Differentiate \(f\left( x \right) = \sqrt[3]{{{x^4}}}\) .

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Question 3 of 10

3. Question

Find the positive angle \(\theta \), correcting to the nearest degree, that makes the normal and the \(x\)-axis where the gradient of the normal to \(y = {x^3} – 3{x^2} + 7\) at \(x = 1\).

\(\theta = \) \(^\circ\)

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Hint

\(\text{gradient of tangent } \times \text{gradient of normal } = – 1\)

Question 4 of 10

4. Question

Find \(x\) for which \(f\left( x \right) = \dfrac{{2x}}{{x – 3}}\) is a decreasing function.

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Hint

increasing: \(f’\left( x \right) > 0\), decreasing: \(f’\left( x \right) < 0\)

Question 5 of 10

5. Question

Find the turning point and the nature of the turning point of \(y = \sqrt {1 – {x^2}} \).

turning point = ( , ), nature = (minimum or maximum)

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Question 6 of 10

6. Question

Find the maximum \(y\)-value given \(y = – 2{x^2} + 8x + 10\) for \(3 \le x \le 5\).

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Question 7 of 10

7. Question

Find the maximum and minimum values of \(P = 400 + 100\sin \dfrac{{\pi t}}{3}\).

maximum = , minimum =

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Incorrect

Question 8 of 10

8. Question

Find \(p\) and \(q\), if \(\int {{{\left( {\dfrac{{2x}}{3} + 1} \right)}^3}} dx = \dfrac{1}{p}{\left( {2x + 3} \right)^q} + C\).

If a curve has a stationary point \(\left( {1,5} \right)\) and a gradient of \(y’ = 8x + p\), where \(p\) is a constant, find the value of \(p\), and \(y\) when \(x = 2\).

\(p = \) , \(y = \)

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Question 10 of 10

10. Question

Find the value of \(k\) if \(\int_{ – 1}^2 {\left( {3{x^2} + 4x + k} \right)dx} = 30\).