ACTM State Calculus Competition Saturday April 30, 2011
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1 ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward completng questons 1 through 25, attempt the te-breaker questons n sequental order (do #1 frst, followed by #2, and then #3 last). Be sure that your name s prnted on each of the tebreaker questons. Unless otherwse stated, assume all varables are real and all functons are contnuous over relevant domans. Assume all angles are n Radans. Good luck! 1. Consder the followng lmt: Gven, determne the maxmum value for to satsfy the defnton of the lmt. Cannot be determne 2. Determne the slope of the normal lne to the functon at the pont. 3. Apple In t k pr e ha h w exp e ta gr wth e 98 The stock prce can be modeled wth the equaton, where s n years snce What does ths model predct the growth rate per year s ths year (2011)? per year per year 8 8 per year 8 per year Cannot be determned 4. Fnd the oblque asymptote, f one exsts, to the followng functon. No oblque asymptote exsts.
2 ACTM State Calculus Competton Sprng 2011 Page 2 5. Determne by ant-dfferentatng twc 6. Determne the total area bounded by the equaton, the ax, the lnes and. 7. e ta e e ta Unable to evaluat 8. Evaluate the followng ndefnte ntegral: a e t tegrate
3 ACTM State Calculus Competton Sprng 2011 Page 3 9. Determne the maxmum number of nflecton ponts that a quadratc functon of the form may hav t ha e t p t 10. Fnd the average depth of a 2 foot tall water trough wth a cross sectonal area bounded by the equaton a where s n feet. 11. The radus of a crcular ol spll s growng at a constant rate of 2 km per day. At what rate per day s the area of the spll growng 4 days after t began? Assume the thckness s constant. Unable to determne from nformaton gven. 12. Fnd the maxmum volume of a rectangular open-top box made from a pece of cardboard measurng 24 nches long and 9 nches wde by cuttng out dentcal squares from the four corners and turnng up the sdes. 13. What s/are the regon(s) where the followng functon s concave up? The functon has no regons where t s concave up.
4 ACTM State Calculus Competton Sprng 2011 Page Consder the functon wth the doman. Determne the coordnate of the mnmum One of the most famous symbols n all of scence fcton s the double parabola shape found n the Star Trek seres. The equatons are 9 and. Fnd the area bounded between these two parabolas Calculate the volume of the sold of revoluton generated by the regon bounded n the frst quadrant by the curve y x x, the y axs, and the lne y, rotated around the y axs. Use the shell metho t t t t t 17. Determne the sum of the followng: The summaton has no value 18. The concentraton of a certan drug n the bloodstream can be approxmated by the equaton, where s the concentraton (n percent) and t s n hours after takng the medcn Fnd the rate of change of concentraton per hour at hours decrease per hour decrease per hour 8 decrease per hour ncrease per hour 8 decrease per hour
5 ACTM State Calculus Competton Sprng 2011 Page The hyperbolc functons are analogous to the standard trg functons, but they are defned n terms of a hyperbola nstead of a crcl One of these functons s hyperbolc tangent, whch s defned as. Calculate the frst dervatve of the functon. 20. Determne the doman of the followng functon: 21. L Hôpta r e a e app e t solve lmts nvolvng ndetermnant forms. Whch of the followng s not an ndetermnant form? 22. One day whle Cooke Monster was sttng n the top of a 25-meter tree, he dropped a cook Assumng that hs cooke falls only under the nfluence of gravty, what s the acceleraton at the nstant t hts the ground? 9 8
6 ACTM State Calculus Competton Sprng 2011 Page Calculate the followng ant-dervatve usng a trg substtuton. ta 24. Consder the functon, Evaluate the followng lmt 9 The lmt does not exst. 25. The Mean Value Theorem states that for any functon f (x) that s contnuous on [a, b] and dfferentable on (a, b), there exsts some c n the nterval (a, b) such that the secant lne connectng the ponts (a, f (a)) and (b, f (b)) s parallel to the tangent lne to f at (c, f (c)). The followng functon satsfes the Mean Value Theorem on the gven nterval. Fnd a sutable value for that satsfes the concluson of the Mean Value Theorem. the ter a
7 ACTM State Calculus Competton Sprng 2011 Page 7 ACTM State Calculus Competton Name Te Breaker Questons Aprl 30, 2011 School/Teacher Remnders: Attempt these tebreaker questons after you have fnshed all the multple-choce questons. Attempt the tebreaker questons n sequental order (Do #1 frst, followed by #2, and then #3 last). 1. Gven the functon, fnd the values of and such that a th ex t 2. Gven the formula for a general cubc equaton,. Derve a formula to fnd the x-value(s) of the relatve extrem What are the condtons requred to guarantee that there are two dstnct extrema?
8 ACTM State Calculus Competton Sprng 2011 Page 8 ACTM State Calculus Competton Name Te Breaker Questons Aprl 30, 2011 School/Teacher 3. Fnd the equaton(s) of the tangent lne(s) to the followng equaton. Use Implct Dervatves. Assume. at
9 ACTM State Calculus Competton Sprng 2011 Page 9 ACTM State Calculus Competton SOLUTIONS Aprl 30, 2011 Multple Choce Answers 1....A 2....E 3....B 4....D 5....E 6....E 7....D 8....B 9....E C C C D E A B A E C B C D A A C
10 Te Breaker Queston 1 Soluton ACTM State Calculus Competton Sprng 2011 Page Gven the functon, fnd the values of and such that a th ex t Ths s a system of equatons problem n dsgus Snce and both exst, the lmts to the left and rght must be equal. Thus, check the followng equatons: Substtute from the rght column, 8 8 Substtute from the left column, The fnal functon s, Te Breaker Queston 2 Soluton 2. Gven the formula for a general cubc equaton,. Derve a formula to fnd the relatve extrem. Fnd the dervatve of the formula and set t equal to 0. use the quadratc formula, wth a Ths formula wll gve the values for the extrem What are the condtons requred to guarantee that there are two dstnct extrema?. The dscrmnate must be postve for the quadratc equaton to have two dstnct solutons. Thus, the condton s that or
11 ACTM State Calculus Competton Sprng 2011 Page 11 Te Breaker Queston 3 Soluton 3. Fnd the equaton(s) of the tangent lne(s) to the followng equaton. Use Implct Dervatves. Assume that. at. Frst, fnd the coordnate by substtutng and solvng. Thus there are two coordnates that satsfy ths equaton,. Second, fnd the mplct dervatv Solve for,. Use the mplct dervatve to fnd the slopes, v. Now gven these ponts and slopes, fnd the y-ntercepts, v. Now pull everythng together nto the fnal equatons, These are the two tangent lnes at.
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