Practice Midterm II - 2

# Practice Midterm II - 2 - 15:05 IFAX...

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Unformatted text preview: 06/11/2004 15:05 IFAX [email protected] —> Connie Ooster' I001/009 KM; McMaster University Mathematics 1A3/1N3E Prof: C Brady Duration of examination: 75 minutes June 3, 2004 Name: Student Number: Tutorial # or Instructor: This examination is 10 pages. Part A is simply ﬁll in the blank. One mark for each answer. Part B is complete answer questions. For full marks show all your work with clear and concise explanations. Be sure to have proper mathematical form. NO CALCULATORS are allowed. There is a blank page at the back for any rough work. 06/11/2004 15:05 IFAX [email protected] —> Connie Ooster I002/009 Math 1A3/1N3 Name: Test #2 page 2 of 10 Student Number: Part A: Fill in the blanks (1 MARK EACH) 1. in ln|x| = 1' Y 2. D23in(m) = “ §(\/\. K 3. If f”(c) < 0 then when 9:: c the function is Camccu/f own 4- ﬁCEXw): «C/cogm—Fmg/CK) . [911581 5. If the tangent lines of two relations are perpendicular at all points of intersection we call the functions 0 r ‘i’k%ma«0 . 6. If a function, f, is increasing on (a, b) then for any c e (a, b) M) 2 o . :: € 7. StateﬂaERiffi cos(:c)=0. 0" K’Tl' / I( Z. »—S:;~(a)-—o 2a: ifmS—l , . \$2__3 ifx> _1.Statealla:swheref(a:)1s continuous. R V , . 8. Consider f(:e) = 9. With reference to f(a:) in question #8, state all w's where f (x) is differentiable. (‘00,‘0 We! -—f w 10. If f is continuous everywhere and 3423' f (z) = 0 and 72%| f (x) < 0 thenwecansay f(a.) isa 6% féﬂgé M . 06/11/2004 15:05 IFAX [email protected] » Connie Ooster I003/009 Math 1A3/1N3 Name: Test #2 Student Number: pg 3 of 10 Part B. Answer all questions in with full and clear reasoning for full marks 1. [9 marks] Find % for the following: 2/ 3/ ‘5 z a)y=‘3/m2+2‘lm3 : X +QX 911—; Z—XJ/g-f-BXK \/ dz 3 = 3371—33‘ 0/ b) \/':E+—zy=1+a:2y2 (was; + w 1 , éc¥+y5"‘.(/+y’) = 2w 12" 2W / “ 1/ (“Wig “/I) = LFYVI + SIX yy / / byz— ; 7- 7‘ / (“774+ y (“7) ' 4“” ‘L 9% 7y 4/1 I 2 __ CK+y> / 06/11/2004 15:05 IFAX [email protected] —> Connie Ooster I004/009 Math 1A3/1N3 Name: Test #2 Student Number: pg 4 of 10 2. [6 marks] a) State the Mean Value Theorem. fag Mme/J w! MM)/ “a“; “514% Moe/é CEGIA) 5% b) Verify that the following function satisﬁes the hypotheses of the Mean Value Theorem on the given interval, then ﬁnd all numbers, 0, that satisfy the conclusion of the Mean Value Theorem. f (as) = x3 + a: — 1 on the interval [0, 2] 06/11/2004 15:06 IFAX scanner®mail.math.mcmaster.ca Math 1A3/1N3 Name: Test #2 Student Number: pg 5 of 10 a Connie Ooster arks] Find the following limits. a) lim 5%»ZJW exzco 3 X-yw z x1 ru+ g~ 2x b) lim £1726“: : .-—/ = ya-a: -X / : \$1 2x Xé‘cao -/ pm 3 We x me x [KM/L 31+, —H wg’M 5f. 1‘ r”.- xaoo .€K+X ” xaco c"+l ' {—500 ﬂy. . A; = C 3 - A“ ; mo 43 3 X-Poo : e I005/009 06/11/2004 15:06 IFAX [email protected] —> Connie Oostel" I006/009 / e Math 1A3/1N3 Name: Test #2 Student Number: pg 6 of 10 4. [8 marks] a) Using L'ngital, ﬁnd the following: D“!!! 911; ii) lim coax—1i AW» _S;"nX: O :c—>0 .._— 1" z—m 9’ X9 0 A Six 1'! A €01," V / xao X W¢ b) Using the results of part a) prove the following from ﬁrst principles. dieing: = cosx. 11 06/11/2004 15:06 IFAX [email protected] » Connie Ooster I007/009 Math 1A3/1N3 Name: Test #2 Student Number: pg 7 of 10 5. gmrks] Find Wteuf the following functions. a) y = 52% 12/ , [a A "u'l ’Zﬂ" X+ V:_CO L ‘ U0 Us 01% ﬁll/4 +60 Y'Z — K4460 [-2246 / x—» — ~ +2 —'-—"— - S UHF x—ZS :‘+Ox+l'2 \ X“: x 2‘ M: Ya 2" - M -(‘K+Z>: xaco TC"; :22: = f. /(o 6. For the following functions ﬁnd the domain, intercepts, symmetry, asymptotes, intervals of increasing/decreasing, local max/min,concavity and points of inﬂection and sketch the graph of the function. 06/11/2004 15:06 IFAX [email protected] —> Connie Ooster I008/009 Math 1A3/1N3 Name: Test #2 Student Number: pg 8 of 10 ____________________________._.____ 63) continued... r 3 -: i 3 ” xé/WL X - I _ ‘ --_l .— X ._ i L/ Ys§oo X31— 1 .. K; 1-03 ' ’ 3) Ho: I2>< 3H: .25? 2 3 1 cw?” : zQKH/azx—zéxﬂ Imam” (x3+ ()3 (163+ ()3 06/11/2004 15:07 IFAX scanner- mal mat mcmaster ca 00:0 (AGO/*0 ("I O) (o w (“/ﬁxx» 01% w ...
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