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chapter 1 36 - 36 D CHAPTER1 FUNCTIONS AND MODELS 36(a y...

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Unformatted text preview: 36 D CHAPTER1 FUNCTIONS AND MODELS 36. (a) y : sin(\/E) (b) y = sin(m2) This function is “0‘ periodic; it oscillates This function oscillates more frequently as |x| increases. less frequently as 5'3 increases. Note also that this function is even, whereas sin as is odd. 1.5 M A H V 4.5 37. The graphing window is 95 pixels wide and we want to start with a: : O and end with :1: 2 27r. Since there are 94 “gaps” between pixels, the distance between pixels is 275310 . Thus, the art—values that the calculator actually plots are x : O + 35% - n, where n = 0, 1, 2, . . . , 93, 94. For y = sin 2x, the actual points plotted by the calculator are ($27: -n, sin(2- % 77.)) for n = 0, 1, . . . , 94, Fory = sin96at, the points plotted are (3—2' -n, sin(96 - all - 11)) forn:0,1,...,94.But sin(96-—ffi-n) :sin(94-g—Z-n+2~:—Z.n) =sin(27m+2-§—1~n) : sin(2 - g—Z -n) [by periodicity of sine], n = 0, 1, . . . . 94 So the y—values, and hence the points, plotted for y = sin 9635 are identical to those plotted for y : sin 2m. Note: Try graphing y : sin 94x. Can you see why all the y—values are zero? 38. As in Exercise 37, we know that the points being plotted for y = sin 45w are (3—1 - n, sin(45 . ‘3 - n)) for n = O, 1,...,94.But ' sin(45-§—g-n) =sin(47-?9—g.n—2-§—§~n):sin(m—2-g—g~n) : sin(mr) cos(2 ‘ 2’1 . n) — cos(mr) sin(2 ~ g—Z - n) [Subtraction formula for the sine] 94 :0-cos(24§—: -n) —(i1)sin(2.%§ ~71) =:l:sin(2-g—1— -n), n:0,1,...,94 So the y—values, and hence the points, plotted for y = Sin 4523 lie on either y : sin 21: or y = — sin 2m. 1.5 Exponential Functions 1. (a) f(a:) 2am, a>0 (b)lR (c) (0, 00) (d) See Figures 6(c), 6(b), and 6(a), respectively. 2. (a) The number 6 is the value of a such that the slope of the tangent line at m = 0 on the graph of y : am is exactly 1. (b) e 8 2.71828 ((2) flan) = e‘” ...
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