# Sample Problems for the Final of Math 121, Fall, 2005

Size: px
Start display at page:

Transcription

1 Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd of the text which constitute only prt of the common Mth Finl. Since the finl is comprehensive, this collection should be complemented by the smple problems for the midterm nd the rel midterm of this semester when prepring for the finl. Another file posted seprtely contins smple Mth Finl, which gives concrete exmple of rel finl nd lso provides good set of review problems. Note tht the skills needed to solve the Gtewy Exm problems re expected from you when you tke the finl. In prticulr, you re expected to hve memorized the derivtives of the bsic functions like x, x, log (x), sin(x), sin (x), cos(x), cos (x), tn (x), tn (x), etc. Also note tht most of the problems in this list cn nd my be formulted, in the ctul finl, in form (s multiple-choice, true-flse, or essy problem) different from how they re formulted here. (The following problems re collected/edited from vrious sources including previous review mterils from KU professors nd the textbook.) Except for multiple-choice problems nd true-flse problems, you hve to show your work nd find the exct solution.. Find the following limits. (i) lim x + ln (x + ) cot (x). (ii) lim x ( + 5x) / ln(x).. Let y = tn (x ). (i) Find the differentil dy. (ii) Evlute dy nd y if x = nd x = dx =.. (iii) Use dy to estimte tn (.), i.e. the y-vlue t x =... Evlute (i) x dx nd (ii) x + x ( x ) dx, if convergent. x6. For f (x) =, find (i) the most generl ntiderivtive of f on the intervl (, ), x nd (ii) the ntiderivtive F of f with F (/) =. 5. Wht constnt ccelertion (in mi/h ) is required to increse the speed of cr from 5 mi/h to 6 mi/h in seconds? 6. If f (x) = 6x + x, f () =, nd f () =, find f. 7. If f (x) dx =, 7 5 f (x) dx =, nd 7 f (x) dx =, find 5 f (x) dx. 8. Find the re of the region (in the first qudrnt) bounded by the y-xis nd the curve x = y / y (whose y-intercepts re nd ). 9. Compute. Differentite x () x + dx, (d) x ln (x) dx, x (g) x x dx. x (b) (e) cos (t) x t dt, nd cos t dt. x + x + x + dx, xe x+ dx, (c) dx, x (f) sin (x) cos (x) dx,

2 . If f is continuous nd f (x) dx = 8, find x f ( x ) dx.. Find the length of the prmetric curve (x, y) = ( t sin (t), ln (t) ) with t s concrete definite integrl without ctully computing its vlue.. A spring exerts restoring force proportionl to the distnce it is stretched from its nturl length. We stretch it feet beyond its nturl length nd mesure the restoring force t pounds. How much work, in foot-pounds, is done in stretching the spring n dditionl feet (thus from feet beyond its nturl length to 5 feet beyond its nturl length)?. A ft ldder is lening ginst the wll. If the bottom of the ldder is being pulled wy (from the wll) t the constnt rte of ft/sec, how fst is the top coming down when the top is 6 ft bove the ground? 5. Two crs strt moving from the sme point. One cr trvels north t mph nd the other cr trvels est t mph. Let d(t) denote the distnce between the crs t time t (the number of hours fter the crs leve the initil point). How fst is the distnce between the crs incresing two hours lter? 6. A mn strts wlking north t 5 ft/s from point P. Ten seconds lter womn strts wlking est t ft/s from point ft due est of P. At wht rte re the people moving prt seconds fter the womn strts wlking? 7. A tnk of the shpe of circulr cone with its vertex pointing downwrd (nd its top horizontl) is being filled with wter. Assume tht the rdius of its circulr top is 6 m nd its height (i.e. the distnce from the vertex to the top) is m. Let h (t) be the wter level (i.e. the distnce from the vertex to the wter surfce) in the tnk t time t in minutes. If the wter is being pumped into the tnk t the rte of 5 m /min strting from t =, how fst is the wter level rising t time t =? 8. Find the derivtives of the functions (i) F (x) = (iii) H (x) = ln(x) tn (t) dt. x sin ( t ) dt, (ii) G (x) = 9. For function f with continuous derivtive f on the whole rel line, lim h h () f (), (b) f(), (c) f(), (d) f (), (e) none of the bove.. Find continuous function f nd constnt such tht x x x x +h f (t)dt = 6x 9. + t dt, nd f (x)dx =. () Give the itertive formul for Newton s method for pproximting root of n eqution f(x) =, where f is differentible function on the rel line. (b) Use Newton s method with first guess x = to pproximte the solution of the eqution x x 8 = by listing the first numbers in the sequence of pproximtions obtined by the Newton s method.. Given function f with continuous derivtive f on the rel line R nd with f () =, f () = 5, lim x f (x) =, nd lim x f f (x) (x) =, evlute the following limits: () lim x x, (b) lim x f (x) ln (x), (c) lim cos (x) x f (x).. A prticle moves long the y-xis so tht its velocity t ny time t is given by v(t) = t cos t. At time t =, the position of the prticle is y =. () For wht intervls of t, t 5, is the prticle moving upwrd? (b) Write n expression for the ccelertion (t) of the prticle in terms of t. (c) Write n expression for the position y(t) of the prticle in terms of t. (d) Find the position of the prticle t the

