2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).
- Find the equation of the tangent to the curve at the point \(( 2,9 )\).
- Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
- Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.
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\caption{Fig. 11}
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Fig. 11 shows a sketch of the cubic curve \(y = \mathrm { f } ( x )\). The values of \(x\) where it crosses the \(x\)-axis are - 5 , - 2 and 2 , and it crosses the \(y\)-axis at \(( 0 , - 20 )\). - Express \(\mathrm { f } ( x )\) in factorised form.
- Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
- Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 .
Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
- State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\).
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\includegraphics[alt={},max width=\textwidth]{4e8d7217-61f7-4ae4-96dd-d34e37c4d623-3_768_1023_223_598}
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\caption{Fig. 11}
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Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\). - Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
- Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
- Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
- Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
- Sketch the curve.
A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
- Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
- Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).