- Topic
- Taylor polynomials
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Taylor polynomials practice question 5
A function has , and . Use the quadratic approximation of centred at to estimate .
The quadratic approximation is the tangent line plus one curvature term. What does that extra term use, and what must it be divided by?
.
With : .
A decimal, to two decimal places.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
The quadratic approximation adds one term to the tangent line: half the second derivative times the square of the step.
Working
Second-order Taylor polynomial at 1.
The linear estimate alone would be 2.8; the curvature term adds 0.03.
Answer
The answer is 2.83.
Check
At x = 1 the polynomial gives 3, its derivative −2 + 6(x − 1) gives −2, and its second derivative is 6, matching all three given values.
Where this goes wrong
Using f″(1) = 6 as the coefficient instead of 6/2 = 3, which gives 2.86. The second-order coefficient is always the second derivative divided by 2.
Leans on
Linear approximation, from Linear approximation
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