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business world. The maximums and minimums of the curve of a
function have the slope of !, so the derivative of the e"uation is set
e"ual to ! to determine the input at the function#s optimal state. $or
example, in the case of an apartment complex company, a function
can be determined to %nd out the amount of money produced by the
number of apartment buildings sold, an e"uation of a simple
relationship. &eeping in mind the domain 'the number of apartments
actually available(, the function#s derivative reveals the critical points
that can be determined to be absolute maximums, absolute
minimums, or neither with the reinsertion of the values into the original
function. The use of derivatives and their role in optimization reveals
that the maximum pro%t does not necessarily come from renting out
allthe apartment buildings, as maintenance costs and other variables
need to be considered. )onse"uently, the analysis from the derivative
e"uations alerts the business owners to %nding a means of raising rent
costs or lowering maintenance costs to get maximum pro%t to full
resource capacity. The economic principle of marginal analysis is
bene%cial to brea-even analysis and pro%t maximization. The pro%t-
maximizing "uantity of output is achieved at the point where marginal
cost, the extra cost of production of a single unit of output, and
marginal revenue, the extra revenue generated by the sale of an
additional unit of product, are e"ual. The marginal revenue is formally
de%ned as the change in total revenue over the change in one unit of
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"uantity and can be represented as a derivative. The total revenue is
e"ual to the price demanded multiplied by the "uantity, so through the
product rule the marginal revenue is e"ual to the price demanded plus
the "uantity times the derivative of the price demanded. *ith this
information, the owners of a %rm can ascertain the pro%t-maximizing
"uantity based on its role as a monopoly or a competitive %rm and to
e+ciently organize its factors of production and exercise specialization.
Because the derivative gives the slope of a curve, it can be
applied to the curves commonly utilized in the business conference
room revenue and cost curves and stoc maret curves. graph of a
%rm#s total revenue for a period of time depicts changes in total
revenue while that of the cost depicts changes in costs with relation to
the level of output. The slope of the curve at a single value is
determined by the function#s derivative, and it can be used to
determine an increase or decrease in sales and proections for the
future. The attainment of slopes and instantaneous rates of change
can be used for stoc maret curves as well. The price history curve for
a %rm or corporation reveals the price of its shares at dierent points in
time, and the slope at a speci%c point indicates growth or decline,
further denoting the potential future success or failure of a business.
The same concept wors for other curves relevant in the business
arena, such asyield curves, which capture the overall movement of interest rates on
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bonds of different maturities. Curves can be evaluated by the important differential
calculus applications.
Business calculus can be applied to determine the e"uation and
graphical shape of a total cost curve, a function#s derivative, and other
aspects necessary or convenient for a business to now. /n addition to
measuring a function#s change with respect to that of its input, a
derivative is also a %nancial instrument that plays a role in the stoc
maret. /t is derived from an underlying asset, which is usually an
asset, event, index, or value. *ith the purchase of a stoc comes the
ownership of a small part of the company, an ownership that provides
an intrinsic value. 0erivatives traders over time exchange cash or
assets based upon the underlying asset, and they can in1ate stoc
values and increase their ostensible values.
2o matter which occupational %eld a student is interested in,
calculus and mathematics in general will play a role, apparent or
implicit, in its success. 3o instead of hoping to leave that calculus class
and dismissing its teachings, one must embrace calculus
wholeheartedly. To have a competency in mathematics is to be armed
with the proper tools to thrive in any %eld.
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B$B'$()*AP+
-Business Applications. /0 Sept 122.Pauls Online Math Notes. 14 (ct 1225
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Clayton, )ary :. :conomics principles ; practices. erivative. 122?. The Financial Dictionary. 14 (ct 1225 6http788financial=
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'ane, Margaret. $nterview. 14 (ct 1225.
'au, effrey. -"otal *evenue and :lasticity. 32 un /55. University of Hawaii. 1? (ct
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"atum, Malcolm. -hat is a >erivatives Mar&etD // Sept 1225. Wise Geek. 13 (ct1225 6http788www.wisegee&.com8what=is=a=derivative=mar&et.htm9.
agner, +ans. -Bond ield Curve and the Stoc& Mar&et. 14 May 1225.Daily Markets.
13 (ct 1225 6http788www.dailymar&ets.com8stoc&s8122582481?8bond=yield=curve=and=the=stoc&=mar&et89.