IB Mathematics SL 5.2 Increasing and decreasing function AI SL Paper 1- Exam Style Questions- New Syllabus

(a)
Evaluate the rate of change at \( t = 2 \):
\[ \begin{aligned} \frac{dP}{dt} &= 3t^2 – 8t \\ \text{At } t = 2: \quad \frac{dP}{dt} &= 3 \times 2^2 – 8 \times 2 \\ &= 3 \times 4 – 16 = 12 – 16 = -4 \end{aligned} \]
Since \( \frac{dP}{dt} = -4 < 0 \), the profit is decreasing. M1 A1
[2 marks]

(b)
Integrate \( \frac{dP}{dt} \):
\[ \begin{aligned} \frac{dP}{dt} &= 3t^2 – 8t \\ P(t) &= \int (3t^2 – 8t) \, dt \\ &= 3 \cdot \frac{t^3}{3} – 8 \cdot \frac{t^2}{2} + c \\ &= t^3 – 4t^2 + c \end{aligned} \] A1 A1
Use the condition \( P(1) = 4 \):
\[ \begin{aligned} P(1) &= 1^3 – 4 \times 1^2 + c = 4 \\ 1 – 4 + c &= 4 \\ c &= 7 \end{aligned} \] (M1)
Thus, \( P(t) = t^3 – 4t^2 + 7 \). A1
[4 marks]