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soln 1

soln 1 - MATH 137 ‘ Assignment 1 Solutions 1 If —2 g a...

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Unformatted text preview: MATH 137 ‘ Assignment 1 Solutions 1. If —2 g a: 3 7r / 2, use the triangle inequality to show that I221:3 — 3:2 + 31: — sinasl g 27. ! 83> “m waéj‘mfk w e ; Wt Mam g w‘xmwﬁx w )MMQ CM' W +» ism; w J§ mm “ﬁ‘” mmwwmmwm 2“ 2. The Heaviside function H is the piecewise deﬁned function given by the formula: 2 4’ H(m)_ 1 ifoO, _ 0 ifx<0. (a) Prove that = 33H — \$H(—x) for every real number x. Hint: Consider what happens when so = 0, 3: > 0, and a: < 0. (b) Sketch the graph y = H(x4 + x3 — 7x2 — a: + 6). Hint: You need to factor the polynomial ﬁrst. For that you can start by looking for roots, and then use the factor theorem you learned in high school. (0) Sketch the graph y = (cos + 7r/2) — H(x — 7r/2)). (a) :56 «cam M<r)-~><+H~x): 04-0.1: on M rgﬁwojam «HM-w Mam: w! who :1: ix) (if- X40, 7mm MHW"X HM): ’X'O-‘x-h-x =~ Hi. I“ 0“ 6mm) w; ism“ x M) ~=» m . ’ KWe “£ng “z 25“" war é Q tth m a. < O . * was watt ﬁw i Wait “‘E'xttaata__ “ﬂaw a ‘ 5:": t- MM? W M my {fat x3 wma‘wxré if" CMﬁXXr‘!) (Wiliiklj FA+-w 3% a r mwaWtﬂ MO I; , 2 . lg M H 04?qu “a A it Q} '2: E ﬁttﬁ‘ WM /- ‘ dismiss hwy w m it M t 7 My t 3) ‘ (\:0% '14 “‘5 “3 \wig x v iwwiﬁmﬂﬂx-‘spéwggﬁ {3 5,?ng @U~“&%M5“*‘ H&d~ﬁ£§FEEfEW#QF?EE “PA, wBAx~‘ («+33 (X+l><¥")(>€”&> <0 Ahwa Wﬁ>0 &J%wrﬂ%Tmﬂ%dw MK 4(1) § W Hfﬁﬂzw’é ’ / I, w W v > ‘5‘ \$0M wl< yahwf [Mme] (Mg/(5+13M INK QT; \ , “71\$ kﬁw \$2M ?’ {J W“ « =3 as r w ‘34»? ,4 {Sim mix-<92, mm»? gméwaméﬁﬁww‘ﬂx {Mg '3‘) Q a? EMM%N%%§?Q%%€)\$Q [\$3 I Tu M4MMWW % H2 Mg, ha L3", ; Wag, x I. may 79%, ' \$4 >91“? mm 41m. Mgﬁmowﬁ Visa 2 ;L a”; g: HOD) “*1 €D{i”l> :0 (12‘ x :51, M {w W+Izo =5 w]: :«ww 9k 52 a~ m ‘5“ w»:on if; ’x (“V—5:) Wm ﬁiwiza % aéwféo '1 2 3» W w WWW» 3‘ ’3 3% g ““ W <H*WE.,>* m’w'ﬁgﬂ :2: , ) w ﬁg“; 0%? 3m M%>g~ ﬁg WEN) ‘ \$1) W“[H 5mg) “‘ Frivvsz (Lam zomﬂ {mg} 3.3, W 3. In the Qty—plane, sketch the region of points (3:, 3/) that satisfy 2|xl + 3|y| g 6. G Wﬁ MM X" 404/52. ‘W at? x at: 5w (:36 t 3 3 arm a a,“ I M, g a PM, «30,;26)‘m kw mm #15th 8o gammy L, Magma; a. W 331% Q t CWW 4*? may 4. Letf(a:)=ln(a:+\/:c2+1). 7‘ (a) Explain brieﬂy why the domain of f is all of R. (b) Prove that f(:1:) + f (—x) = 0, and thereby deduce that f is an odd function. (0) What is the range of f? (d) Ifs < t, explain why f(s) < f(t). (e) A function with the property in (d) is called an increasing function. Increasing func- tions are one-to—one, and thereby have an inverse function. By solving the equation y = ln(:c + x/ 11:2 + 1) for :1: in terms of y, ﬁnd a formula for the inverse function of f. (“J %m KX’LMEO) l; {1-H 3/: x, ‘ M if“ , wt >6 reafﬁrm” ’6/a(x+ml Mk r \ - WM 5 V.“ 'r‘ (a) Mm Ammm, W A}; 37; \$3 82)?“ lg f4: My m, d? 4C a O) ' M5. \$0 ’X MW G “a c333 #1, W-m 4 ﬁn? w. Wm ACHW) Wéw %WW\ ’. (ya ; ¥ ﬂwp ~ Ax g. AJQ g 949 ‘ ) 3P , W ,, 31"? ~§ 4 gm”.ng m k _ j é "" ‘ ’ f . 71% 51 ‘ r2 ""‘ _ e r v ' " ‘ -- kl masz f W I . Q/W My ) A4 5 ix. 7% 3 E: M m\ w» 50 {m 4<0 W W m {mg-{plow k _\ 3 5. Let f(x) = 3+2€_m. _ ’Ot (a) As a: rises from —00 to +00, the function 3 + 26”” decreases from +00 towards the asymptotic value of 3. What does f do as x rises from —00 to +00? (b) Using your answer from part (a), sketch the graph of f, and indicate the y-intercept. (c) Find a formula for the inverse function of f, and indicate the domain of this inverse function. r 57, 6. Suppose f is an even function and g(:c) is an odd function. Determine if eadgh of the following functions is always even or always odd. Give a proof if it is, and give a counterex— ample if it is not. (a) The sum f(\$) + 9(23) (b) The product f (c) The composition (f o : f(g(a:)) (d) The composition (g o = g(f(33)) gm: x gal )gémﬂ some, KM. Mm ...
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soln 1 - MATH 137 ‘ Assignment 1 Solutions 1 If —2 g a...

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