3 moment when its velocity becomes zero for the first time fter the beginning of motion.. A prticle with velocity t ny time t given by v(t) = e t moves in stright line. How fr does the prticle move from time t = to t =? 5. In three hour trip, the velocity of cr t ech hlf hour ws recorded s follows: Time (Hours) Velocity (MPH) Estimte the distnce trveled using the Simpson s pproximtion S 6 nd estimte the verge velocity of the cr during this trip. 6. Express s concrete sum of numbers the pproximtion T 6 to x + dx obtined by the Trpezoidl Rule. 7. Evlute the following integrls: () e sin x cos x dx, (d) x sin (x) dx, 8. Let F (x) = π/ (g) cos x sin x dx, (j) (xe x + e +x ) dx, f(x) (b) (e) 6 (h) (k) x + x + x + 5 dx, 6x(x + ) dx, x x dx, dt 9t +, (c) ( x) dx, (f) x + x dx, (i) (l) (x + x ) dx, x(ln x) dx. tn (t ) dt for differentible function f. Then F (x) = () tn (x), (b) tn (x) f (x), (c) f (tn (x)), (d) sec (x), (e) none of the bove. 9. If k (kx x ) dx = 8, then k = () 9, (b), (c), (d) 9, (e) none of the bove.. If the function g hs continuous derivtive on [, c], then c g (x) dx = () g(c) g(), (b) g(x) + c, (c) g(x) g(), (d) g(c), (e) none of the bove.. Given the following grph of function f, define the function g (x) = ech of the following sttements is true or flse. x 5 f (t) dt. Determine whether

4 y 5 5 x figure T F () g ( ) =. T F (b) g ( ) >. T F (c) g () >. T F (d) g () >. T F (e) g ( ) <. T F (f) g () <.. For the function g defined in Problem, find the points x in the open intervl ( 5, 5) t which g hs locl mximum, nd the points t which g hs n bsolute mximum over the closed intervl [ 5, 5].. Let < c < b nd let g be differentible on [,b]. Which of the following is NOT necessrily true? () b b g(x) dx = c g(x) dx + b c g(x) dx, (d) lim x c g(x) = g(c), (e) If k is constnt, then. If f is n even nd continuous function, then () f(x) dx, (b) g(x) dx, (b) There exists d in [, b] such tht g (d) = g(b) g(), (c) f(x) dx, (c), (d) /, (e) f(x) dx + b k g(x) dx = k f(x) dx = f(x) dx, (f) none of the bove. b b g(x) dx. 5. A publisher estimtes tht book will sell t the rte of r(t) = 6, e.8t books per yer t the time t yers from now. Find the totl number of books tht will ever be sold (up to t = ). 6. Let R be the region in the first qudrnt enclosed by the y-xis nd the grphs of y = sin x nd y = cos x, for x π/. () Set up the definite integrl for the re of R nd evlute it exctly. (b) Find the centroid of (x, y) of R. (c) Set up the integrl for the volume of the solid generted when R is revolved bout the x-xis nd evlute it exctly. (d) Set up definite integrls to compute the perimeter of R. Do not compute the integrls. 7. For function f with continuous derivtive f on the rel line, the integrl x f ( x ) dx = () x f ( x ) x f ( x ) dx, (b) xf ( x ) f ( x ) dx, (c) uf (u) du with u = x, (d) xf ( x )

