Calculus: Analytic Geometry and Calculus, with Vectors

40 Analytic geometry in two dimensions

37 With the aid of the quadratic formula, show that the point (x,y) lies on
the graph of the equation x2 + xy + y2 = 3 if and only if -2 5 x 5 2 and y

y is one of the two numbers

Figure 1.592

-x - 3(4 - x2) -x -{- 1/3(4 - x2)

2 2

which are equal only when x = -2 and when x = 2.

Formulate and prove an analogous statement in which

the roles of x and y are interchanged. Find the coordi-

nates of the eight points in which the graph intersects the

lines having the equations x = -2, x = 2, y = -2,

y = 2, y = x, and y = -x. Remark: The graph is an

oval which is shown in Figure 1.592 and which is, as Chapter 6 will show us, an

ellipse.

38 Sketch a graph of y = sin x over the interval 0 5 x <= 2ir and then, with

the aid of simple arithmetic facts like 02 = 0, (0.4)2 = 0.16, (0.8)2 = 0.64, use

the result to obtain a graph of y = sine x.

39 Sketch graphs of y = cos 2x and y = (1 - cos 2x)/2. Remark: Because

of the trigonometric identity

1 - cos 2x

sine x = 2

the answers to this and the preceding problem are the same.

40 Perhaps the classic guns-and-butter interpretation of the formula

x+y=M

should not be overlooked. It is supposed that a chief has control of M man-

hours of human energy. The chief may preempt x man-hours to provide pressure

and power to keep his subjects in line and to preserve or extend his authority.

Then, even when x = M and y = 0, there remain y man-hours part of which

may be used for production of food, shelter, education, and sundries. Sketch

a graph which shows how x and y are related. Hint: Do not ignore the basic

idea that x > 0 and y 0.

41 Sketch graphs of the three equations = lx-, y = x, y2 = x2 and

make some relevant comments.

42 Let a > 0. Show that the equation

(1)

holds if and only if 0 5 x 5 a and

(2) y = (1/a - \/ x)2 = a -f- x - 2 1'-a-x.

Without making onerous calculations, sketch rough graphs of (1), (2),

(3)

and

(4)

y = a + x + 2 1/ax,

(y-a-x)2=4ax.

Remark: Chapter 6 will reveal the fact that the graph of (4) is a parabola. The

graphs of (1), (2), and (3) are parts (subsets) of the parabola.