QCAA Maths Methods Differential Calculus Mini Test 1
External Assessment Paper 1 — Technology-free
Number of marks: 10
Perusal time: 1 minute
Writing time: 15 minutes
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.
a) Determine the second derivative of \(y = x^3 - 3x^2\). [2 marks]
b) Use your result from Question 11a) to calculate the value of the second derivative when \(x = -1\). [1 mark]
c) Determine the x- and y-coordinates of the point on the graph of \(y = x^3 - 3x^2\) for which the rate of change of the first derivative is zero. [3 marks]
A person enters the lowest carriage of a miniature Ferris wheel with a six-metre diameter. The bottom carriage is one metre off the ground. When top speed is reached, it takes three seconds for a carriage to travel from the lowest to the highest point of the ride. It is claimed that:
The vertical motion of the Ferris wheel produces a maximum vertical acceleration on each rider that is more than half the acceleration of free fall.
Free fall occurs when gravity is the only force acting, resulting in an acceleration of 9.8 ms\(^{-2}\).
Evaluate the reasonableness of the claim.
END OF PAPER
