M408C_Makeup1_Solutions

M408C_Makeup1_Solutions - “A[1 17,6,1 Exam 1 M h 408C...

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Unformatted text preview: “A [1 17,6 (,1. Exam 1 M h 408C Name Fall 2009 TA Discussion Time: TTH You must show sufficient work in order to receive full credit for a problem. Do your work on the paper provided. Write your name on this sheet and turn it in with your work. Please write legibly and label the problems clearly. Circle your answers when appropriate. No calculators allowed. Discussing any part of this exam with a classmate who has not yet taken the exam is considered scholastic dishonesty. 1. Calculate the limits below, if they exist. In the case that a limit doesn’t exist, indicate whether the limit approaches 00, —00, or neither. You must show sufficient work to get credit for a correct answer. 1 — 23: 10 ' t l’ (——————) (a) ( pom S) 221:1; 3 — 5:1: —— 2x2 (b) (10 points) lim t—t—fl t—>1 t—1 4 1 (C) (10 pom) .1311.” + 33) (a? — x + 1) 2. (10 points) Determine whether the function f below is continuous at as = 1. Use the definition of continuity to justify your answer. 7r cos< ), 2:21 Lt—l—l f (15) = 1r — 1 < 1 m I22 — 1r 1 3. (10 points) Use the definition of derivative to calculate f’ (x) for f (5r) : 3$ _ 1. (You must use the definition, not the differentiation rules, in order to get credit for this problem.) 4. Let y = sin6cos0. (a)(10 points) Find all values of 0 in the interval [0,27T] where the tangent to the curve is horizontal. d2 (b) (10 points) Find fit Simplify your answer to a reasonable degree. 5. (10 points) An object moves along a straight line, its position at time t given by t . . . . 803) = for 0 S t, where 8 is measured 1n meters and t 1n seconds. Determine Vfl+8 all times t (if any exist), when the velocity of the object is i ft / sec. [MORE ON BACK] 4 tan(a:2) + 1 6. (10 points) Find the equation of the tangent to the curve 3; = + 2 x $20. when mlw 7. (10 points) Find y’ if (:1: + 3/) degree. : 312:5 + 2. Simplify your answer to a reasonable Bonus (5 points): Calculate the limit below, fully justifying all your steps, or explain Clearly why it does not exist. lim ; "H0 sec (i) + a: WW 4331‘ : L” "9% Xa 1,1 3-cx—9><" X-Wz— (I~ax)(3+7<) .5 i l :1 = — / 3(a); 3+x 3+”; 7' (L) M Q'T'V't4—3 : Q-Ut4g vQ+\/t43 )(q) t —) X-?\ e; *4 91+ v5"; = M L} ~1t+33 )W 1- E W 3 \ .. = I t7| a+¥§3 an”? =~—‘; (C3 JAM. (sz) ( i _. L) ‘ PM" (Xz+x)(" X‘?“ X: 1+! ‘Y‘V" : JIM» Y(7&+0(‘+¥+‘!-¥") _ Q 4%+‘f~x x—‘a—I y‘lx-H) y-a—I )c —_— -’-§+‘I—(-I)7‘ _ I x4“ {-317 M £(K) .: )l/Vv— .35: = M- X" .._._. Y‘?’ Y-fil— Ix~l’ X9! --—[x4) 30 M 1:55) M w MES/f M $0 £ ILS We} Y-fll MMW J‘Xz,‘ «IA—90 ,a‘ g Q! _.L____ .. __.L_ 13,40 3km)" 32M ($(n+h)-I)(3x-‘) ’33 Md ”1‘40 3x ~I - (SIKHn) a) 4x (3(K+k)~I)/3x~i) = Xv» -3JA .. 3 Ik—wo ——“——"'———- ._. M m ’2‘ (“M“MVSX") Jae (3(K+k)‘l)(3X-l) .. - 3 " f1 (3m 4'. 33WQC¢A9 (“$02, = 3iw9(~S/\8~Q) 1— (>059qu d9 : -S~k?9+ws?9 03 .2 z z s: 34/“ :CpQ ow 7 9 a #m’Q: =3 Masjl :79.- 1179, am, “74,704 “’3 a”: _ :13" — 249(‘W79 + c.0293 : — awe m9 «Laws? C-WQ) .-.- ~45~L9m9 ...
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