Math 116 Final Exam April 26, 2013


 Helen Singleton
 1 years ago
 Views:
Transcription
1 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 re not of equl difficulty, so you my wnt to skip over nd return to problem on which you re stuck. 3. Do not seprte the pges of this exm. If they do become seprted, write your nme on every pge nd point this out to your instructor when you hnd in the exm. 4. Plese red the instructions for ech individul problem crefully. One of the skills being tested on this exm is your bility to interpret mthemticl questions, so instructors will not nswer questions bout exm problems during the exm. 5. Show n pproprite mount of work (including pproprite explntion) for ech problem, so tht grders cn see not only your nswer but how you obtined it. Include units in your nswer where tht is pproprite. 6. You my use ny clcultor except TI92 (or other clcultor with full lphnumeric keypd). However, you must show work for ny clcultion which we hve lerned how to do in this course. You re lso llowed two sides of 3 5 note crd. 7. If you use grphs or tbles to find n nswer, be sure to include n explntion nd sketch of the grph, nd to write out the entries of the tble tht you use. 8. Turn off ll cell phones nd pgers, nd remove ll hedphones. Problem Points Score Totl
2 Mth 6 / Finl (April 26, 23) pge 2 You my find the following expressions useful. Known Tylor series (ll round x = ): sin(x) = n= ( ) n x 2n+ (2n + )! = x x3 3! + + ( )n x 2n+ + for ll vlues of x (2n + )! ( ) n x 2n cos(x) = (2n)! n= = x2 2! + + ( )n x 2n + for ll vlues of x (2n)! e x = n= x n n! = + x + x2 2! + + xn + for ll vlues of x n! ( ) n+ x n ln( + x) = n n= = x x2 2 + x3 3 + ( )n+ x n + for < x n ( + x) p = + px + p(p ) 2! x 2 + p(p )(p 2) 3! x 3 + for < x < x = x n = + x + x 2 + x x n + for < x < n=
3 Mth 6 / Finl (April 26, 23) pge 3. [ points] Indicte if ech of the following is true or flse by circling the correct nswer. No justifiction is required.. [2 points] Let < q <, then q n = q + q 2 + q q n + = n= q q. series n= True Flse Since n= qn = q ( n= qn ), then using the formul for geometric r n = r with r = q yields the result. b. [2 points] Let F (t) be n ntiderivtive of continuous function f(t). If the units of f(t) re meters nd t is in seconds, then the units of F (t) re meters per second. True Flse The Second Fundmentl Theorem of Clculus sys tht if F (t) is n ntiderivtive of f(t), then F (t) = t f(x)dx. The units of definite integrl re the unites of f(t) times the units of t. In this cse, the units of F (t) re meters times seconds. c. [2 points] If the motion of prticle is given by the prmetric equtions x = t + t 3, y = t2 + t 3 for >, then the prticle pproches the origin s t goes to infinity. True Flse Since lim x(t) = lim y(t) =, then the prticle pproches the origin t t s t goes to infinity. d. [2 points] Let n be sequence of positive numbers stisfying lim n =. Then n the series converges. n n= If n = n, then lim n n =, but n= e. [2 points] Let f(x) be continuous function. Then n True diverges by pseries test. Flse f(2x)dx = 2 f(x)dx.
4 Mth 6 / Finl (April 26, 23) pge 4 Using the substitution u = 2x, you get f(2x)dx = 2 2 f(u)du. True Flse
5 Mth 6 / Finl (April 26, 23) pge 5 2. [6 points] Let the sequence n be given by =, 2 =. [ point] Find , 3 = 5, 4 = 7 = , 5 = 9, 6 = 6 b. [3 points] Write formul for n. n n = ( ) n 2n. c. [2 points] Does the sequence n converge? If so, find its limit. Yes, it converges to.
