e™. Its derivative evaluated at the inside function is e”˜,“. Multiplying this by the

derivative of the inside function u(x), we conclude that

E.wnr[˜ie 5.5 Lsing Theorem 5.2, wc compute the following derivatives:

E.wmpi˜˜ 5.6 The density function for the standard normal distribution is

96 EXPONENTS AND LOGARITHMS [51

Let™s use calculus to sketch the graph of its core function

&) = ,-x2/2,

We first note that g is always positive, so its graph lies above the x-axis every-

where. Its first derivative is

&&) = -g”/>.

Since em”” is always positive, g™(x) = 0 if and only if x = 0. Since g(0) = 1,

the only candidate for max or min of g is the point (0, 1). Furthermore, g™(x) > 0

if and only if x < 0, and g™(x) < 0 if and only if x > 0; so g is increasing for

x < 0 and decreasing for x > 0. This tells us that the critical point (0, 1) must

be a max, in fact, a global max.

So far, we know that the graph of g stays above the x-axis all the time,

increases until it reaches the point (0,l) on the y-axis, and then decreases to the

right of the y-axis. Let™s use the second derivative to fine-tune this picture:

&+) = (+*2/Z)™ = &-x™/2 ,-i/z = p _ I) ,˜.2/2,

! Since e-.˜/z > 0, g”(x) has the same sign as (x™ - 1). In particular,

g”(0) < 0, g”(x) = 0 t) x = il.

and (8)

The first inequality in (8) verifies that the critical point (0, 1) is indeed a local

max of R, Using the second part of(S), we note that

-m<x<-I j SW > 0,

-1<x<+1 * &?(I-) < 0,

1<x<+= i g”(x) > 0;

this implies that g is concave up on (p m, 1) and on (1, m) and concave down

on (- 1, + 1). The second order critical points occur at the points (- 1, e-l”)

and (1, em”?). Putting all this information together, WC sketch the graph of ,g in

Figure 5.5.

The graph of g is the graph ofthc usual bell-shaped probahility distribution.

Since / is simply g times (2˜) -I™* = .3Y. the graph off will be similar to the

graph of g but closer to the x-axis.

We now use equation b in Example 5.5 to compute the derivative of the general

exponential function y = b™.

For any fixed positive base b,

Theorem 5 . 3

(h™)™ = (In b)(W). (9

1

L5.61 97

APPLICATIONS

t

Figure

of e-x™/2. 5.5

The graph

Proof Since b = elnh, then ff = (e™““)” = et™“h*. By equation b in Example

/ 5.5,

I .

(b”)™ = (@)“) = (In b)(e(˜“hpx) = (In b)(b™).

Example5.7 (IF)™ = (In 10)(W)

only if Inb = 1, that is, if and only if b = P. In

Note that (bx)™ = Pi if and

fact, the exponential functions y = k6 are the only functions which are equal to

their derivatives throughout their domains. This fact gives another justification for

e being considered the natural base for exponential functions.

EXERCISES

5.8 Compute the first and second derivatives of each of the folluwing functions:

0) xe™“. h) &-, c) I”(.$ + 2)˜, d) ;, c) &, f) $.

5.9 USC calculus lo sketch the graph of each of the following functions:

u) x@. h ) .rr I_ <) cash(x) = (e+ + e “)/2.

5.10 Use the equation IOL”” = x. Example 5.7, and the mcthud of the proof of Lemma

5.3 to derive- a formula for the derivative ofy = Logx.

4PPLICATIONS

5.6

Present Value

Many economic problems entail comparing amounts of money at different points

of time in the sane computation. For example, the benefit/cost analysis of the

construction of a dam must compare in the same equation this year™s cost of con-

struction. future years™ costs of maintaining the dam, and future years™ monetary

98 EXPONENTS AND LOGARITHMS I51

benefits from the use of the dam. The simplest way to deal with such comparisons

is to USC the concept ofpresent val˜re to bring all money figures back to the present.

If we put A dollars into an account which compounds interest continuously at

rater, then after f years there will be

dollars in the account, by Theorem 5.1. Conversely, in order to generate R dollars

f years from now in an account which compounds interest continuously at rate r,

we would have to invest A = Be-” dollars in the account now, solving (10) for A

in terms of B. We call Be-” the present value (PV) of B dollars f years from now

(at interest rater).

Present value can also be defined using armual compounding instead of con-

tinuous compounding. In an account which compounds interest annually at rate

I, a deposit of A dollars now will yield B = A(1 + r)l dollars f years from now.

Conversely, in this framework, the present value of B dollars f years from now is

R/(1 + r)™ = B(l + r)-™ dollars. Strictly speaking, this latter framework only

makes sense for integer f™s, For this reason and because the exponential map err is

usually easier to work with than (1 + ?-)I, we will use the continuous compounding

version of present value.

Present value can also be defined forflows of payments. At interest rate I, the

present value of the flow--BI dollars 11 years from now, BL dollars I? years from

now,. , B,, dollars r,, years from now- is

p” = Ble ˜(˜1 + B# )I, + + B ,/ e r˜/>y (11)

Annuities

An annuity is a sequence of equal payments at regular intervals over a specified

period of time. The present value of an annuity that pays A dollars ar the end

of each of the next N years, assuming a constant interest rate I™ compounded

continuously. is

p” = Ae-˜l + ,&-,.™ + + AE-I.\