Find equation of normal

5 A curve has the equation \(\mathrm { y } = \frac { 3 } { 2 \mathrm { x } ^ { 2 } - 5 }\).
Find the equation of the normal to the curve at the point \(( 2,1 )\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

7 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_766_589_251_778} The diagram shows part of the curve \(y = 2 - \frac { 18 } { 2 x + 3 }\), which crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(A\) crosses the \(y\)-axis at \(C\).

1. Show that the equation of the line \(A C\) is \(9 x + 4 y = 27\).
2. Find the length of \(B C\).

11 The point \(P ( 3,5 )\) lies on the curve \(y = \frac { 1 } { x - 1 } - \frac { 9 } { x - 5 }\).

1. Find the \(x\)-coordinate of the point where the normal to the curve at \(P\) intersects the \(x\)-axis.
2. Find the \(x\)-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. {www.cie.org.uk} after the live examination series. }

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }\)
• the point \(P ( 4,12 )\) lies on \(C\)
  1. find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found,
  2. find \(\mathrm { f } ( x )\), giving each term in simplest form.

1. The curve \(C\) has equation

$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point \(P\) on \(C\) has \(x\) coordinate 3
Find an equation of the normal to \(C\) at the point \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).

1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
3. Find the \(x\)-coordinate of \(R\) in exact form.

3 Find the equation of the normal to the curve \(y = \frac { x ^ { 2 } + 4 } { x + 2 }\) at the point \(\left( 1 , \frac { 5 } { 3 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1

1. Differentiate \(\ln x\) with respect to \(x\).
2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).