Calculus: Analytic Geometry and Calculus, with Vectors

494

Y

1

01 1 z

Figure 9.263

e3 4 X

correct to 5 or 10 decimal places

Exponential and logarithmic functions

which we can now obtain very easily

by evaluating the integrals. Each

of these formulas actually deter-

mines e, and the second one says

that e is so determined that the area

-4 of the region shaded in Figure

9.263 is 1. Many other formulas

involving e will appear later, and one

of them will enable us to calculate e

with surprisingly little effort. Mean-

while, we can remark that many persons remember the 16D (16-decimal,

but only 15 after the decimal point) approximation to e by mentally

grouping digits in the form

(9.264) e = 2.7 1828 1828 45 90 45

so we can visualize the repeated 1828 followed by 45 and twice 45

and 45.

The number e is the natural base of exponentials and logarithms.

Other bases sometimes appear. The base 10 is used when we give a

number n and say that "there are 10^ atoms in the universe," but not

even Eddington suggested that this should be differentiated with respect

to n. We never differentiate or integrate a is e. If, as never

or rarely happens outside misguided examinations in calculus, we are

called upon to differentiate or integrate a1:1 where a e, we write

(9.265) ak1z = ekta

and take logarithms with base e to obtain kix log a = k2x and hence

k2 = kl log a. We then work with ek'z instead of aI. z.

Our theory of exponentials and logarithms enables us to provide the

promised proof of the power formula which, for convenience of reference,

we put in a theorem.

Theorem 9.27 If n is a constant, integer or not, then

dxn = nxn-1

when x>0.

To prove this theorem, let y = xn and take logarithms to obtain

log y = n log x. Since log y is differentiable, we can, because y= eb0y,

conclude that y itself is differentiable andcan use the chain rule to obtain

(1/y)dy/dx = n/x so dy/dx= nx". This proves Theorem 9.27. We

are now able to prove the last of the basic limit theorems, Theorem 3.288,

which we now restate.