Hint
The required derivatives exist near zero. Differentiate once and evaluate at zero.
Worked solution
(a)
\(A=f'(0)=1\)
Answer: \(1\).
Answer: \(1\)
(b)
\(B=f''(0)/2!=-4\)
Answer: \(-4\).
Answer: \(-4\)
(c)
\(C=f^{(3)}(0)/3!=- \frac{13}{3}\)
Answer: \(- \frac{13}{3}\).
Answer: \(- \frac{13}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Differentiate once and evaluate at zero.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A derivative value is not necessarily the same as its Taylor coefficient.
(b) · 2 marks
- 1 mark: Divide the second derivative at zero by 2!.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not omit mixed products when multiplying two series.
(c) · 2 marks
- 1 mark: Divide the third derivative at zero by 3!.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A derivative value is not necessarily the same as its Taylor coefficient.
Total: 6 marks.