029
OF ENGINEERING FOBMUT Æ.
Diffebknttal and Integral Calculus—continued.
Maxima and Minima.
If a quantity increases continuously, and then decreases, its values,
at the limits of increase or decrease, are the maxima or minima
respectively. If it decreases (or increases) continually it has no
maxima or minima.
du
The function u is a minimum or maximum when — = 0.
If the second differential coefficient a negative quantity the
value of w is a maximum.
If be positive, u is a minimum.
d x2 _ _
The point of contrary flexure (in a curve whose equation is
d2y
f(xy) = 0) occurs when = 0 or = co .
Simple Example of the Application of the Principles
of the Calculus to Increase in Area.
x = Radius = 12.
(? x = Rate of increment of x.
z — Area of figure.
d z =s Rate of increment of area
— hatched portion in dia-
gram.
Differential Calculus.—A figure increases at the rate dx = 2 when
x = 12; at what rate d z does the area increase when z = ax2 and
a — = 1.5708 ? dz = 2ax2~1 dx — 2ax.dx = 15.4.
Example in Integration..—A figure is found to
increase in the ratio — = 2 a x. Find the f, or
ax
2 Eil
function;/dz =/2a xdx=-----------------= ax2.
Note.—In this and other diagrams given to exemplify the prin-
ciples of the calculus the increments are shown as having considerable
magnitude, otherwise they cannot be shown in diagram; but properly
the increments should be indefinitely small.