22M.1.SL.TZ1.12

pestleMathematicsAISLPaper 122M· sl-5-2-increasing-and-decreasing-functions, sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

A company’s profit per year was found to be changing at a rate of

$\frac{d P}{d t} = 3 t^{2} - 8 t$

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.

Determine whether the profit is increasing or decreasing when $t = 2$ .

[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]

Markscheme / solution

METHOD 1

(when $t = 2$ )

$\frac{d P}{d t} = - 4$ OR $\frac{d P}{d t} < 0$ (equivalent in words) OR $3 {(2)}^{2} - 8 (2) = - 4$ M1

therefore $P$ is decreasing A1

METHOD 2

sketch with $t = 2$ indicated in 4th quadrant OR $t$ -intercepts identified M1

therefore $P$ is decreasing A1

[2 marks]

$(P (t) =) t^{3} - 4 t^{2} (+ c)$ A1A1

$4 = 1^{3} - 4 {(1)}^{2} + c$ (M1)

Note: Award M1 for substituting $(1, 4)$ into their equation with $+ c$ seen.

$c = 7$

$P (t) = t^{3} - 4 t^{2} + 7$ A1

[4 marks]

Examiners’ report

Even some weaker candidates were able to score in this part of the question as many candidates understood the process required to determine whether the profit is increasing or decreasing.

Many candidates failed to recognize that they needed to integrate the original function. Of the few that attempted to find the value of the constant the vast majority substituted 4000 rather than 4. So, a correct final expression for $P (t)$ was rarely seen.