M137.F08.assmt3 - Math 137 ASSIGNMENT 3 Fall 2008 Submit...

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Unformatted text preview: Math 137 ASSIGNMENT 3 Fall 2008 Submit all boxed problems and all extra problems by 8:20 am. October 3. All solutions must be clearly stated and fully justified. Use the format given on UV‘VACE under Content, in the file Assignment Format for Math 135 and Math 137. TEXT PROBLEMS: Section 1.6 — 16, , 59a), 60a), 63b), 64b), , 73a) Section 2.2 — 7, 9, 14, 20, 28, 30, 32 (if you are uncertain about the limit concept, do these problems.) Section 2.3 — 2, 10, , , 25,, Section 2.6 — 9, 11,, 22, 34, 36,, Appendix D — 42, 46,, 56, 61, 69, , , 76, 85, EXTRA PROBLEMS: E31 a) Use graphical operations on the basic graphs y : arctan a: , y = arcsina; , y = fl to sketch graphs of the given functions. (1) = arctan — (ii) 9(a)) = 2a1‘csin (a: + (iii) h(x) : v.2: — 2 b) For the functions g(:L‘) and h(a:) in part a), find g‘1(:r) and h‘1(a:) and their domains, and sketch y = 9-103) and y = h‘l(a:) on your graphs in a) (ii) and a) (iii). 0) Evaluate each expression: (i) sin(arcsin(%)) (ii) arcsin(sin(5%)) (iii) arctan(tan(3%)) E32 a) Find the domain of each function, the ranges for f and h, and Show that —1 S g(a:) S 1 for all to. Use inequalities which express what you know about the component functions. (1) f (a?) = GHWM (ii) 9(96) b) Sketch a qualitative graph of [HINTz See Example 1(iii) on pages 25-26 of your Course Notes] 6””2 cosx (iii) 11(2)) 2 arctan(e‘1) E33 a) Use the diagram (right) to find 1 f (2:) = sin(arctan State the domain and range of f. 3" b) Use your result from a) to show that lim f(:c) = 1 and lim f(:r:) = —1. \ xd—oo 1'—'OO (HINT: Recall that x/x— =| a: |, so for :I:< 0, WE: —:1:.) E34 Find the vertical and horizontal asymptotes of each function. 5” Wyn—21 b) 9($)=% c) hurlgiifi E36" The horses on a Carousel (Merry-go—Round) move up and down vertically according 2 to h(t) = 0.7sin(§t) + 1, where h(t) n1 is the height of the horses’ backs above the platform, as viewed from your location, and t is time in seconds. a) What is the period of the vertical motion? b) You want to photograph your nephew, who is riding one of the Carousel horses, Your view is partially obstructed by a safety fence, so you can only take the picture if 1 5 h S 1.5 m. Approximate the first two possible time intervals for this photo. c) The edge of the circular platform is moving at 1.2 m/s. If the horses go through 5 complete periods vertically during one revolution, what is the radius r of the Carousel? NOTE: When writing solutions involving limits, it is important to referenee any theorems used. Convenient abbreviations are: limit sum rule (LSR); limit product rule (LPR); limit quotient rule (LQR): limit composite rule (LCR). MAPLE PROBLEM: (Optional: If suggested by your instructor, submit this problem directly to him/her in class on October 3. Do not submit to the drop box.) a) Use Maple to plot the graphs of each function for —37r S m s 371' (i) sin(arcsin(:r)) (ii) arcsin(sin(:r)) b) One of the graphs in a) is correct and the other is not. By examining the graphs carefully, explain which is which. Comment: Computer plotting routines are not infallible! ...
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