Calculus: Analytic Geometry and Calculus, with Vectors

3.9 Increments and differentials 199

is equivalent to the other two. Use these formulas to obtain

(2)

dT =

.(:)-bigdL 2Ldg

g g

(3) 2TdT =4rzgdL 2Ldg

(4) T dT

= 2 L dL gdg.

Making a suitable application of the fact that 1,41 < JBI + ICI whenever 4, B,

and C are numbers (not necessarily positive) for which 4 = B + C or 4 = B - C,

use (2) to show that if IdLI 5 (p/100)L and (dgl S (q/100)g, then JdTj <_

[-(p + q)/100]T. Repeat the process by use of (3) and (4). Remark: The

first of the formulas (1) is a standard formula for the period T (a number of

seconds) of a pendulum of length L which oscillates in a world where the scalar

acceleration of gravity is g. Our result shows that if errors in measurement of L

and g do not exceed p and q per cent, respectively, then the error in T will not

exceed ''T(p + q) per cent.

8 A pendulum clock gains 3 minutes in 24 hours. By what per cent should

the pendulum be lengthened? 4ns.: 0.42 per cent.

9 Under appropriate conditions the pressure p and the volume V of confined

gas satisfy the relation

(*) pv"=C,

where ry (gamma) and C are constants that depend upon the gas and the condi-

tions. Obtain the formula

(**) dp+ydV0

p T'

in two different ways. First, differentiate the members of () as they stand.
Then operate upon the equation obtained by taking logarithms of the members
of (). Remark: It is so often desirable to take logarithms before differentiating
that the process is named logarithmic differentiation. The derivative of the
logarithm of a function is called the logarithmic derivative of the function.
10 Apply the procedure of the preceding problem to the relation

pV = nRT,

in which n is the number of gram-moles of a gas, R is a universal proportionality

constant known as "the gas constant," and T is the absolute, or Kelvin, tempera-

ture. It is now supposed that p, Y, and T are all functions of t and the relation

dp dV dT

p + V T

is to be obtained.

(^11) For dense projectiles fired short distances over a horizontal plane, the
range R is calculated from the formula
z
R =
v--°
sin 2a,
g