Math 116 Practice for Exam 2


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1 Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem on which you are suck.. Do no separae he pages of he exam. If any pages do become separaed, wrie your name on hem and poin hem ou o your insrucor when you hand in he exam. 3. Please read he insrucions for each individual exercise carefully. One of he skills being esed on his exam is your abiliy o inerpre quesions, so insrucors will no answer quesions abou exam problems during he exam. 4. Show an appropriae amoun of work (including appropriae explanaion) for each exercise so ha he graders can see no only he answer bu also how you obained i. Include unis in your answers where appropriae. 5. You may use any calculaor excep a TI9 (or oher calculaor wih a full alphanumeric keypad). However, you mus show work for any calculaion which we have learned how o do in his course. You are also allowed wo sides of a 3 5 noe card. 6. If you use graphs or ables o obain an answer, be cerain o include an explanaion and skech of he graph, and o wrie ou he enries of he able ha you use. 7. You mus use he mehods learned in his course o solve all problems. Semeser Exam Problem Name Poins Score Winer Winer 7 6 Venice Beach 3 Fall Fall 6 racking chip 4 Winer 6 3 O ghan 3 Toal 58 Recommended ime (based on poins): 58 minues
2 Winer, 7 Mah 6 Exam 3 Problem 4 Soluion
3 Mah 6 / Exam (March, 7) page 6 6. [3 poins] Anderson and Glen decide o ake a road rip saring from Venice Beach. They have no worries abou geing anywhere quickly, as hey enjoy each oher s company, so hey ake a very inefficien roue. Suppose ha Venice Beach is locaed a (,) and ha Anderson and Glen s posiion (x,y) (measured in miles) hours afer leaving Venice Beach is given by a pair of parameric equaions x = f(), y = g(). A graph of f() and a formula for g() are given below. Noe ha f() is linear on he inervals [,.5], [.5,.5], and [.5,3]. f() (mi) 3 g() = (hr) Noe: For each of he following, your final answer should no involve he leers f and g. a. [ poins] If heir roadrip las 3 hours, wha are he x and y coordinaes of heir final desinaion? Soluion: Noe ha a ime = 3, we have x = f(3) = and y = g(3) = 9. So he coordinaes of heir final desinaion are (, 9). b. [3 poins] A wha speed are hey raveling hours ino heir rip? Soluion: We have dx d = f dy () = and = d = g () = 5. = So heir speed a ime = is +5 = 5 miles per hour. c. [4 poins] Wrie, bu do no compue, an expression involving one or more inegrals ha gives he disance hey raveled, in miles, in he firs half hour of heir rip. Soluion: On he inerval (,.5), we see ha f() = 6, so on his inerval, we have f () = 6 and g () = The parameric arc lengh formula hen implies ha he disance hey ravelled from.5 = o =.5 is (6) +( 3 + 3) d miles. d. [4 poins] Wrie down a pair of parameric equaions using he parameer s for he line angen o heir pah a =.75 hours. Soluion: Noe ha df f(.75) =.5, d =, g(.75) = , and =.75 dg d =.85 =.75 There are many possible paramerizaions. There is no need o have his mach wih he parameer from earlier, so he answer below has he line passing hrough(.5, ) a s =. Answer: x(s) = s.5 and y(s) =.85s Winer, 7 Mah 6 Exam Problem 6 (Venice Beach) Soluion
