Find equation of tangent

7 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_490_665_1793_740} The diagram shows the curve \(y = x ( x - 1 ) ( x - 2 )\), which crosses the \(x\)-axis at the points \(O ( 0,0 )\), \(A ( 1,0 )\) and \(B ( 2,0 )\).

1. The tangents to the curve at the points \(A\) and \(B\) meet at the point \(C\). Find the \(x\)-coordinate of \(C\).
2. Show by integration that the area of the shaded region \(R _ { 1 }\) is the same as the area of the shaded region \(R _ { 2 }\).

2 A curve has equation \(y = \frac { 2 + 3 \ln x } { 1 + 2 x }\).
Find the equation of the tangent to the curve at the point \(\left( 1 , \frac { 2 } { 3 } \right)\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4 Find the equation of the tangent to the curve \(y = \frac { \mathrm { e } ^ { 4 x } } { 2 x + 3 }\) at the point on the curve for which \(x = 0\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

3 The equation of a curve is \(y = x \sin 2 x\), where \(x\) is in radians. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 4 } \pi\).

3 A curve has equation $$y = \frac { 2 - \tan x } { 1 + \tan x }$$ Find the equation of the tangent to the curve at the point for which \(x = \frac { 1 } { 4 } \pi\), giving the answer in the form \(y = m x + c\) where \(c\) is correct to 3 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{d53a2d6b-4c5e-4bc6-8aa1-587e97c87920-2_371_839_1409_651} The diagram shows the part of the curve \(y = 3 \mathrm { e } ^ { - x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and the stationary point \(M\).

1. Find the equation of the tangent to the curve at the origin.
2. Find the coordinates of \(M\), giving each coordinate correct to 3 decimal places.

4 The equation of a curve is \(y = x \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right)\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The tangent to the curve at the point where \(x = 2\) meets the \(y\)-axis at the point with coordinates \(( 0 , p )\). Find \(p\).

1. The curve \(C\) has equation

$$y = \frac { 5 x ^ { 3 } - 8 } { 2 x ^ { 2 } } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form. The point \(P ( 2,4 )\) lies on \(C\).
2. Find an equation for the tangent to \(C\) at \(P\) writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. The curve \(C\) has equation

$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(R \left( 2 , \frac { 13 } { 2 } \right)\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(R\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

4. (i) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \cos 3 x\)
2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\) (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) which has equation $$y = \mathrm { e } ^ { x \sqrt { 3 } } \sin 3 x , \quad - \frac { \pi } { 3 } \leqslant x \leqslant \frac { \pi } { 3 }$$

1. Find the \(x\) coordinate of the turning point \(P\) on \(C\), for which \(x > 0\) Give your answer as a multiple of \(\pi\).
2. Find an equation of the normal to \(C\) at the point where \(x = 0\)

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

1. $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$ Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

6. (i) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(ii) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(iii) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

1 Find the equation of the tangent to the curve \(y = \frac { 5 x + 4 } { 3 x - 8 }\) at the point \(( 2 , - 7 )\).

1 Find the equation of the tangent to the curve $$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$ at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).

5 In this question you must show detailed reasoning.

1. Show that the gradient of the curve \(\mathrm { y } = \sqrt { \mathrm { x } } \left( \frac { 1 } { \mathrm { x } ^ { 2 } } - 2 \mathrm { x } \right)\) at the point \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) is \(- \frac { 99 } { 2 }\).
2. Find the equation of the tangent to the curve at \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

1. (a) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that

$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(b) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.

6. (a) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(b) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(c) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).

Find the equation of the tangent to the curve $$y = 3x^2(x + 2)^6$$ at the point \((-1, 3)\), giving your answer in the form \(y = mx + c\). [5]