Hint
Use a separate constant after each integration. The slope condition determines the first constant.
Worked solution
(a)
Integrate once to \(\displaystyle \frac{dy}{dx}= 6x^{2}- 4x+ A\). At \(\displaystyle x=1\) the slope condition gives \(\displaystyle 5=6-4+A\), hence \(\displaystyle A=3\).
Answer: \(\displaystyle \frac{dy}{dx}= 6x^{2}- 4x+ 3\).
(b)
Integrate again to \(\displaystyle y=2x^{3}-2x^{2}+3x+B\). At \(\displaystyle (2,10), 10=16-8+6+B=14+B\), so \(\displaystyle B=-4\). The two constants are independent.
Answer: \(y=2x^3-2x^2+3x-4\).
(c)
At \(\displaystyle x=1\) the curve ordinate is \(\displaystyle 2-2+3-4=-1\) and the tangent slope is the given value 5. The normal slope is \(\displaystyle - \frac{1}{5}\), giving \(\displaystyle y+ 1= - \frac{x- 1}{5}\).
Answer: \(y+1=-\tfrac15(x-1)\), equivalently \(\displaystyle y= - \frac{x}{5}- \frac{4}{5}\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Integrate the second derivative correctly.
- 1 mark: Introduce and determine the constant \(\displaystyle A=3\) from the slope condition.
- 1 mark: Give the complete first derivative \(\displaystyle 6x^{2}-4x+3\).
Alternative methods: Any equivalent correct first derivative satisfying the gradient condition is accepted.
Common error: Applying the curve point to the first derivative.
(b) · 3 marks
- 1 mark: Integrate the complete first derivative correctly and introduce B.
- 1 mark: Use (2,10) to obtain \(\displaystyle B=-4\).
- 1 mark: Give the specified curve equation.
Alternative methods: Equivalent algebraic forms that satisfy all three original conditions are accepted.
Common error: Reusing the value \(\displaystyle A=3\) as the second integration constant.
(c) · 2 marks
- 1 mark: Use the recovered point (1,−1) and normal slope \(\displaystyle - \frac{1}{5}\).
- 1 mark: Give the correct normal equation.
Alternative methods: Any equivalent exact line equation is accepted.
Common error: Using slope 5 for the normal or taking the point ordinate as 5.
Total: 8 marks.