Equation of tangent line

4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.

9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).

1. Find the equation of the tangent to the curve at \(A\).
   The normal to the curve at \(A\) intersects the curve again at \(B\).
2. Find the coordinates of \(B\).
   The tangents at \(A\) and \(B\) intersect each other at \(C\).
3. Find the coordinates of \(C\).

1 A curve has equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 3 x - 4 x ^ { \frac { 1 } { 2 } } + 4\). Find the equation of the tangent to the curve at the point \(( 4,0 )\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x = \left( 9 + 16 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
The curve crosses the \(x\)-axis at the point \(A\).

1. State the coordinates of \(A\).
2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\), in terms of \(y\).
3. Find an equation of the tangent to the curve at \(A\).

1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ).

3 Fig. 8 shows the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\). P is the point on this curve with \(x\)-coordinate 1 , and R is the point \(\left( 0 , - \frac { 7 } { 8 } \right)\). \begin{figure}[h] \end{figure}

1. Find the gradient of PR.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that PR is a tangent to the curve.
3. Find the exact coordinates of the turning point Q .
4. Differentiate \(x \ln x - x\). Hence, or otherwise, show that the area of the region enclosed by the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is \(\frac { 59 } { 24 } - \frac { 1 } { 4 } \ln 2\).

5. (i) Given that $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } }$$ (ii) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

10. \begin{figure}[h] \end{figure} In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 3 } x ^ { 2 } - 2 \sqrt { x } + 3 \quad x \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 4
The line \(l\) is the tangent to \(C\) at \(P\).

1. Show that \(l\) has equation $$13 x - 6 y - 26 = 0$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the curve \(C\), the line \(l\) and the \(x\)-axis.
2. Find the exact area of \(R\).

8. A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).

1. Show that the tangent to the curve at the point where \(x = 1\) has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
2. Show that \(0.3 < \alpha < 0.4\)
3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
4. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 5 decimal places. END

4 A curve has equation $$y = \frac { x ^ { 2 } } { 8 } + 4 \sqrt { x }$$ 4

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 4
2. The point \(P\) with coordinates \(( 4,10 )\) lies on the curve.
   Find an equation of the tangent to the curve at the point \(P\) □
   4
3. Show that the curve has no stationary points.

\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.

1. Find the equation of the tangent to the curve at \(A\). [5]
2. Find, showing all necessary working, the area of the shaded region. [5]
3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]

A curve has equation \(y = 6x\sqrt{x} + \frac{32}{x}\) for \(x > 0\)

1. Find \(\frac{dy}{dx}\) [4 marks]
2. The point \(A\) lies on the curve and has \(x\)-coordinate 4 Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis. [5 marks]