5 f ( x ) dx, (e) none of the bove. 8. The mount of pollution in lke x yers fter the closing of chemicl plnt is P (x) = /x tons (for x ). Find the verge mount of pollution between nd yers fter the closing. 9. Consider the function f(x) = + x on the intervl [, ]. Find number c in [, ] so tht the re of the rectngle with bse on [, ] nd height f(c) is equl to the re under the grph of f in the given intervl.. Compute the length of the curve given by x = e t sin t nd y = e t cos t, for t π.. A prticle is moved long the x-xis by force tht mesures x pounds t point x feet from the origin. Find the work done in moving the prticle over distnce of ft. from the origin.. A crne is lifting 5 lb trnsformer from the ground level to the third floor which is feet bove ground level. A 6 foot cble connects the trnsformer to the top of the crne. The cble weighs 5 lb per liner foot. How much work is done in lifting the trnsformer feet bove the ground?. The grph of continuous function f on the closed intervl [ 5, 5] is shown in the following figure, where the rc is semicircle. Let h(x) = x f(t) dt for 5 x 5. () Compute h(5) nd h ( ). (b) Compute h ( ) nd h (). (c) Find the set of points x t which h is well-defined. (d) On wht intervl or intervls is the grph of h concve upwrd? (e) Find the vlue(s) of x t which h hs its bsolute mximum nd minimum on the closed intervl [ 5, 5] figure. It is observed tht long stright highwy from city A to city B, cr pssed city A t speed mph (miles per hour) t :.m. nd pssed city B t speed 5 mph (miles per hour) t :.m. on the sme dy, where cities A nd B re miles prt. T F () At certin moment between :.m. nd :.m., the cr s speed hs to be t lest 7 mph. T F (b) At certin moment between :.m. nd :.m., the cr s speed hs to be t lest 66 mph. T F (c) Between :.m. nd :.m., the cr s speed cn never exceed 7 mph. T F (d) At certin moment between :.m. nd :.m., the cr s ccelertion hs to be t lest mi/h. T F (e) Between :.m. nd :.m., the cr s ccelertion cn never be greter thn 5 mi/h. T F (f) Between :.m. nd :.m., the cr s ccelertion cn never be negtive. 5

6 5. Find the volume of the solid obtined by revolving, bout the line x =, the region R in the first qudrnt nd bounded by the curves y = x nd y = x /5. 6. Find the volume of the solid S tht hs the region { (x, y) : x /5 y } in the xy-plne s its bse nd hs ll of its cross-sections perpendiculr to the y-xis being squres. 7. Find the volume of the solid obtined by revolving, bout the y-xis, the region { R = (x, y) : x nd y e x}. (Hint: Use the method of cylindricl shells.) 8. An qurium 5 m long, m wide, nd m deep is full of wter. Find the work needed to pump hlf of the wter out of the qurium over its top. Note tht the density of wter is kg/m nd the grvittionl ccelertion is 9.8 m/s. 9. A swimming pool is m wide nd 5 m long, nd its bottom is n inclined plne, the shllow end hving depth of m nd the deep end m. If the pool is full of wter, find the hydrosttic force on () the deep end, (b) one of the two (trpezoidl) sides, nd (c) the bottom of the pool. 5. Let y = ln ( x + e ). (i) Find the differentil dy. (ii) Use dy t x = to estimte ln ( e +. ) in terms of the constnt e. 5. The edge of cube is mesured to be cm with possible error in mesurement of. cm. Use differentil to estimte the mximum possible error in computing the surfce re of the cube. 5. A mn strts wlking north t ft/s from point P. Five seconds lter womn strts wlking est t ft/s from point ft due est of P. At wht rte re these two persons moving prt seconds fter the womn strts wlking? 5. Find f (x) for x >, if f () =, f () =, nd f (x) = x + x. 5. With wht constnt negtive ccelertion (ft/s ) by brkes cn cr be brought to full stop from speed of 6 mi/h within exctly distnce of 5 feet? ( mi. = 58 ft.) 55. Find the volume of the solid S with flt bse which is the region R bounded by y = x nd y = x on the xy-plne nd with its intersection with ny plne x = c, c R, being either n equilterl tringle or n empty set. 56. If f (x) dx =, 8 6 f (x) dx =, nd 8 f (x) dx =, find (i) 6 f (x) dx nd (ii) Find the re of the region R bounded by the curves y = x x nd y = x x. (f (x) sin (x)) dx. 58. A log meters long is cut t -meter intervls nd the dimeters, in meters, of its (circulr) cross sections t these 9 cuts from one end to the other re.5,.8,.6,.7,.8,.,.9,.8, nd.9. Use Simpson s Rule to estimte the volume of the log. x 59. Compute (i) x dx, (ii) x x + dx, (iii) + x dx, (vi) (x ) x dx, nd (vii) 9 6. Which of the following improper integrls is convergent? () x / sin (x) dx, (b) bove, (f) none of the bove. xe x dx, (c) sin (x) cos (x) dx, (iv) ( ) x x + ln ( x + ) dx. sin (x) dx, (v) x ( + x ) dx, (d) x / ( x) dx, (e) ll of the 6

