- Topic
- Linear approximation
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Linear approximation practice question 7
Kleiber’s law models a mammal’s resting metabolic rate as kilocalories per day, where is its mass in kilograms. A sheep of 81 kg loses 2 kg over the winter. Use a linear approximation at to estimate its resting metabolic rate at 79 kg.
Give the answer in kilocalories per day.
Why is 81 a friendly mass for the power ? And does the step from 81 kg to 79 kg go up or down?
Linearise at , where makes both and exact.
, and the step is .
A number.
Theory UBC Math 100 §9.2CLP-1 §3.4Key results
Worked solution
Strategy
Linearise at the nearby mass where the fractional power is exact, 81 = 3⁴, then step back 2 kg along the tangent line.
Working
811/4 = 3, so 813/4 = 3³ = 27.
Power rule: near 81 kg, each kilogram is worth about 17.5 kcal a day.
The step is negative because the mass falls. The true value is 1854.9.
Answer
The answer is about 1855 kilocalories per day.
Check
Directly, 70 · 793/4 ≈ 1854.9 kcal/day, so the estimate is off by only about 0.1.
Where this goes wrong
Taking the step as +2 because the mass changed by 2 kg. The step is the new mass minus the base mass, 79 − 81 = −2, so the estimate falls by 35 to 1855 rather than rising to 1925.
Leans on
Laws of exponents with fractional powers, from Power functions and first sketches
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