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—fﬁn) :sin(94g—Zn+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§—gn) =sin(47?9—g.n—2§—§~n):sin(m—2g—g~n) : sin(mr) cos(2 ‘ 2’1 . n) — cos(mr) sin(2 ~ g—Z  n) [Subtraction formula for the sine] 94
:0cos(24§—: n) —(i1)sin(2.%§ ~71) =:l:sin(2g—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‘” ...
View
Full Document
 Spring '10
 Ban
 Calculus

Click to edit the document details