Hint
Keep every original denominator exclusion. Keep the uncancelled denominator zero.
Worked solution
(a)
\(x=9\) because the proper fraction has a nonzero numerator.
Answer: \(9\).
Answer: \(9\)
(b)
\(y=x+2\)
Answer: \(2\).
Answer: \(2\)
(c)
\(\displaystyle f'(x)= 1- \frac{16}{(x- 9)^{2}}= 0\implies x= 9\pm 4\)
Answer: \(5\).
Answer: \(5\)
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: Keep the uncancelled denominator 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: Do not multiply an inequality by an unknown-sign denominator without cases.
(b) · 2 marks
- 1 mark: Let the proper fraction tend to zero as |x| grows.
- 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: An oblique asymptote need not be a line that the curve never crosses.
(c) · 2 marks
- 1 mark: Set the derivative to zero, preserving both branches.
- 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 multiply an inequality by an unknown-sign denominator without cases.
Total: 6 marks.