Calculus: Analytic Geometry and Calculus, with Vectors

38

Figure 1.58

Analytic geometry in two dimensions

y = sin x and y = cos x. The graphs are shown in Figure 1.58. We

must always know that, except at the points of tangency, the graphs lie

between the lines having the equations y = -1 and y = 1. Moreover,

7 is a little bit greater than 3, and this must be fully recognized when the

graphs are sketched. When we want to sketch the graphs, the first step

is to draw guide lines one unit above and one unit below the x axis. The

next step is to hop three units and a bit more to the right of the origin to

mark a, and make another such hop to mark 2a. We must be able to do

this and sketch reasonably accurate graphs of y = sin x and y = cos x

in a few seconds, and we must be able to look at the graphs and see

answers to trigonometric questions just as we look at dogs and see answers

to questions about canine structure. We cannot tolerate doubts about

the assertions sin 0 = 0, cos 0 = 1, sin a/2 = 1, cos 7r/2 = 0, and dogs

have two ears. The table on the back cover of this book can be used to

produce very accurate graphs, but this is seldom necessary.

Finally, we are never too young to be informed that substantial parts

of scientific lives are devoted to learning about and using equations akin

toy = e and y = log x. Graphs of these equations are shown in Figures

1.581 and 1.582. The exponentials and logarithms have base e, and e is a

number that we shall encounter very often. Here again the tables on the

back cover of this book can be used. While we should have basic informa-

tion about graphs before we start our study of functions, limits, and the

Figure 1.581 Figure 1.582

61Y

4 2

y

a

2

1

-1

y=log x

1 2 e 3 4 x

-2

-2 -1 0 1 2 x