Frctions Wht you re finding: When you ttempt to integrte frction, typiclly you let u be the epression in the denomintor nd hope tht du will be in the numertor. When this doesn t hppen, the technique of prtil frctions my work. One form of this type of problem is where + m + n fctors into two + m + n non-repeting binomils. How to find it: Use the Heviside method. Fctor your denomintor to get. You need to ( + ) ( + b) write ( + ) ( + b) s + +. To find the numertor of the + epression, cover up the + in + b epression, nd plug in =. To find the numertor of the + b epression, cover up the ( + ) ( + b) + b in epression, nd plug in = b. From there, ech epression cn be integrted. + ( + b) = A. ln C B. ln C C. 5ln + ln + + C D. ln + 5ln + + C E. ln C + C. u - substitution doesn't work here. u = + +,du = ( + ) = + + ( +) = + + ( +) = 5ln + ln + + C. Use the substitution u = cos to find sin cos ( cos ). cos A. lncos + C B. lncos + C C. ln cos cos D. ln cos + C E. ln cos cos + C D. u = cos,du = sin du uu u du = ln u ln u = ln u u u cos = ln + C cos + C Demystifying the BC Clculus MC Em

6 . = A. + ln + + ln + C B. + C C. ln + C D. + ( ln + ln +) + C E. ln C + ) D. = + + = = + ln + ln + = + ln + ln + + C ( could lso be done with u - sub). Region R is defined s the region between the grph 9 of y =, = nd the -is s shown in the + figure to the right. Find the re of region R. A. ln B. + ln C. ln D. 6ln E. infinite 9 D. A = + = ( + ) = 9 ( + ) ( + ) = ln ln + = ln + = ln ln = ln ( ln ) = ln = ln = 6ln Demystifying the BC Clculus MC Em

7 C. Improper Integrls Wht you re finding: An improper integrl is in the form be in the form continuous. b f ( ) or f ( ) or f ( ). It lso cn f ( ) where there is t lest one vlue c such tht c b for which f How to find it: Improper integrls re just limit problems in disguise: b f ( ) = lim into two pieces: b b is not f ( ) = lim f ( ) or b b f ( ). In the cse where there is discontinuity t = c, the improper integrl is split f ( ) = lim f ( ) + lim k k c with re nd volume problems. 5. Which of the following re convergent? b k c + k f ( ). Improper integrls usully go hnd-in-hnd I. II. III. A. I only B. II only C. III only D. II nd III only E. I, II nd III 6. e = A. 6 B. 6 D. = which is divergent = = = = C. -6 D. 6 E. infinite B. u = v = e du = dv = e e = e 6e = 6 = 6 e + e = Demystifying the BC Clculus MC Em

8 7. The region bounded by the grph of y =, the line = nd the - is is rotted bout the -is. Find the volume of the solid. A. B. C. D. 6 E. infinite C. V = = 6 = 6 V = + = 8. ( +) = A. B. C. D. E. infinite B. u = du = + = =, u =, =,u = + = = tn tn = = ( +) = du u + = tn u 9. To the right is grph of f ( ) =. Find the vlue of f ( ). A. B. C. D. 6 E. Divergent = E. = ( ) + + Both clcultions re divergent Demystifying the BC Clculus MC Em

9 D. Euler s Method Wht you re finding: Euler s Method provides numericl procedure to pproimte the solution of differentil eqution with given initil vlue. How to find it: ) Strt with given initil point (, y) on the grph of the function nd given =. ) Clculte the slope using the DEQ t the point. ) Clculte the vlue of using the fct tht. ) Find the new vlues of y nd : y new = y old + nd new = old + 5) Repet the process t step ). There re clcultor progrms vilble to perform Euler s Method. Typiclly, Euler Method problems occur in the non-clcultor section where only one or two steps of the method need to be performed.. Let y = f ( ) be the solution to the differentil eqution f size of.5? = + y with the initil condition tht =. Wht is the pproimtion for f ( ) if Euler s Method is used, strting t = with step A..5 B..5 C..5 D. 5.5 E..5 B. y Let y = f ( ) be solution to the differentil eqution = y with initil condition f k constnt, k. If Euler s method with steps of equl size strting t = gives the pproimtion f ( ), find the vlue of k. = k, A. B. C. D. - E. A. y k k k k k k k k + k k + k = k( + k) = k =,k = Demystifying the BC Clculus MC Em