6 Mth 6 / Finl (April 26, 23) pge 6 3. [ points] A bot s initil vlue is $, ; it loses 5% of its vlue ech yer. The bot s mintennce cost is $5 the first yer nd increses by % nnully. In the following questions, your formuls should not be recursive.. [2 points] Let B n be the vlue of the bot n yers fter it ws purchsed. Find B nd B 2. B = $, (.85). B 2 = $, (.85) 2. b. [3 points] Find formul for B n. B n =, (.85) n c. [2 points] Let M n be the totl mount of money spent on the mintennce of the bot during the first n yers. Find M 2 nd M 3. M 2 = 5( +.) M 3 = 5( +. + (.) 2 ) d. [3 points] Find closed form formul for M n. M n = 5 ( (.)n ).
7 Mth 6 / Finl (April 26, 23) pge 7 4. [ points] The lifetime t (in yers) of tree hs probbility density function for t. (t + ) p f(t) = for t <. where > nd p >.. [4 points] Use the comprison method to find the vlues of p for which the verge lifetime M is finite (M < ). Properly justify your nswer. The verge lifetime M is given by the formul M = Since then We know tht t (t + ) t p t = for t >, p t p t (t + ) dt p t p dt t (t + ) dt. p converges precisely when p > ( p > 2) by the ptest, tp so the first integrl converges precisely when p > 2. This implies tht the verge lifetime M is finite for p > 2. Note: We use the inequlity t (t + ) dt dt since the inequlity p tp ( ) dt is not useful dt = for ll vlues of p. tp tp You do not need to discuss the convergence of the integrl this integrl is not n improper integrl. t (t + ) p t dt since (t + ) p
8 Mth 6 / Finl (April 26, 23) pge 8 b. [4 points] Find formul for in terms of p. Show ll your work. We know tht = (t + ) p dt. We use usubstition with u = t + to clculte the integrl: Therefore = dt = lim (t + ) p b b b+ = lim b = lim b (since p > ) = p., so = p. p (t + ) p dt b+ du = lim u p du up b u p+ ( p + ) b+ = lim b+ b ( p + )up c. [2 points] Let C(t) be the cumultive distribution function of f(t). For given tree, wht is the prcticl interprettion of the expression C(3)? C(3) is the probbility tht given tree lives t lest 3 yers.
9 Mth 6 / Finl (April 26, 23) pge 9 5. [4 points] A skydiver jumps from plne t height of 2, meters bove the ground. After some time in freefll, he opens his prchute, reducing his speed, nd lnds sfely on the ground.. [5 points] The grph of the skydiver s downwrd velocity v(t) (in meters per second) t seconds fter he jumped is shown below. Sketch the grph of the ntiderivtive y(t) of v(t) stisfying y() =. Mke sure your grph reflects the regions t which the function is incresing, decresing, concve up or concve down. It is importnt to notice tht y (t) exist for ll vlues of t since y (t) = v(t). b. [3 points] Write down righthnd sum with 4 subintervls in order to pproximte the verge downwrd velocity of the skydiver during the time the skydiver is in freefll. Show ll the terms in your sum. The verge downwrd velocity is 2 v(t)dt. We pproximte this 2 s 2 2 v(t)dt 5( ) 2 c. [2 points] Is your estimte in (b) gurnteed to be n underestimte or overestimte of the verge velocity of the skydiver, or there is not enough informtion to decide? Justify. It s gurnteed to be n overstimte, becuse v(t) is incresing throughout [, 2]. d. [4 points] Find formul for the height H(t) (in meters) bove the ground of the skydiver t seconds fter he jumped. H(t) = 2, t v(s)ds = 2, y(t).
10 Mth 6 / Finl (April 26, 23) pge 6. [ points] At hospitl, ptient is given drug intrvenously t constnt rte of r mg/dy s prt of new tretment. The ptient s body depletes the drug t rte proportionl to the mount of drug present in his body t tht time. Let M(t) be the mount of drug (in mg) in the ptient s body t dys fter the tretment strted. The function M(t) stisfies the differentil eqution dm dt = r M with M() =. 4. [7 points] Find formul for M(t). Your nswer should depend on r. We use seprtion of vribles dm r 4 M = dt. Using usubstition with u = r /4M, du = /4dM for the lefthndside, we ntidifferentite: 4 ln r 4 M = t + C. Therefore, ln r 4 M = t/4 + C 2 nd r 4 M = e t/4+c 2 = C 3 e t/4. Therefore /4M = r C 3 e t/4 nd M(t) = 4r C 4 e t/4. With M() =, we conclude tht C 4 = 4r, so we get M(t) = 4r 4re t/4. b. [ point] Find ll the equilibrium solutions of the differentil eqution. M = 4r. c. [2 points] The tretment s gol is to stbilize in the long run the mount of drug in the ptient t level of 2 mg. At wht rte r should the drug be dministered? You need 4r = 2, then r = 5 mg/dy.
