1.07c Sketch gradient function: for given curve

5 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-04_700_727_260_242} The diagram shows a curve \(C\) for which \(y\) is inversely proportional to \(x\). The curve passes through the point \(\left( 1 , - \frac { 1 } { 2 } \right)\).

1. 1. Determine the equation of the gradient function for the curve \(C\).
   2. Sketch this gradient function on the axes in the Printed Answer Booklet.
2. The diagram indicates that the curve \(C\) has no stationary points. State what feature of your sketch in part (a)(ii) corresponds to this.
3. The curve \(C\) is translated by the vector \(\binom { - 2 } { 0 }\). Find the equation of the curve after it has been translated.

10 In this question you must show detailed reasoning.

1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.

12 A function is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - x\).

1. By considering \(\frac { f ( x + h ) - f ( x ) } { h }\), show from first principles that \(f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1\).
2. Sketch the gradient function \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that the curve \(y = f ( x )\) has a single point of inflection which is not a stationary point.

10 The diagram below shows the curve \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-07_942_679_1500_242} Sketch the graph of the gradient function, \(y = f ^ { \prime } ( x )\), on the copy of the diagram in the Printed Answer Booklet.

4 The graph of $$y = \mathrm { f } ( x )$$ where $$f ( x ) = a x ^ { 2 } + b x + c$$ is shown in Figure 1. \begin{figure}[h] \end{figure} Which of the following shows the graph of \(y = \mathrm { f } ^ { \prime } ( x )\) ? Tick \(( \checkmark )\) one box. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_2272_437_429_557}
□ \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_117_117_1151_1133}
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A curve has equation \(y = f(x)\) where $$f(x) = x(6 - x)$$

1. Find \(f'(x)\) [2 marks]
2. The diagram below shows the graph of \(y = f(x)\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes. [3 marks] \includegraphics{figure_9}

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_6}

1. Sketch the graph of \(y = f(-x)\) [2 marks]
2. Sketch the graph of \(y = 2f(x) - 4\) [2 marks]
3. Sketch the graph of \(y = f'(x)\) [3 marks]

The diagram shows the graph of \(y = f(x)\), where \(f(x)\) is a quadratic function of \(x\). A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_2}

1. On the copy of the diagram in the Printed Answer Booklet, draw a possible graph of the gradient function \(y = f'(x)\). [3]
2. State the gradient of the graph of \(y = f''(x)\). [1]

A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).

1. Find \(\frac{dy}{dx}\). [1]
2. Hence sketch the gradient function for the curve. [4]
3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]