YORK UNIVERSITY Faculty of Science and Engineering Faculty of Liberal Arts and Professional Studies MATH 1013 3.00 B Test #1 September 28, 2012 Surname (print):------------ Given Name: Student No: -------------- Signature: -------- INSTRUCTIONS: 1. Please write your name, student number and final answers in ink. 2. This is a closed-book test, duration - 50 minutes. 3. No calculator is permitted. 4. There are 6 questions on 6 pages. Answer all questions. Fill in answers in designated spaces, and in multiple choice questions, circle the correct answer(s). Your work must justify the answer you give. Show your work on the space provided. If you need more space, use the back of a page and clearly indicate this fact on an original page each time when you use the back of a page for your work. 5. Remain seated until we collect all the test papers. 6. Do the easiest questions first, GOOD LUCK! j Question I Points I Scored I 1 12 2 10 3 5 4 9 5 9 6 10 Total: 55
Name: Student No: 1. {12 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this question, each correct answer is worth 1.5 point and each incorrect answer is worth -0.5 (negative half!) point. If the number of incorrect answers is more than three times greater than the number of correct ones, then the total mark will be zero. If you don't know the answer, don't write anything. For this question only, you do NOT need to explain your answer or show your work. I Statement I TRUE/FALSE I If a function f(x) is even and function g(x) is odd, then function (Jg)(x) is odd. rfru..e_ If a function g(x) is odd, then (! o g)(x) is odd, for any function f(x). t=aq.se_ The slope of the tangent line to the graph of y = lnx at the point (1, 0) is 1. Every polynomial function is invertible. Graph of a nonzero function may be symmetric about the x-axis. rrv-1-ul Fais.e.. F&~ If lim f(x) exists, then the both lim f(x) and lim f(x) must exist. x--7a x--7a- x--7a+ 1'v-~ If lim f(x) and lim g(x) both exist, then lim fix; exist. x--72 x-+2 x-+2 g X Fa sl Let J(x) be a function such that lf(x)l :::; 9x 2. Then lim j(x) = 0. x-+0 'l'v'~ 1,..
Name: Student No: 2. (4 + 6 pts) Find the domain of each of the following functions: 1- x 2 (a) J(x) = x2-4x+3; ~ -')C.'"Z... -= ) c"):_-~) c")t_-'3,) urtu_ u cs JtLj;_'J w1~ea/e-y' ~ ').:_- L -t- 0 9/- X-- 3 i=-- 0 ANSWER: ('c._-~} ex-~)* o ~) X-:f=- ~ ~ )C -=f:- 3. flt-vt._z ) t>qwl (-F) = { 'KE IR )(_ ~ t1!,l_ ex:+-- :s ~ -= c- eo)~; v c~) 3) v ( 31 Oo;. (b) g(x) = )3x- x2. ~ (q_) = { xe-r O~-)(_.._ >-"O 1.ANSWER 3 ')<: -- )C_ 'L -= ')C... c ~ -~). SJ) 3'X_- ')(_ ~ ~ 0 ~ rv~..e,v- 'X. ~ 0 ~ ( '3- 'X..-J ~ 0 GY' JC ~ 0 ~ '6 -)<- ~ 0 JC.~O ~X.~3 ~»1,0S'~~' 1kre&-e_} :how>. c~ ') ~ ~ ~E--IR 2._ o ~X~ 3, ~ -=- (o > 3 ]. 2
Name: Student No: 3. (5 pts) A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 58 m/sec. Express the radius of this circle as a function of the time t (in seconds) and find a composite function A o r, where A is the area of this circle as a function of the radius. r (f) -=- 1Y. t J+ ( V'l -=:: Jf V' L (!o v)(-t) = :f(v(i}) = 1t[v-C-t)]'- ='!r(st-t)'l ANSWER: 3
Name: Student No: x 2 if x < 0 4. (3 + 6 pts) Let j(x) = 2x -1 ifo::; x < 2 { -x 2 + 7 if x 2: 2 (a) Sketch the graph of j(x); "'.,. lr. 1 / ' J ~ Lt-::.---1 )Vi) I<J II I... 0 v t ' ~ l t (b) Find each of the following limits lim f(x), lim f(x), lim f(x), lim f(x), lim f(x) x~o- x~o+ x~2- x~2+ x~o and lim f(x) if it exists, or explain why it does not exist. x~2 R.iw. +c1e.-) ~ t i.1 ')C.. 2. -= o; ANSWER: -x ~ o- ")C._~ o- e:~.f ('X.-) -=: flw, ( J. ){._-f) -=- - i. ">c..~o +.,~- o+ > QL W ~ (>~) -=-.e.: u-, ( ot )(_- 1) -==- 3. )t_~ 2- }C:...'"' 2..- ) t:_w ~(1t-) -=- R.iW (-')( '2...+ i-) -=: 3; j(... ~ ~ + )t._ --'> 2 + e_:m tc>v) ZJJJE) &~ film +-c)e-) ::::0*- -1-== ~~ +('t) ~~o- ~~o+ ' ~ &~ ~~ +c K-)-=- 3 ") )(._~ 2.- 4
Name: Student No: 5. (4 + 5 pts) Use the limit laws to find each 9f the following limits if it exists, or state that it does not exist. Justify your answer. Be sure to clearly indicate the corresponding limit law each time when you use it. () l. (2+h)2-4 a 1m h-)0 h ' 4 +Lt ~ + ~ ~- 4 "' _~ ~_(4--- k) ~ ANSWER: (b) lim x2-2x + 1. X-)l x 3 - X ")::. -'J> " = e.:""' (:x: - 1 ) u Ri faa_ "k.~ -1 'Jc_(~ (")C--1 J "':::::: ~ ~ ' ~-1#1 (x-~) [1-.4] '"X..-" ANSWER: (_~'1M. ) t_ ).&.I.Vl ( lc. -t- q ) ')C.. ~., ")c.. --.>;) " [ l-.!{)!] 5
Name: Student No: 6. (5 + 5 pts) (a) Show that hm. vt+x x-t-1 X does not exist. ~VVI ",C.~--~ ){--X. ANSWER:.0_1M. ')t... ~WI )C_ ')L ")(...~-~ +!(;, 1M, "' 1 -t- ) ( ' '"X---> - -1 - }L vju'(}- ')C_ < - ~... t_v Y. ~ x... 7JAI E. )C.-":> -1 (b) Prove that lim v'xesin( ;-) = 0. x-to+ ")L ~ -I t-~---~,--~ -==- "~~-1+ t + ~~~l~ - ~0 -= 0 L.V-1 ')(... - " }C..-t --1. + Have a nice weekend! 6