Calculus: Analytic Geometry and Calculus, with Vectors

5.3 Second derivatives, convexity, andfexpoints 309

and hence that the line having the equation y = x is an asymptote of G.

graph of (1) is shown on a small scale in Figure 5.391.

The

Y

I I /

2

-2 -1 / 1 2

00,

1 -21

x

Figure 5.391

13 Determine the natures of the graphs of the equations

(a) y4 = x(1 + x2)

(c) y4 = x2(x2 - 1)

x

(e) y2 = 1 + X2

1

(8) y = (x+ 3) (x - 4)

14 Sketch graphs of y

(b) y4 = x(x2 - 1)

(d) y4 = x(1 - x)

Y2 2 x2

=1.+x2

(h) y2- (x + 3) (x - 4)

= x, y = sin x, and then y = x + sin x. Then make
repairs in the last graph that may be necessary to make it agree with formulas
for the slope and the derivative of the slope.
15 Persons interested in themselves and the surrounding world should not
neglect opportunities to learn about the honorable Gauss (or normal) probability
density function '' defined by the formula

(1)

(x-M)'

I(x) = 1 e jr^0

\/21r o

in which o- (sigma) and M are constants for which a > 0. We should know that
e° = 1, and we can cheerfully accept the facts that

(2) e = 2.71828 , e-34 = 0.60653

We want to determine the manner in which the graph of y = t(x) depends upon
the constants M (which is called the mean of 4)) and v (which is called the stand-
ard deviation of 4)). Show that

1

(x-M)2

(3) '(x) =, as-1 (x - M)e 2a'.