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

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 12/31/2011 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

Page1 / 12

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

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online