∫ cos xdx
The Calculus
A Definition for Dieters
dx
dy
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
dx
∂y
∂x
f '( x)
∫ f ( x )dx
The handles and hieroglyphs attached to the techniques of calculus may look
alien, but the ideas behind them are easily recognized.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Being a rate-of-change, a derivative means, simply, the speed of a process;
so many miles per hour or feet per second if it refers to change in position;
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
…so many pounds per week if it refers to a triumph in dieting;
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
… so many geniuses per childbirth if it refers to I.Q. statistics;
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
… so much cornstalk per candle power of sunshine if it refers to the growth of
a corn crop.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The integral corresponding to each of these derivatives would be the miles
traveled, pounds lost, geniuses gained or length of cornstalk grown.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
When used abstractly in an equation, a derivative can most readily be thought
of in terms of the curve which represents this equation on a graph.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
y
x
At any point, the curve is rising or falling at a rate of so many y-units per x-unit.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
y
slope =
rise dy
=
run dx
x
This slope up or down is the exact geometric equivalent of the rate of
change—the derivative—of y with respect to x.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Engineers often express the grade of a hill, the pitch of a roof or the
steepness of a airplane’s climb in identical terms: as so much altitude gained
per unit of horizontal distance traversed.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
But in these applications the slope is normally conceived of as being
measured over some definite span of distance.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
dx
In calculus, on the other hand, the derivative is thought of as a instantaneous
slope at a single point of a curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
That this elusive concept of instantaneous slope is no figment of the
mathematical imagination can be seen in an artillery shell as it arcs toward
target.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
At any single moment the shell is moving in definite direction. This direction is
an instantaneous slope with respect to the ground, a rate of change in the
shell’s altitude with respect to its horizontal position.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
v=
dy
dt
In terms of a graph, the speed of the shell, moving up or down, can also be
considered as an instantaneous slope on a curve—a rate of change in the
shell’s altitude with respect to the time that has been elapsing since the shell
was fired.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
A mathematician would normally write such a derivative—the velocity of climb
for fall, or the rate of change of vertical distance– as dy/dt, in which the t stands
for time.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The counterpart of a derivative, an integral, can also be visualized in terms of
a graph. Suppose that y equals some expression of x and that this equation is
plotted as a curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Then the integral of y is the area between the curve and the horizontal line,
or axis, running below it. Why this is so can be seen by imagining that the
area under the curve is filled by a picket fence with a scalloped top…
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
… as the fence is being built, each new picket adds to the area of fence. In
fact, the height of each added picket is a measure of the rate at which the
area of the fence is growing; a six-foot picket, for instance, adds twice as
much area as a three-foot picket.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The integral of the rate of change, therefore, must be the actual factor in the
situation that is changing: namely, the area of the fence itself. The geometric
equivalent of each picket is simply the height of the curve—the vertical, or y
coordinate of each point on a curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Integrating y must, therefore, give the total area under the curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Many of most practical applications of calculus stem from integration’s ability
to sum-up y-length pickets and determine areas.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Through it a mathematician can determine the volume of all manner of
irregular shapes, such as airplane fuselages or oil-storage tanks; he can also
find areas of curved surfaces—the amount of sheet metal in a molded car
body or the lifting surface on the wings of a jet.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
There is one major difficulty in the process of integration—a difficulty so
enormous and so recurrent that most of our largest computers today have
been built specifically to cope with it.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
This is the problem of so-called “boundary conditions.” When the area of a
picket fence is measured, the boundary conditions are established by the two
pickets that mark the two ends of the fence.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
This is the problem of so-called “boundary conditions.” When the area of a
picket fence is measured, the boundary conditions are established by the two
pickets that mark the two ends of the fence.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
But there are no ends to many of the curves that represent equations.
The area beneath this type of curve may be indefinitely large.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
0
10
To give it boundaries, the mathematician erects the equivalent of end posts to
mark off the particular part of the area he is interested in. He then integrates
the equation represented by the curve between these two verticals.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
?
0
10
Often, however, in the case of equations which are arrived at experimentally,
the proper way to interpret such an equation cannot be found except by
integrating it, and the boundary conditions necessary for integrating it cannot
be found except by understanding how to interpret it.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
?
0
5
10
To avoid this impasse, the scientist, in effect, chooses arbitrary picket posts
and lets a computer run off dozens of laborious solutions, which give him
insight into the equation and the process of change which it symbolizes.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
0
5
10
To work out the rules of calculus, Newton visualized what would happen if one
point on a graph-curve slid down into a point nearby.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
0
5
dx
10
As the slide begins, the average slope of the curve between the two points
is the number of y-units separating then vertically, divided by the number of
x-units separating them horizontally.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
0
5
dx
10
As the slide continues, both distances in this fraction diminish toward zero
and finally vanish when the two points merge. But this does not mean that
the fraction itself vanishes. A ratio of 1:2, for instance, need not suddenly
become zero just because its numerator and denominator become
indefinitely small.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
When last heard from, as they disappear arm in arm into the fastness of
infinity, the numerator may be one zillionth and the denominator two
zillionths, but the ratio between them is still 1 to 2.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Finding the value which a fraction approaches as its numerator and
denominator both diminish toward nothingness is called “taking a limit.”
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
If the numerator equals half the denominator, the limit is one half. If the
numerator equals 10 times the denominator, the limit is 10.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
0
5
dx
10
As two points on a curve slide together, the vertical and horizontal
distances between them remain coupled, even as they fade away, by the
relationship of y to z expressed in the original equation of the curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
dy
0
5
dx
10
As they merge, therefore, the ratio of their distances approaches a definite
limit which can be evaluated in terms of y and x.
This limit, ½ or 10 or whatever it may be, is the slope of the curve at the
precise spot where the two points merge—the rate of change of vertical y
with respect to horizontal x or, put another way, the derivative of y with
respect to x.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The subtle train of reasoning which enable Newton to differentiate
equations and find the derivative or limiting value of the ratio—written dy/dx
or dy/dt—is the fundamental process of calculus.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
It can be roughly paraphrased as follows: in a developing situation, the
difference between the state of affairs at one moment and the state of
affairs at the next moment is an indication of how the situation is shaping
up; and the ratio of the net changes that take place between the two
moments is evaluated as a limit—a limit approached when the interval
between the two moments is imagined as diminishing toward zero– then
that limit shows how fast developments are taking place.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The logic of calculus can be applied to moments of time, points on a curve,
temperatures in a gas or any state of affairs which can be related by
equations; the same rules of differentiation apply to all of them.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)