Calculus: Analytic Geometry and Calculus, with Vectors

9.2 Derivatives and integrals of exponentials and logarithms^499

Note that considerations very similar to those in Problem 8 establish validity
of the formula

(6) e2 =

Jim

fin' x = 1 + x +xX2-{ X3 X

x

n-,kO2. 3i .-I-4.^4 -I-...

when x < 0. For some purposes, (4) and (5) and their extensions are more useful
than (6).
10 Have another look at the formula

e2=1+x-}-2i-I-3 +4i+

and remember it. Then write the formula and use it to obtain an approximation
to e3 which agrees with the idea that (e½)2 = e.
11 Prove one of the inequalities
e2 > 1 + x, log (1 + x) <- x, log x 5 x - 1

and show that each implies the other two. Hint: If no better idea appears,
find the minimum value of e2 - 1 - x.
12 Prove that if x > 0 and k is a positive integer, then

0<et<(k

ixk=(k

zl)l

Use this result to show that

13

(1)

(2)

lim

x k

= 0.

We can become accustomed to the formulas

1 -to=1+t+t2+ ... +t"-1

1-t

1 1 t = +t+t2+ ... +tn-1+1

if we see them often enough. Prove that if -1 5 x < 1, then integration from
0 to x gives the formula

2 3 n
(3) log l-x=x++3+..+n+Rn
where

(4) R. = f oy 1 totdt.

In case 0 5 x < 1, show that

(5)

foxltnxdt=(n+1)(1-x)<(n+1)(1-x)

and in case -1 5 x < 0, show that

(6) JRn[=lL 1-tdtl5l fxot^dtI=nx+1-n+I