Question
SLPaper 1
A company’s profit per year was found to be changing at a rate of
$$ \frac{\mathrm{d}P}{\mathrm{d}t} = 3t^2 - 8t $$
where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.
1.
Determine whether the profit is increasing or decreasing when $t=2$.
[2]Verified
Solution
METHOD 1
- Substitute $t=2$ into the derivative: $\frac{\mathrm{d}P}{\mathrm{d}t} = 3(2)^2 - 8(2)$ OR $\frac{\mathrm{d}P}{\mathrm{d}t} = -4$ M1
- Therefore $P$ is decreasing A1
METHOD 2
- Sketch of $\frac{\mathrm{d}P}{\mathrm{d}t}$ with $t=2$ indicated in 4th quadrant OR $t$-intercepts identified M1
- Therefore $P$ is decreasing A1
2 marks total
NoteAward M1 for correct substitution or for finding the correct value of the derivative.2.
One year after the company was founded, the profit was $4$ thousand dollars.
Find an expression for $P(t)$, when $t \geq 0$.
[4]Verified
Solution
- Integrate the expression: $P(t) = \int (3t^2 - 8t) \, \mathrm{d}t$
- $P(t) = t^3 - 4t^2 (+c)$ A1 A1
- Substitute $(1, 4)$ into the equation with $+c$ seen: $4 = 1^3 - 4(1)^2 + c$ M1
- $c = 7$
- State the final expression: $P(t) = t^3 - 4t^2 + 7$ A1
4 marks total
NoteAward A1 for $t^3$ and A1 for $-4t^2$.NoteAward M1 for substituting $(1, 4)$ into their equation with $+c$ seen.