Normal or tangent line problems

2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \left( \frac { 1 } { 2 } x - 1 \right) ^ { - 4 }\). It is given that the curve passes through the point \(P ( 6,4 )\).

1. Find the equation of the tangent to the curve at \(P\).
2. Find the equation of the curve.

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { } ( 6 - 2 x ) }\), and \(P ( 1,8 )\) is a point on the curve.

1. The normal to the curve at the point \(P\) meets the coordinate axes at \(Q\) and at \(R\). Find the coordinates of the mid-point of \(Q R\).
2. Find the equation of the curve.

5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find

1. the equation of the normal to the curve at \(P\),
2. the equation of the curve.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 4 x - 3 ) }\) and \(P ( 3,3 )\) is a point on the curve.

1. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\).
2. Find the equation of the curve.

6. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve with equation $$y = \sqrt { 4 x - 7 }$$ The line \(l\), shown in Figure 3, is the normal to the curve at the point \(P ( 8,5 )\)

1. Use calculus to show that an equation of \(l\) is $$5 x + 2 y - 50 = 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and \(l\).
2. Use algebraic integration to find the exact area of \(R\).

11

1. Use the substitution \(u ^ { 2 } = x ^ { 2 } + 3\) to show that \(\int \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 4 } { 3 } \left( x ^ { 2 } - 6 \right) \sqrt { x ^ { 2 } + 3 } + c\).
2. In this question you must show detailed reasoning.
   \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_620_951_1836_317} The graph shows part of the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 2 } }\).
   Find the exact area enclosed by the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } }\), the normal to this curve at the point \(( 1,2 )\) and the \(x\)-axis.