9e.4.3.Pr61. The cost C(x) of producing the first x units of a
commodity is C(x) = e0.2 x.
a.
Find the marginal cost [dC/dx].
b.
Determine the level of production x for which the
marginal cost equals average cost A(x) = [C(x)/x] = [dC/dx].
c.
Determine the level of production x for which the
average cost A(x) is minimized.
d.
Here is the graph of C(x) = e0.2 x (I have
changed x to q).
For various values of q, use a straight edge or ruler to
represent lines through O = (0,0) and (q,C(q)). The slope of
these lines is C(q)/q = A(q). As you move q to the right, the
slope of the line from 0 to (q, C(q)) decreases and then
increases. Which of these lines has minimal slope? How does your
answer correspond the the result in parts (b.) and (c.)?
2.
9e.4.2.Pr55.Variation. Worker Efficiency. Worker efficiency is
measured by (hourly) output Q and is given by a formula of the
form
Q(t)
= 500 − A e−kt,
where t is the "experience" measured by the time in months
after initial training.
a.
What is limt → ∞ Q(t)?
b.
It is observed that a newly trained worker has an
hourly output of 300 units, and that after 6 months, a worker
has an hourly output of 410 units. Find the function of the form
Q(t)
= 500 − A e−kt,
which fits the data.
Hint: What is the form of the function F(t) = 500 − Q(t)?
3.
9e Example 4.3.14 Variation. Total net profit comes from wheat,
steel, and oil.
At a certain time,
The profit W from wheat accounts for 20% of net profit and is
increasing at a rate of 2%.
The profit S from steel accounts for 30% of net profit and is
increasing at a rate of 3%.
The profit O from oil accounts for 50% of net profit and is
decreasing at a rate of 5%.
At what percentage rate is total net profit increasing
or decreasing?
Hint: When the data is taken, find
a.
[W/(W + S + O)] =
b.
[([dW/dt])/W] =
4.
9e.4.3.Pr66. Money is deposited in a bank offering interest
at an annual rate of 6% compounded continuously (CC). Find the
percentage rate of change with respect to time.
5.
9e.4.3.Pr33. If g(u) = ln(u + √{u2 + 1}), find
g′(u). Simplify your answer.
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