10 . Consider the differentil eqution = y between the ect vlue of f 8 with initil condition f =. Find the difference nd n Euler pproimtion of f ( 8) using step of.5. A. B. C. D. 5 E. 5 A. y y = ln y = ln + C y = e ln +C y = C = C C = y = y = 5 = 5 Difference = 5 5 = Students should see tht ech y will be twice the vlue of nd not perform Euler over nd over gin.. (Clc) Consider the differentil eqution = cos with initil condition f =. Find the difference between the ect vlue of f nd n Euler pproimtion of f using two equl steps. A. B.. C.. D..555 E..77 C. y = = = cos y = sin + C = sin + C C = y = sin y = sin = Difference = + 8 = Demystifying the BC Clculus MC Em

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. The curve hs distinctive S-shpe where the initil stge of growth is eponentil, then slows, nd eventully the growth essentilly stops. How to find it: Logistic growth is signled by the differentil eqution dp = kp( P t). While this DEQ C cn be solved into Pt =, 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 =. Also students need to know tht the curve hs horizontl symptote mening limpt = C ( the crrying cpcity). t. A popultion of students hving contrcted the flu in school yer is modeled by function P tht stisfies the logistic differentil eqution with dp = P 6 P. If P =, find limpt. 8 t A. B. 8 C.,6 D., E.,8 C. Since this is logistic, there is horizontl symptote to P( t) nd thus dp dp = P 6 P = P P = = 5. A popultion is modeled by function G tht stisfies the logistic differentil eqution dg = G e G. If G e A. B. e C. e D. e E. e =, for wht vlue of G is the popultion growing the fstest? C. dg = G e G = G e e d G 6. Consider the differentil eqution differentil eqution with f A.,L G e = e G e = eg = e P = e hlf the crrying cpcity = ky( L y). Let y = f ( ) be the prticulr solution to the =. If, find the rnge of f ( ). B. (,) C. ( L,] D. [,L) E. [,kl) D. You must recognize this DEQ s logistic one. So = ky L y = y = nd y = L. So the grph of f line y = L. But since, the rnge is [,L). Not recognizing the logistic eqution would involve you solving the DEQ which is time consuming nd unnecessry. hs horizontl symptotes long the - is nd the Demystifying the BC Clculus MC Em

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 = to = b. For this section, we will only concentrte on curves tht re defined in function form. Functions defined prmetriclly, in polr or in vector-vlued forms hve their own formuls. How to find it: The rc length of continuous function f b [ ] over n intervl [, b] is given by L = + f ( ). Most problems involving rc length need clcultors becuse of the difficulty of integrting the epression. 7. (Clc) An nt wlks round the first qudrnt region R bounded by the y-is, the line y = nd the curve f ( ) = 6 s shown in the figure to the right. Find the distnce the nt wlked. A..9 B.. C..85 D..88 E..85 E. First, it is necessry to find the intersection point of the two curves. 6 = t =. The distnce between (,) nd (,6) is 6 nd from (,) to (,) is 5 [ ] D = f ( ) + 5 [ ] D = = D = = If the length of curve from = to = 8 is given by + 8, nd the curve psses through the point (-, ), which of the following could be the eqution for the curve? 8 A. y = 9 B. y = C. y = 7 + D. y = E. y = 9 5 [ ] = 8 f ( ) =±9 nd f ( ) =± + C C. f f =± + C = + C = nd C = 7 or + C = nd C =. So y = 7 + or y = Demystifying the BC Clculus MC Em

13 9. (Clc) The yellow bird in the populr gme Angry Birds flies long the pth y = + when. When = (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 -is, find the totl distnce the bird flies. A. 6.8 B.. C.. D. 5.6 E. 8. =. y( ) = + 8 = 8 E. y = y Tngent line : y 8 = ( ) y = Length : L = + ( ) + + = 8.. (Clc) The grphs of i) y =, ii) y = nd iii) y = ll pss through the points (,) nd (,). Find the difference in rc length from the lrgest rc length to the shortest rc length of these functions on the intervl [,]. A..58 B. 8 C..8 D..6 E..9 A. i) d = + =.79 ii) d = + =.79 iii) + ln =..79. =.58.. Find the rc length of the grph of = ( y + ) for y. A. B. C. 6 D. E. A. = ( y + ) ( y) = yy ( + ) L = + y y + = y + y + = y + = y + L = y + y = 9 + = Demystifying the BC Clculus MC Em

