- Topic
- Taylor polynomials
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Taylor polynomials practice question 8
By Kepler’s third law, a planet at an average distance of astronomical units from the Sun takes years to go round it. Use the second-order Taylor polynomial of about to estimate , as a fraction.
Why is a convenient centre for ? A second-order polynomial needs three numbers from there: which three?
and . At 4 every power is exact, because .
Then .
An exact fraction, or a decimal to two places.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Centre the polynomial where the fractional power is exact, at a perfect square, and add the curvature term to the tangent line.
Working
The power rule twice, then evaluate at the centre.
The second-order coefficient is T″(4)/2! = 3/16.
The step is a − 4 = 1. The true period, 5√5 ≈ 11.180 years, is close to 179/16 = 11.1875.
Answer
The answer is 179 over 16 years, about 11.19 years.
Check
5√5 ≈ 11.180, so 179/16 = 11.1875 is off by less than 0.01 of a year.
Where this goes wrong
Using T″(4) = 3/8 itself as the coefficient of (a − 4)², which gives 91/8. The quadratic term is the second derivative over 2!, times the square of the step.
Leans on
The power rule with fractional exponents, from Differentiation rules
Try a similar question
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