Tangent and normal to curve

4. \begin{figure}[h] \end{figure} The points \(P\) and \(Q\), as shown in Figure 2, have coordinates ( \(- 2,13\) ) and ( \(4 , - 5\) ) respectively. The straight line \(l\) passes through \(P\) and \(Q\).

1. Find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found. The quadratic curve \(C\) passes through \(P\) and has a minimum point at \(Q\).
2. Find an equation for \(C\). The region \(R\), shown shaded in Figure 2, lies in the second quadrant and is bounded by \(C\) and \(l\) only.
3. Use inequalities to define region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}
   

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10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).

8.
\includegraphics[max width=\textwidth, alt={}, center]{98667bd4-a612-4b16-a75b-8d8637e5976d-3_611_828_251_392} The diagram shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
2. Find an equation for \(l\).
3. Find the coordinates of the point where \(l\) intersects \(C\) again.

12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h] \end{figure}

1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \(^ { 2 }\).
3. Find the area of the region bounded by the curve and the \(x\)-axis.

1. Calculate \(\sum _ { r = 1 } ^ { 20 } 5 + 2 r\)
2. Find \(\int 5 x + 3 \sqrt { x } d x\)
3. (a) Express \(\sqrt { } 80\) in the form \(a \sqrt { } 5\), where \(a\) is an integer.
   (b) Express \(( 4 - \sqrt { 5 } ) ^ { 2 }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
4. The points \(A\) and \(B\) have coordinates \(( 3,4 )\) and \(( 7 , - 6 )\) respectively. The straight line \(l\) passes through \(A\) and is perpendicular to \(A B\). Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. (5)

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( 0,1 )\) and \(( 3,0 )\). The maximum point on the curve is \(( 1,2 )\). On separate diagrams, sketch the curve with equation

10 A curve has equation \(y = ( 2 x - 1 ) ( x + 3 ) ( x - 1 )\).

1. Sketch the curve, indicating the coordinates of all points of intersection with the axes.
2. Show that the gradient of the curve at the point \(P ( 1,0 )\) is 4 .
3. The line \(l\) is parallel to the tangent to the curve at the point \(P\). The curve meets \(l\) at the point where \(x = - 2\). Find the equation of \(l\), giving your answer in the form \(y = m x + c\).
4. Determine whether \(l\) is a tangent to the curve at the point where \(x = - 2\).

10 A curve has equation \(y = ( x + 2 ) ^ { 2 } ( 2 x - 3 )\).

1. Sketch the curve, giving the coordinates of all points of intersection with the axes.
2. Find an equation of the tangent to the curve at the point where \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\). \section*{OCR}

10 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 ).

1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
2. 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.
3. 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.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find f \({ } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).

9. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
2. Find an equation for \(l\).
3. Find the coordinates of the point where \(l\) intersects \(C\) again.

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
2. Find an equation for \(l\).
3. Find the coordinates of the point where \(l\) intersects \(C\) again.
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