14 G. Prmetric Equtions Wht you re finding: Prmetric equtions re continuous functions of t in the form = f ( t) nd y = gt. Tken together, the prmetric equtions crete grph where the points 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 stu of vectors prllels the stu 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 = slope of C t the point (, y) is given by =,. The nd derivtive of the curve is given by d y = The rc length is given by L = t = b t = + d = d. f ( t) nd y = gt, then the. The curve must be smooth nd my not intersect itself.. Wht is the re under the curve described by the prmetric equtions = cost nd y = sin t for t? A. B. 8 C. D. E. B. cost = cos t = nd sin t = y Since sin t + cos t = + y = y = When t =, = nd when t =, = A = = Demystifying the BC Clculus MC Em = 6 6 ( + ) = 8. A position of prticle moving in the y-plne is given by = t 6t + 9t + nd y = t 9t. For wht vlues of t is the prticle t rest? A. only B. only C. only D. nd only E., nd = t C. = t t + 9 = t t + = 6t 8t = 6t t = nd = t t =. = t =,t = ( t ) = t =, t =

15 . A curve C is defined by the prmetric equtions = t t nd y = t 7t. Which of the following is the eqution of the line tngent to the grph of C t the point (, )? A. y = 6 B. y + = C. 5 y + = D. y = E. No tngent line t (, ) D. = t t = t t 6 = ( t ) ( t + ) = t =,t = y = t 7t = t 7t 6 = = Tngent line : y = = t 7 t 7 7 = [ t = ] 6 = y = 5. Describe the behvior of curve C defined by the prmetric equtions = + t nd y = t + t t t t =. Only t = stisfies this eqution. A. Incresing, concve up B. Decresing, concve up C. Incresing, concve down D. Decresing, concve down E. Incresing, no concvity B. = d y = d = t + t t ( ) = t = t t + t t =t t + t < so C is decresing [ t =] 5 +t t d y > so C is concve up [ t =] 6. Find the epression which represents the length L of the pth described by the prmetric equtions = sin ( t) nd y = cos( t ) for t. A. L = sint cost sint B. L = sin t + 9sin 9t C. L = 6sin t cos t + 9sin 9t D. L = sin t cos t + 9sin t E. L = 6sin t cos t + 9sin t E. L = = sint cost = sint Demystifying the BC Clculus MC Em 6sin t cos t + 9sin t

16 H. Vector-Vlued 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 t nd y( t), its velocity vector ( t) nd y ( t) or its ccelertion vector ( 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: vt = ( t) [ ] + [ y ( t) ]. The speed is sclr, not vector. 7. A prticle moves on plne curve such tht t ny time t >, its -coordinte is t t + t while its y-coordinte is t. Find the mgnitude of the prticle s ccelertion t t =. A. B. C. D. E. E. vt = t + t,( t )( t) = t + t,t 8t t = 6t,t 8 =, = + = 8. The position of n object moving in the y-plne with position function r( t) = + sint,t + cost, t. Wht is the mimum speed ttined by the object? A. B. C. D. E. C. vt = ( t) [ ] + [ y ( t) ] = cos t + sint vt = cos t + sint + sin t = sint Clculus techniques could be used to mimize this epression but since sint hs minimum/mimum vlues of nd, the speed will rnge from to =. 9. A y-plne hs both its nd y-coordintes mesured in inches. An nt is wlking long this plne with its position vector s t,t, t mesured in minutes. Wht is the verge speed of the nt mesured in inches per minute from t = to t = minutes? A. B. C. D. E. 9 B. L = + = 9t + 9 = t + = t + [ ] = 8 = Avg velocity = L = in min Demystifying the BC Clculus MC Em