4 Mah 6 / Final (December 9, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 8 8. [ poins] In his problem, we consider he parameric curve given by x = f() y = g() for all, where f and g are wicediffereniable funcions. Some values of f and g and heir derivaives are given in he ables below f() g() f ()  3 g () a. [poin] Inhespaceprovided, wrieaninegralhagiveshearclenghofheparameric curve from = o = 5. 5 (f ()) +(g ()) d Arc lengh = b. [3 poins] Use a midpoin sum wih as many subdivisions as possible o esimae your inegral from par a. Wrie ou all he erms in your sum, and do no simplify. Soluion: The midpoin sum is ( +( ) ). c. [3 poins] Find he Caresian equaion for he angen line o he parameric curve in he xyplane a =. Soluion: In poinslope form, he angen line is given by y 5 = (x+3). d. [poins] Considerheangenlinesoheparamericcurveahevalues =,,3,4,5. Are any of hese lines perpendicular o each oher? If so, lis any wo values for which he angen lines are perpendicular. If no, wrie NO. Soluion: The angen lines corresponding o = and = 4 are perpendicular. e. [poins] Asrangesfromo5, hecorrespondingparofheparamericcurveinersecs he line y = x exacly once. Which inerval conains he value for which he curve inersecs he line y = x? Circle your answer. You do no need o show any work. (,) (,3) (3,4) (4,5) Fall, 6 Mah 6 Exam 3 Problem 8 Soluion
5 Mah 6 / Miderm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 9. [4 poins] Fearing ha she is losing auhoriy over her robo ward, Dr. Duran has insalled a racking chip in Seph s mainframe. The chip gives Seph s locaion separaely in x and ycoordinaes, where he unis of he axes are miles, Dr. Duran s office corresponds o he origin (x,y) = (,), he posiive yaxis poins norh, and he posiive xaxis poins eas. On nigh, Dr. Duran noiced unusual levels of aciviy; hours afer midnigh, Seph began moving according o he parameric equaions x = f() y = g(), where f() and g() are ploed below for 5. f() g() (,.5) (4,.7) (,) (5,.7) a. [ poins] When was Seph farhes norh and souh on nigh? Wrie your answers in he blanks provided. You do no need o show your work. Soluion: Norh: 4 a.m. Souh: a.m. b. [3 poins] Wha was Seph s speed a = 4.9 on nigh? You may use he fac ha f (4.9) =. Include unis. Soluion: Her speed is ( ) +(.4) = 6.76 mi/hr. c. [ poins] Wha direcion was Seph moving a = on nigh? Circle only one answer. NORTH AND EAST EAST ONLY SOUTH AND EAST NORTH AND WEST WEST ONLY SOUTH AND WEST Fall, 6 Mah 6 Exam Problem (racking chip) Soluion
6 Mah 6 / Miderm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page (coninued). Recall ha on nigh, Seph s posiion was given by he parameric equaions x = f() y = g(), where f() and g() are ploed below for 5. As before, Dr. Duran s office is a he origin (x,y) = (,), he posiive yaxis poins norh, and he posiive xaxis poins eas. f() g() (,.5) (4,.7) (,) (5,.7) d. [3 poins] How far away was Seph from Dr. Duran s office a = on nigh? Soluion: Seph was +.5 = 3.5 mi away. On nigh, Seph s movemens were even sranger, following he parameric equaions x = f(s)ds y = g(s) ds. e. [ poins] Wha direcion was Seph moving a = on nigh? Circle only one answer. NORTH AND EAST EAST ONLY SOUTH AND EAST NORTH AND WEST WEST ONLY SOUTH AND WEST f. [ poins] Did Seph come o a sop beween midnigh and 5 a.m. on nigh? If so, a wha ime(s) did she come o a sop? Soluion: Yes; she came o a sop a 3 a.m. Fall, 6 Mah 6 Exam Problem (racking chip) Soluion
7 Mah 6 / Exam (March, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 6 3. [3 poins] Oguk s playful son, Oghan, is running on he xyplane. His posiion seconds afer he begins running is x = y = sin()+. Assume x and y are in meers. a. [3 poins] Does Oghan pass hough he origin? Briefly jusify. Soluion: x = when = so when =. For his value of, y = sin()+. So he didn pass hrough he origin. b. [4 poins] How fas is Oghan running a = 5? Give your answer in exac form (i.e. no decimal approximaions). Include unis. Soluion: ( 5 ) +(cos(5)) m s c. [6 poins] Find an equaion, in xycoordinaes, of he angen line o his pah a =. Soluion: The slope of he angen line is given by m = dy/d dx/d = cos() = cos() The equaion of he angen line is y sin() = cos()(x ) or equivalenly y = cos()x+sin() Winer, 6 Mah 6 Exam Problem 3 (O ghan) Soluion