Page 219 - How To Solve Word Problems In Calculus

EXAMPLE 2
A publisher estimates that t months after he introduces a new
magazine, the circulation will be
2
C(t) = 150t + 400t + 7000 copies. If this prediction is correct,
how fast will circulation increase 6 months after the magazine
is introduced?

Solution

C (t) = 300t + 400
C (6) = 300(6) + 400 = 2200 copies per month

Sometimes it is useful to understand how a quantity’s
change is related to variables other than time. For example, a
manufacturer’s profit may change with the number of items
produced and sold or his production cost may change with
the availability of raw materials. In general, the derivative
y (x)or dy/dx measures the rate of change of the variable y

with respect to x.

EXAMPLE 3
The demand q for a certain commodity expressed as a function
2
of its selling price p is q(p) = 200p − 100p + 5000 units. At
what rate is q changing with respect to p when the selling
price is $5? Is demand increasing or decreasing at this price?

Solution

q (p) = 200 − 200p

q (5) =−800
The rate of change when p = 5is −800 units per dollar. Since
this number is negative, demand is decreasing.

EXAMPLE 4
A computer manufacturer’s total cost in dollars when q units
are produced is given by the function
C(q) = 500q + 5000e −q/10 . At what rate is his cost changing
when 20 units are produced?

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