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

### 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

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### 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.

### Math 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

### Section 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

### Final Exam - Review MATH Spring 2017

Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

### 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,

### Math 0230 Calculus 2 Lectures

Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

### 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)

### 7.6 The Use of Definite Integrals in Physics and Engineering

Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

### a < 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

### Math 113 Exam 1-Review

Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 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

### 5 Accumulated Change: The Definite Integral

5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel

### 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

### The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas

Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous

### [ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

### 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

### Math 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 x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### Definite integral. Mathematics FRDIS MENDELU

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

### Unit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NON-CALCULATOR SECTION

Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NON-CALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)

### FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

### Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value

The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

### Math 1132 Worksheet 6.4 Name: Discussion Section: 6.4 Work

Mth 1132 Worksheet 6.4 Nme: Discussion Section: 6.4 Work Force formul for springs. By Hooke s Lw, the force required to mintin spring stretched x units beyond its nturl length is f(x) = kx where k is positive

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### ( ) 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

### APPLICATIONS OF THE DEFINITE INTEGRAL

APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through

### ( ) 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

### 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:

### Indefinite Integral. Chapter Integration - reverse of differentiation

Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

### 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

### 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

### Chapters 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

### ( ) 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

### The 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

### 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

### MATH 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

### MAT187H1F 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

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 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

### Math 116 Calculus II

Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

### ( ) 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

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### 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

### Unit #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

### SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

### MATH 115 FINAL EXAM. April 25, 2005

MATH 115 FINAL EXAM April 25, 2005 NAME: Solution Key INSTRUCTOR: SECTION NO: 1. Do not open this exm until you re told to begin. 2. This exm hs 9 pges including this cover. There re 9 questions. 3. Do

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

### Math 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

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### 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

### 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

### APPM 1360 Exam 2 Spring 2016

APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### 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

### 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

### 13.4 Work done by Constant Forces

13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

### Math 113 Exam 2 Practice

Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

### 1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

### 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

### Minnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017

Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.

### Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A

Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new

### 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

### 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 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

### 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

### MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

### ( ) 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

### 1 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

### Mathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions

Mthemtics 9A; Fll 200; V Ginzburg Prctice Finl Solutions For ech of the ten questions below, stte whether the ssertion is true or flse ) Let fx) be continuous t x Then x fx) f) Answer: T b) Let f be differentible

### MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

### HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue

### 7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

### Not for reproduction

AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

### 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

### Form 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6

Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

### Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

### Trigonometric Functions

Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

### Review Exercises for Chapter 4

_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

### AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review

AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle

### 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

### 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

### MATH1013 Tutorial 12. Indefinite Integrals

MATH Tutoril Indefinite Integrls The indefinite integrl f() d is to look for fmily of functions F () + C, where C is n rbitrry constnt, with the sme derivtive f(). Tble of Indefinite Integrls cf() d c

### The 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

### 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

### 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

### 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

### Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.

4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37

### 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

### Summary Information and Formulae MTH109 College Algebra

Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

### JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS

JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### 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