(^310) Functions, graphs, and numbers
Figure 5.392
Tell why it is true that, as Figure 5.392 indicates, (P is increasing over the inter-
val x <= M, cb is decreasing over the interval x M, 'I has a maximum at M,
and '(M) = 1/'v. Show that
(x-M)2
(4) CD "(x) _ 0.5[(x - M)2 - Q2Je 2,2
Tell why it is true that, as the figure indicates, cb'(x) is increasing and the graph
is bending upward when x < M - v and when x > M + v, whereas V(x) is
decreasing and the graph is bending downward when M- a< x < M +o
Finally, show that, as the figure indicates,
(5) (P(M - o) = -1)(M + v) =
0.60653
= 0.60653 (M).
1/2ir or
Remark: The index will tell where this and other bits of information about Gauss
probability functions are concealed. We shall learn that
(6)
I0ofi(x) dx = 1,
and budding scientists are never too young to start hearing that, in appropriate
circumstances, the number
(7) fab D(x) dx
is taken to be the probability that a number x lies between a and b.
(^16) Sketch rough graphs of y = cos x, y = 2 cos x, y = cos 2x, and then
(1) f(x) = 2 cos x - cos 2x
over the interval 0 5 x S i. Find the maxima and minima of f and the flex-
points of its graph. Make the results agree.
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(lu)
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