(^496) Exponential and logarithmic functions
is a continuous increasing function for which L(e) = 1, the formula
(9.281) must be correct. Thus we have an "indirect" proof of (9.281).
Replacing x and h by their reciprocals shows that (9.281) implies the
more interesting formulas
\ \
(9.282) lim (1 + x
)hI5= e, lim (1 + x)h= ez.
Ihhm h IhI-/
\ h/
The most famous of these formulas, obtained by setting x = 1 and con-
sidering h to be a positive integer, is(9.283) lim 1 +
I )n- = e.
n-. m n
Since 1 + 1/n > 1, the result in (9.283) furnishes a very interesting
compromise between the exponent n whose growth tends to make the
quantity large and the term 1/n whose decrease tends to make the
quantity near 1.
Problems 9.29
1 When we know some calculus, we can quickly dispose of problems that
were troublesome. Show that the functions having values eZ and log x are
increasing.
2 With the aid of information yielded by first and second derivatives, sketch
a graph of y = ex and the tangent to the graph at the point (0,1).
3 With the aid of information yielded by first and second derivatives, sketch
a graph of y = log x and the tangent to the graph at the point (1,0).
4 Let y = xe Z. Show thaty'(x) x)e Z. -iX
Show that y'(1) = 0, y'(x) > 0 when x < 1, and y'(x) < 0 when x > 1. Show
that
y"(x) = (x - 2)e Z.
Determine when the slope of the graph is increasing and when it is decreasing.
Use your information to sketch the graph.
5 Show that(a) e-' = -2xe ' b - e'Z = eaiaZ cos x
(e)
dxel' = er+- (d) dx e-' = abet-+-, z
6 Show that(a) d 2logx d^1
dx(log x)2
x (b) log log x =x log x
(c) log sin x = cot x (d) log (1 + x') = 1
+ x2(^7) Textbooks in differential equations provide substantial information about
some of the situations in which a "population" or something else depends upon