Problem 3.4.17. Price p and demand q
are related by the equation p = 49 − q. The total cost of
producing q units is C(q) = (1/8) q2 + 4q + 200
(a)
revenue function R(q) = ?
(b)
profit function P(q) = ?
(c)
marginal revenue function R′(q) = ?
(d)
marginal cost C′(q) = ?
(e)
average cost A(q) = (C(q)/q) = ?
(f)
derivative of average cost A′(q) = ?
(g)
Find the q for which A(q) is minimized. For this q, compare
the marginal cost C′(q) and A(q).
Problem 3.4.22 Price p and demand q
are related by the equation p = 81 − 3q. The total cost of
producing q units is C(q) = (q + 1)/(q + 3)
(a)
revenue function R(q) = ?
(b)
profit function P(q) = ?
(c)
marginal revenue function R′(q) = ?
(d)
marginal cost C′(q) = ?
(e)
average cost A(q) = (C(q)/q) = ?
(f)
derivative of average cost A′(q) = ?
(g)
Find the q for which A(q) is minimized. For this q, compare
the marginal cost C′(q) and A(q).
Problem 3.4.39. PRICE ELASTICITY OF DEMAND When a particular
commodity is priced at p dollars per unit, consumer demand q
units, where p and q are related by q2 + 3 p q = 22.
(a)
Find the price elasticity of demand, [p/q] [dq/dp], for this commodity. Hint: Use implicit
differentiation wrt p.
(b)
For a unit price of p = $3, is the demand
elastic, inelastic, or of unit elasticity? Hint: q > 0.
Problem 3.5.5 variation: LINEAR PRICE-DEMAND MODEL A store
has been selling a popular computer game at the price of $40 per
unit, At this price, players have been buying 50 units per month.
The owner of the store wishes to raise the price of the game and
estimates that for each $1 increase in price, 3 fewer units will
be sold (Hint: [dq/dp] = …). If each unit
costs the store $25, at what price p should the game be sold to
maximize profit?
(a)
In this problem, express the profit function as a quadratic function of
p.
(b)
Use differentiation wrt p to find the critical number (p = …)
of the profit function.
(c)
The graph of the profit function is a parabola. Where is the
vertex of the parabola?
Problem 3.5.10. There are 320 yards of fencing available to
enclose a rectangular field. How should this fencing be used so
that the enclosed area is maximized? What is the shape of the
optimal field ?
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