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


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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 multiplechoice, trueflse, 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 multiplechoice problems nd trueflse 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 yvlue 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 yxis nd the curve x = y / y (whose yintercepts 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 footpounds, 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 yxis 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 yxis 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 xxis 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 xxis 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 welldefined. (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 xyplne s its bse nd hs ll of its crosssections perpendiculr to the yxis being squres. 7. Find the volume of the solid obtined by revolving, bout the yxis, 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 xyplne 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
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