Hint
Arcosh has real domain [1,∞), with range [0,∞). Use the real logarithmic inverse-sinh expression.
Worked solution
(a)
\(\ln(a+\sqrt{a^2+1})=2.99822295029797\)
Answer: \(2.99822295029797\).
Answer: \(\approx 2.99822295029797\)
(b)
\(\ln(a+\sqrt{a^2-1})=2.99322284612638\)
Answer: \(2.99322284612638\).
Answer: \(\approx 2.99322284612638\)
(c)
\(\displaystyle \frac{1}{2}\ln [\frac{1+ p}{1- p}]= 1.15129254649702\)
Answer: \(1.15129254649702\).
Answer: \(\approx 1.15129254649702\)
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: Use the real logarithmic inverse-sinh expression.
- 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: Sinh⁻¹x is not \(\displaystyle \frac{1}{\sinh x}\) when inverse notation is intended.
(b) · 2 marks
- 1 mark: Choose the nonnegative inverse-cosh branch.
- 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 drop the negative solution of \(\displaystyle \cosh u=c\) without a branch condition.
(c) · 2 marks
- 1 mark: Use the logarithmic form with |p|<1.
- 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: Sinh⁻¹x is not \(\displaystyle \frac{1}{\sinh x}\) when inverse notation is intended.
Total: 6 marks.