11 Mth 6 / Finl (April 26, 23) pge 7. [9 points] A tnk hs the shpe of circulr cone. The cone hs rdius 2 m nd height 7 m (s shown below). The tnk contins liquid up to depth of 4 m. The density of the liquid is δ(y) = y 2 kg/m 3, where y mesures the distnce in meters from the bottom of the tnk. Use the vlue g = 9.8 m/s 2 for the ccelertion due to grvity.. [6 points] Find definite integrl tht computes the mss of the liquid in the tnk. Show ll your work. Let r(y) be the rdius t height y. By similr tringles, 2/7 = r/y, so r = 2y. The pproximte mss of thin slice t height y is 7 π(2/7y)2 ( y) 2 y, so the nswer is 4 π(2/7y) 2 ( y 2 )dy. b. [3 points] Find definite integrl tht computes the work required to pump the liquid 2 meters bove the top of the tnk. Show ll your work. We wnt to lift ech thin slice (9 y) feet. The work to lift slice is 9.8(9 y)π(2/7y) 2 ( y 2 ) y, so the integrl is 4 9.8(9 y)π(2/7y) 2 ( y 2 )dy.
12 Mth 6 / Finl (April 26, 23) pge 2 8. [8 points] Consider the power series n= 2 n n (x 5)n. In the following questions, you need to support your nswers by stting nd properly justifying the use of the test(s) or fcts you used to prove the convergence or divergence of the series. Show ll your work.. [2 points] Does the series converge or diverge t x = 3? At x = 3, the series is n= series test, since / n is decresing nd converges to. ( ) n n, which converges by the lternting b. [2 points] Wht does your nswer from prt () imply bout the rdius of convergence of the series? R 2. Becuse it converges t x = 3, we know tht the rdius of convergence c. [4 points] Find the intervl of convergence of the power series. Using the rtio test, we hve lim n 2 n+ n+ x 5 n+ 2 n x = n 5 n 2 x 5 = L, so the rdius of convergence is 2. Now we hve to check the endpoints. We know from prt () tht it converges t x = 3. For x = 7, we get, which diverges. n Thus, the intervl of convergence is 3 x < 7. n=
13 Mth 6 / Finl (April 26, 23) pge 3 9. [4 points] Determine the convergence or divergence of the following series. In questions () nd (b) you need to support your nswers by stting nd properly justifying the use of the test(s) or fcts you used to prove the convergence or divergence of the series. Circle your nswer. Show ll your work.. [4 points] n= 2n n5 + Converges Diverges You cn use either the limit comprison test or the comprison test. We simply use the comprison test. We know tht < 2n n5 + 2n n 2 5/2 n. 3/2 Becuse n converges by the pseries, the series 2n 3/2 n5 + n= the comprison test. converges by b. [4 points] n 2 e n3 Converges Diverges n= Since the function f(x) = x 2 e x3 is positive nd decresing for x >, we cn use the integrl test to determine the convergence or divergence of n 2 e n3. To do this, we use usubstitution. Let u = x 3, du = 3x 2 dx. Therefore n= Hence n= x 2 e x3 dx = lim x 2 e x3 dx = lim b 3 = lim b 3 eu b = 3 3. b b n 2 e n3 converges by the integrl test. b 3 e u du c. [6 points] Determine if the following series converge bsolutely, conditionlly or diverge. Circle your nswers. No justifiction is required. ). n= sin(3n) n 6 + Converges bsolutely Converges conditionlly Diverges b). ( ) n+ n 3n + n= Converges bsolutely Converges conditionlly Diverges
14 Mth 6 / Finl (April 26, 23) pge 4. [9 points]. [3 points] Find the first three nonzero terms in the Tylor series of f(y) = bout y =. Show ll your work. Using the binomil expnsion, this is ( + y) 3 2 ( + y) 3 ( 3/2) ( 5/2) y + y 2 = 3 3/ y y2 b. [2 points] Use your nswer in () to find the first three nonzero terms in the Tylor series of g(x) = bout x =. Show ll your work. ( 2 + x 2 ) 3 2 Fctoring, we hve ( 2 + x 2 ) 3 2 = ( 2 ( + ( x )2 )) 3 2 = ( 2 ) 3 2 ( + ( x )2 ) 3 2 ( x ) 2, Therefore, letting y = we hve 3 ( + ( x )2 ) = ( ) ( 3 3/2 3 ( + y) 3/2 3 2 y + 5 ) 8 y2 = 3 ( + ( x )2 ) 3/2. ( 2 + x 2 ) 3 2 = ( 3 ( x ) 2 5 ( ) x 4 + = ) x x x x4. c. [2 points] For which vlues of x is the Tylor series for g(x) bout x = expected to converge? The Binomil series in ) converges for y <. This implies tht the series for g(x) converges for ll vlues of x stisfying x <, so < x <. Problem continues on the next pge
15 Mth 6 / Finl (April 26, 23) pge 5 Continution of problem. The force of grvittionl ttrction F between rod of length 2L nd prticle t distnce is given by L F = k dx, ( 2 + x 2 ) 3 2 where k is positive constnt. d. [2 points] Use your nswer in (b) to obtin n pproximtion for the force of grvittionl ttrction F between the rod nd the prticle. Your nswer should depend on the constnts k, nd L. Show ll your work. We wnt F = k = k L L dx k 3/2 ) ( 2 + x 2 ) ( L L L5 Hence F kl 3 k 2 5 L3 + 3k 8 7 L x x4 dx
Math 116 Final Exam April 26, 2013
Math 116 Final Exam April 26, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 13 pages including this cover. There are 10 problems. Note that the
More informationMATH 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
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 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 informationThe 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
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 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 informationINTRODUCTION 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
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. Theorem Suppose f is continuous
More informationBig 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:
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 informationPrep Session Topic: Particle Motion
Student Notes Prep Session Topic: Prticle Motion Number Line for AB Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position,
