- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Functions of two variables and partial derivatives practice question 8
Weekly demand for a brand of coffee is bags, where is the price in dollars and is average household income in thousands of dollars. At present and . Use to estimate the change in weekly demand if the price is cut from $4.00 to $3.80 and income stays the same.
is a rate in bags per dollar, with income held fixed. How does a rate times a small change estimate a change in demand, and is this change in price positive or negative?
Hold at 64. Then , and .
The estimate is , and a cut from 4.00 to 3.80 is .
A number of bags: positive for a rise in demand, negative for a fall.
Theory UBC Math 100 §15.1Key results
Worked solution
Strategy
With income fixed, demand is a function of price alone, and ∂q/∂p is its rate of change in bags per dollar. Multiply that rate by the change in price, as in a linear approximation.
Working
200√I is a constant factor when income is held fixed, and p⁻¹ differentiates to −p⁻².
At these values, demand falls by about 100 bags for each dollar the price rises.
The price falls, so Δp is negative and demand rises.
The exact change is close to the estimate, as it should be for a small change in price.
Answer
The answer is Demand rises by about 20 bags per week..
Check
Demand is proportional to 1/p, so a 5% price cut raises it by about 5% of the present 400 bags, which is 20.
Where this goes wrong
Using Δp = +0.20 and predicting a fall of 20 bags. A price cut is a negative change in price; with ∂q/∂p negative, the product is positive.
Leans on
Linear approximation, from Linear approximation
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