QCAA Maths Methods Differential Calculus Mini Test 3
External Assessment Paper 2 — Technology-active
Number of marks: 10
Perusal time: 1 minute
Writing time: 15 minutes
Section 1
Instructions
• This section has 10 questions and is worth 10 marks.
• Use a 2B pencil to fill in the A, B, C or D answer bubble completely.
• Choose the best answer for Questions 1 10.
• If you change your mind or make a mistake, use an eraser to remove your response and fill in the new answer bubble completely.
If \(f(x) = e^{3x}(x+1)^2\) and \(f'(x) = ae^{3x}(x+1)\), determine the expression for \(a\).
- (A) 3x + 5
- (B) 3x + 3
- (C) 5x + 5
- (D) 5x + 3
The number of koalas in a conservation park is modelled by \(N = 15 \ln(7t + 1)\), \(t \ge 1\), where \(t\) represents the time (years) since the park opened. There were 20 koalas in the park when it opened.
Determine the approximate rate of change in the number of koalas when \(t = 3\).
- (A) 46
- (B) 26
- (C) 25
- (D) 5
Section 2
Instructions
• Write using black or blue pen.
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page …
– If you do not do this, your original response will be marked.
• This section has nine questions and is worth 45 marks.
The graph shows the water level under a bridge over a 12-hour period.
a) Determine the equation of the cosine function that models the water level as a function of time after 12:00 am. [1 mark]
b) How long in the 12-hour period shown is the rate of change of water level more than 0.55 metres per hour? [3 marks]
A sandy beach has a fence on one side and ocean on the other. The width of the beach is the distance (in metres) from the fence to the water's edge. The width, \(w(t)\), at a certain point is given by \[ w(t) = a + b \sin\left(\frac{\pi}{6}t - \frac{\pi}{3}\right), \quad 0 \le t \le 24 \]
where \(t\) is time (in hours) since 6 am. The width of the beach is 8 metres at 8 am and 3 metres at 5 pm.
a) Determine \(a\) and \(b\). [2 marks]
b) Determine the rate of change of the width of the beach at 8 am and the first time after this when this rate of change is repeated. [2 marks]
END OF PAPER
