# Math 116 Practice for Exam 2

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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 wice-differeniable 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 xy-plane a =. Soluion: In poin-slope form, he angen line is given by y 5 = (x+3). d. [poins] Considerheangenlinesoheparamericcurveahe-values =,,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 y-coordinaes, where he unis of he axes are miles, Dr. Duran s office corresponds o he origin (x,y) = (,), he posiive y-axis poins norh, and he posiive x-axis 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 y-axis poins norh, and he posiive x-axis 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] O-guk s playful son, O-ghan, is running on he xy-plane. His posiion seconds afer he begins running is x = y = sin()+. Assume x and y are in meers. a. [3 poins] Does O-ghan 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 O-ghan 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 xy-coordinaes, 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

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