17 . An object moving in the y-plne hs position function r( t) = Describe the motion of the object. ( t +),t 6ln t +, t. A. Left nd up B. Left nd down C. Right nd up D. Right nd down E. Depend on the vlue of t E. = which is lwys negtive when t so object moving left t + = ( t + t ) t + ( t ) 6 = t t + = t + t 6 = t + t + t + chnges sign t t = so y chnges direction t t =.. An object moving long curve in the y-plne hs position t, yt = 8t + nd = sint for t. ( ) t time t with At time t =, the object is t position (5, ). Where is the object t t =? B. ( + + 5, +) C. + 5, + A , D., E. ( + 5,) B. = t + t + C = C = 5 = t + t + 5 y = cost + C y = + C = C = + y = cost + + = = + 5 y = (, y) t = = + + 5, = + t time t with. (Clc) An object moving long curve in the y-plne hs position t, yt = t + t + nd = et for t. At time t =, the object is t position (-6, -7). Find the position of the object t t =. A. (.667,.5) B. (-.68,.78) C. (.7, 9.78) D. (.7, 6.5) E. (-.57, -.97) E. = + y = y + ( t ) = =.57 y ( t ) = 7 + e t =.97 (,y ) =.57, Demystifying the BC Clculus MC Em

### 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.

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

### ( ) 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

### AB 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

### ( ) Same as above but m = f x = f x - symmetric to y-axis. 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

### ( ) 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

### ( ) 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

### Topics 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.

### First 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 piecewise-liner function f, for 4, is shown below.

### AP 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

### Calculus 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:

### Improper 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

### Calculus 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..

### Math 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

### Thomas 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

### Chapter 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

### Math 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

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Improper 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:

### MATH 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

### Mathematics 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

### 2 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

### CHAPTER 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 twice-differentile function of x, then t

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### Review 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

### Definition 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)

### Math 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)

### Ch AP Problems

Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

### x 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

### Main topics for the Second Midterm

Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

### Math 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

### f(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

### 4.4 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

### AB Calculus Path to a Five Problems

AB Clculus Pth to Five Problems # Topic Completed Definition of Limit One-Sided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity

### Chapter 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

### ( ) 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

### MAT137 Calculus! Lecture 20

officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find

### lim 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

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/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

### Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

### Logarithmic Functions

Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### Chapter 8: Methods of Integration

Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

### MA 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

### ROB EBY Blinn College Mathematics Department

ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous

### Improper 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

### ( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

### . Double-angle 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?

### critical 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

### along 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

### Sample 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.

### Stuff 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

### Section 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

### 7. Indefinite Integrals

7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

### Properties 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

### Disclaimer: 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

### A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

### . Double-angle 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

### y = f(x) This means that there must be a point, c, where the Figure 1

Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham)

PG. Clculus AB Bile (nd most importnt ook in the world) (Written nd compiled y Doug Grhm) Topic Limits Continuity 6 Derivtive y Definition 7 8 Derivtive Formuls Relted Rtes Properties of Derivtives Applictions

### A 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

### Chapter 6 Techniques of Integration

MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

### Math 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

### We 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:

### MATH 122B AND 125 FINAL EXAM REVIEW PACKET (Fall 2014)

MATH B AND FINAL EXAM REVIEW PACKET (Fll 4) The following questions cn be used s review for Mth B nd. These questions re not ctul smples of questions tht will pper on the finl em, but they will provide

### f 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

### First 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, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

### Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

### 5.7 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

### How 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

### Math 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

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

### Mathematics. Area under Curve.

Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

### Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

### Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

### different 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

### 0.1 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

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

### Homework 11. Andrew Ma November 30, sin x (1+x) (1+x)

Homewor Andrew M November 3, 4 Problem 9 Clim: Pf: + + d = d = sin b +b + sin (+) d sin (+) d using integrtion by prts. By pplying + d = lim b sin b +b + sin (+) d. Since limits to both sides, lim b sin

### Unit 1 Exponentials and Logarithms

HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

### HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

### Integration 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

### f(a+h) f(a) x a h 0. This is the rate at which

M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016

HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M

### Math& 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

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Partial 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 right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

### Lecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is

Lecture XVII Abstrct We introduce the concepts of vector functions, sclr nd vector fields nd stress their relevnce in pplied sciences. We study curves in three-dimensionl Eucliden spce nd introduce the

### MORE 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

### Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

### Lesson 1: Quadratic Equations

Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

### Practice 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

### than 1. It means in particular that the function is decreasing and approaching the x-

6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the