Page 61 - Standard Handbook Of Petroleum & Natural Gas Engineering

50 Mathematics

The transform of a first derivative of f(t) is

S[$f(t)] = sF(s)-f(0')

where f(0') = initial value of f(t) as t + 0 from positive values.
The transform of a second derivative of f(t) is
S[f"(t)] = s2F(s) - sf(0') - f'(0')

and of jf(t)dt is

f-'(O')
+ F(s)
S[jf(t)dt] = 7 -
S
Solutions derived by Laplace transformation are in terms of the complex
variable s. In some cases, it is necessary to retransform the solution in terms
of time, performing an inverse transformation

S-'F(s) = f(t)
Just as there is only one direct transform F(s) for any f(t), there is only one
inverse transform f( t) for any F(s) and inverse transforms are generally deter-
mined through use of tables.

ANALYTIC GEOMETRY
Symmetry

Symmetry exists for the curve of a function about the y-axis if F(x,y) = F(-x,y),
about the x-axis if F(x,y) = F(x,-y), about the origin if F(x,y) = F(-x,-y), and about
the 45" line if F(x,y) = F(y,x).

Intercepts
Intercepts are points where the curve of a function crosses the axes. The x
intercepts are found by setting y = 0 and the y intercepts by setting x = 0.
Asymptotes

As a point P(x,y) on a curve moves away from the region of the origin (Fig-
ure 1-36), the distance between P and some fixed line may tend to zero. If so,
the line is called an asymptote of the curve. If N(x) and D(x) are polynomials
with no common factor, and
y = N(x)/D(x)

where x = c is a root of D(x), then the line x = c is an asymptote of the graph
of y.