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 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 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 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 informationMath 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
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 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
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 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, 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 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 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 informationThe 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
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 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 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 informationFinal 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, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
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 informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
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 informationSection 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
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 informationSection 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
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 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 informationMath 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
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 informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
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 informationTotal Score Maximum
Lst Nme: Mth 8: Honours Clculus II Dr. J. Bowmn 9: : April 5, 7 Finl Em First Nme: Student ID: Question 4 5 6 7 Totl Score Mimum 6 4 8 9 4 No clcultors or formul sheets. Check tht you hve 6 pges.. Find
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More information7.6 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
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
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 informationMath 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
More informationl 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 nsided regulr polygon of perimeter p n with vertices on C. Form cone
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationGoals: 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
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 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 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 informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationDefinite 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
More informationAPPM 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
More informationA. 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
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( ) 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 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 informationDefinite 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
More informationHIGHER 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
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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
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 informationHIGHER 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
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 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 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 information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationCAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.
Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you
More informationAP * Calculus Review
AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril.
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More information1. 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)
More information1 Techniques of Integration
November 8, 8 MAT86 Week Justin Ko Techniques of Integrtion. Integrtion By Substitution (Chnge of Vribles) We cn think of integrtion by substitution s the counterprt of the chin rule for differentition.
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 informationMA123, 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. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
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 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 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 informationn 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
More informationMath 42 Chapter 7 Practice Problems Set B
Mth 42 Chpter 7 Prctice Problems Set B 1. Which of the following functions is solution of the differentil eqution dy dx = 4xy? () y = e 4x (c) y = e 2x2 (e) y = e 2x (g) y = 4e2x2 (b) y = 4x (d) y = 4x
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
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 informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
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 informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationMath 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
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI 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...
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 informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More information