VCE Maths Methods Diff Calculus Mini Test 9

For the curve \(y = x^2 - 5\), the tangent to the curve will be parallel to the line connecting the positive \(x\)-intercept and the \(y\)-intercept when \(x\) is equal to

• A. \(\sqrt{5}\)
• B. 5
• C. -5
• D. \(\frac{\sqrt{5}}{2}\)
• E. \(\frac{1}{\sqrt{5}}\)

A rectangle is formed by using part of the coordinate axes and a point \((u, v)\), where \(u > 0\) on the parabola \(y = 4-x^2\).

Which one of the following is the maximum area of the rectangle?

• A. 4
• B. \(\frac{2\sqrt{3}}{3}\)
• C. \(\frac{8\sqrt{3}-4}{3}\)
• D. \(\frac{8}{3}\)
• E. \(\frac{16\sqrt{3}}{9}\)

The average rate of change of the function \(f\) with rule \(f(x) = 3x^2 - 2\sqrt{x+1}\), between \(x=0\) and \(x=3\), is

• A. 8
• B. 25
• C. \(\frac{53}{9}\)
• D. \(\frac{25}{3}\)
• E. \(\frac{13}{9}\)

Question 4 [2021 Exam 2 Section A Q19]

Which one of the following functions is differentiable for all real values of \(x\)?

• A. \( f(x) = \begin{cases} x & x < 0 \\ -x & x \ge 0 \end{cases} \)


• B. \( f(x) = \begin{cases} x & x < 0 \\ -x & x > 0 \end{cases} \)


• C. \( f(x) = \begin{cases} 8x+4 & x < 0 \\ (2x+1)^2 & x \ge 0 \end{cases} \)


• D. \( f(x) = \begin{cases} 2x+1 & x < 0 \\ (2x+1)^2 & x \ge 0 \end{cases} \)


• E. \( f(x) = \begin{cases} 4x+1 & x < 0 \\ (2x+1)^2 & x \ge 0 \end{cases} \)

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Correct Answer: E
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Question 5 [2021 Exam 2 Section A Q8]

The graph of the function \(f\) is shown below.

The graph corresponding to \(f'\) is

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Correct Answer: E
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Let \(f: [-\pi, \pi] \to R\), where \(f(x) = 2\sin(2x) - 1\).

a. Calculate the average rate of change of \(f\) between \(x = -\frac{\pi}{3}\) and \(x = \frac{\pi}{6}\). 2 marks

b. Calculate the average value of \(f\) over the interval \(-\frac{\pi}{3} \le x \le \frac{\pi}{6}\). 3 marks