QCAA Maths Methods Differential Calculus Mini Test 2
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.
A football coach offered a 12-day intensive training clinic. During the clinic, the height that each player could kick a football was monitored.
One player's kick heights could be modelled by \(H(t) = \log_{10}(10t+10)+5\), \(0 \le t \le 12\), where \(H(t)\) is vertical height (m) and \(t\) is the time (days) spent in training.
a) Determine the initial height that the player could kick the ball. [1 mark]
b) Determine the training time needed for the player to be able to kick the ball to a height of 7 m. [1 mark]
c) Determine the overall improvement in kick height achieved by completing the clinic. [2 marks]
d) Determine the rate of change in kick height when \(t = 1.5\) days. [1 mark]
e) Determine the training time (as a decimal) when the rate of change in kick height is 0.09 m/day. [1 mark]
Determine the derivative of \(f(x) = \ln x^2 + \ln(x-5)^3\). Express the derivative as a single fraction in its simplest and factorised form.
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