- Topic
- Linear approximation
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Linear approximation practice question 3
A workshop’s cost of making chairs a week is dollars. Use the marginal cost to estimate the cost of making the 101st chair, that is, .
Give the answer in dollars.
Making the 101st chair is a step of 1 in , starting from 100. What does a linear approximation say about the change in over a small step?
One more chair is a step of 1 from , so .
.
A number.
Theory UBC Math 100 §9.2CLP-1 §3.4Key results
Worked solution
Strategy
One more chair is a step of 1 along the tangent line at q = 100, so the extra cost is about the slope there: the marginal cost.
Working
Power rule term by term; the fixed cost of 1000 has derivative 0.
The slope of the cost curve at 100 chairs, in dollars per chair.
The exact difference is $10.001: beside 100 chairs, one more is a small step.
Answer
The answer is about 10 dollars.
Check
Directly, C(101) − C(100) = 3010.001 − 3000 = $10.001, within a tenth of a cent of $10.
Where this goes wrong
Dividing the total cost by the number of chairs. C(100)/100 = $30 is the average cost of the first 100 chairs, fixed cost included; the cost of one more chair is the slope, C′(100).
Leans on
The power rule, from Differentiation rules
Saved in this browser only.