Calculus: Analytic Geometry and Calculus, with Vectors

1.5 Equations, statements, and graphs 39

calculus, most of our work with equations and graphs will be done with

the aid of the calculus.

Problems 1.59

Sketch graphs of the following equations and inequalities:

1 yx 2 x0 3 y0

4 y=(x-1)2 5 y=(x+1)2 6 y=x3

7 y=1+x 8 y=1+x2 9xy=-1

10 y+x^111 y(1+x)2^1 12 y1+x2^1

x 1

(^13) y +x2 14 y=x+ 1 15 y=x -I
16 y=IxI 17 y=Ix-2I 18 y= (x+IxI)
19 0<x<1 20 0<y<1 21 0<x+y<1
22 Ixl<1 23 Ix-2I< (^24) <x<
25 y < x2 (^26) lyi < Ixi (^27) Iyi < x2
28 lxi + IyI (^29) IxI + IYI < 1 30 Ixi + IYI > 1
31 With Figure 1.58 out of sight, sketch graphs of y= sin x and y = cos x.
If unsuccessful, glance at Figure 1.58 and try again.
32 Figure 1.591, which features half of an equilateral triangle each side of
which has length 2, shows that
r 1
sin 6 = 2' cos 6 = 2 sin 3 = (^2) cos (^3) = i
AI %\
Cultivate the ability to sketch this figure quickly. Use the I \
information obtained from it to locate points on the graphs a
or y = sin x ana y = cos x. sketch a right triangle in 1
which each leg has unit length and obtain more points on Figure 1.591
the graphs. Finally, sketch graphs of y = sin x and
y = cos x again. Remark: We need familiarity with our graphs, and we need
confidence in them.
33 Sketch graphs of
(a) y = 3 sin x (b) y = sin 2x (c) y = sin (x + 2)
Remark: Graphs of equations of the form y = E sin (wx + a) are called sinusoids,
and we hear very often that E is the amplitude, w is the angular frequency, and
a is the phase angle of the sinusoid.
34 Where are the points (x,y) for which 0 5 x S 2ir and sin x 5 y S cos x?
35 Supposing that h 0 0, find the slope m of the secant line (or chord)
containing the two points of the graph of the equation y = x2 having x coordi-
nates xl and xl + It. 11ns.: 2xi + h.
(^36) It is sometimes quite important to have correct information about the
graphs of y = x8 and y = xK Sketch the graphs over the interval -2 5 x 5 2.