Comments On The Calculus Challenge Exam – June 2014

Transcript

Comments  on  the  Calculus  Challenge  Exam  –  June  2014     Question  #1:     Part  a)  this  part  was  straightforward  and  most  students  solved  it  correctly;  however   some  of    the  explanations  and  presentations  were  poor.   Part  b)  this  was  the  most  difficult  of  the  3  limits.  The  performance  was  not  as  good   as  part  a)  with  about  one  third  of  the  students  solving  it  correctly.   Part  c)  There  were  two  ways  to  solve  this  questions  and  most  students  got  full   points  on  this  question.     Question  #2):     This  question  is  routine  and  most  students  got  full  points  for  all  3  parts.  About  one-­‐ sixth  of  the  student  had  difficulty  taking  the  derivative  of  the  fourth  root  of  x  cubed.     Question  #3):     Most  students  failed  to  write  “if  this  limit  exists”  when  writing  the  definition  of  the   derivative.  Some  students,  in  part  b),  left  the  words,  “lim  as  h  tends  to  0”  even  after   taking  the  limit.     Question  #4):     This  question  was  done  well.  The  most  common  error  was  attempting  to  find  the   tangent  line  at  (9,4)  instead  of  the  curve  y=f(x).  Many  students  who  attempted  to   fine  the  tangent  line  used  3sqrt(x)  instead  of  3sqrt(9)  =  9  for  the  slope.     Question  #5):     Most  students  solved  this  question.  Some  student  made  mistakes  in  calculating  the   integral  and  some  forget  to  use  the  intersection  points  of  the  curves  for  the   endpoints  of  integration  (instead  using  the  x-­‐intercepts).     Question  #6):     The  majority  of  students  did  not  seem  to  understand  the  difference  between  the   existence  of  a  limit  of  a  function,  f(x),  at  a  point,  a,  and  the  continuity  of  f(x)  at  a.  This   was  very  evident  in  part  b)  where  one  needs  to  understand  this  difference  in  order   to  justify  the  steps.  Most  students  simply  equated  the  two  defining  expressions  for   g(x)  and  solved  for  a  without  making  any  reference  to  the  definition  of  continuity.     Question  #7):     Many  students  used  the  function  f(x)  =  x^3-­‐x  rather  than  f(x)=x^3-­‐x-­‐1.     Question  #8):     Overall  this  was  well  done.  Many  students  said  that  any  point  where  the  second   derivative  of  f(x)  is  zero  corresponds  to  an  inflection  point.     Question  #9):     Part  a)  Most  students  solved  this.   Part  b)  In  part  b),  many  students  received  full  marks  for  computing  (V-­‐V_est)/V   directly.  However,  this  does  not  show  that  these  students  understand  the  concept  of   linear  approximation.  Some  students  considered  dx=0.005  while  dx/x  =  0.005.  They   need  to  know  the  definition  of  error.     Question  #10):     Part  a)  Almost  everyone  got  2/2.   Part  b)  Many  used  only  v(t)  or  a(t)  rather  than  using  both  to  determine  where  the   object  was  speeding  up  or  slowing  down.     Question  #11)     Generally  well  done.  A  common  error  was  to  forget  the  negative  sign  for  the  slope  of   the  normal  line.     Question  #12):  Students  are  encouraged  to  carefully  read  the  question.  Many   student  did  not  calculate  how  much  longer  it  would  take  but    rather  the  total  length   of  time  it  would  take  to  cool.     Question  #13):  About  fifty  percent  of  the  students  solved  this  question  correctly.   Many  students  confused  angular  velocity  or  horizontal  velocity  with  vertical   velocity.  Please  note  that  you  do  not  have  to  compute  any  angles  in  order  to  solve   this  problem.     Question  #14):  Mostly  well  done.  The  common  mistake  was  not  including  the   domain  for  the  variable,  x,  (ie  the  distance  form  the  village  to  where  Jane  lands  her   boat).  There  were  a  few  arithmetic  mistakes.     Question  #15):  A  minority  of  students  did  this  problem  very  well.  Many  students  left   it  blank  or  made  no  progress  towards  a  solution.  There  are  several  ways  to  solve   this  problem.