Calc for Business. Must complete by 12/12/17 8:30PM New York Time
Calculus for Business – Signature assignment
Let q d be the number of units of a commodity demanded by consumers at an given time t and let q s denote the number of units of the commodity supplied by producers at a given time t. Let p be the price in dollars of the commodity at time t.
The equilibrium price is defined to be the price where the quantity demanded is equal to the quantity supplied
Suppose the supply and demand functions for a certain commodity in a competitive market are given, in hundreds of units, by
qs = 30 + p + 5 𝑑𝑝
dt
qd = 51 - 2 p + 4 𝑑𝑝
Goals:
1 - Express the m arket equilibrium price as a function of time
2 - Determine whether there is price stability in the market place for this item.
To achieve these goals, answer the following questions. Show all steps of your work.
a. Set the expressions for q s and q d equal to each other .
b. Solve this equation for dp
dt
c. Write this equation in the form: f(p(t)) p ’ = g(t)
d. Integrate both sides with respect to t .
e. Solve the resulting equation for p in terms of t.
f. Suppose at time t = 0, the market equilibrium price is 12 Dollars. Express the market equilibrium price as a function of time.
g. Find lim t → ∞ p , which gives the price we can expect this product to approach . If this limit is finite, then for this item there is price stability in the market place.
If p → ∞ as t → ∞ , then price will continue to increase until conditions
change .