Question

Suppose the demand for a certain item is modeled as D(p) = -3p^2 + 21p + 10, where, p represents the price of the item in dollars. Currently the price of the item is $5. Use marginal demand to estimate the change in demand when the price is increased by one dollar.

Answer 1

given D(p) = -3p² + 21p + 10
=>D'(p) = -3(2)p^2-1 + 21(1) + 0
=>D'(p) = -6p + 21

Currently, the price of the item is $5
=>p=5

D'(5) = -6(5) + 21
=>D'(5) = -30 + 21
=>D'(5) = -9

ΔD=(D'(5))Δp
change in demand when the price is increased by one dollar,
D'(5) = -9,Δp=1

=>ΔD=(-9)(1)
=>ΔD=-9

change in demand when the price is increased by one dollar is -9
demand decreases by 9 units

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total cost ,C(q) = 6q³ - q² + 7q + 90

marginal cost=C'(q)
=>marginal cost= 6(3)q^3-1 - 2q^2-1 + 7(1) + 0
=>marginal cost= 18q² - 2q + 7

when it manufactures 5 DVDs . q=5

=>marginal cost= 18(5)² - 2(5) + 7
=>marginal cost= 450-10+7
=>marginal cost= 447 dollars per DVD

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f(x) = 3x - 4 increases thoughout the interval -2 ≤ x ≤ 3

global minimum =f(-2)
=>global minimum =3(-2)-4
=>global minimum =-6-4
=>global minimum =-10

global maximum =f(3)
=>global maximum =3(3)-4
=>global maximum =9-4
=>global maximum =5