Find normal line equation at given point

5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in simplest form. The point \(P \left( 4 , \frac { 11 } { 3 } \right)\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

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

$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying your answer,
2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\).
2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).

9. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), and \(\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }\). Given that the point \(P ( 4,1 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\) and simplify your answer.
2. Find an equation of the normal to \(C\) at the point \(P ( 4,1 )\).

11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}

10. The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 2 respectively.

1. Show that the length of \(P Q\) is \(\sqrt { 170 }\).
2. Show that the tangents to \(C\) at \(P\) and \(Q\) are parallel.
3. Find an equation for the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \(\_\_\_\_\)

10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   (2)
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
   (1)
   Given that the point \(P ( 2,4 )\) lies on \(C\),
3. find \(y\) in terms of \(x\),
   (5)
4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   (5)
   1. continued

3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 Find the equation of the normal to the curve \(y = \frac { 6 } { x ^ { 2 } } - 5\) at the point on the curve where \(x = 2\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

5 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).

1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the inequalities that define the shaded region, including its boundaries.

17 Show that, for the curve \(y = x ^ { 2 }\), the equation of the normal at the point \(\left( t , t ^ { 2 } \right)\) is \(y = - \frac { x } { 2 t } + t ^ { 2 } + \frac { 1 } { 2 }\), as given in line 27.

2 A curve has equation \(y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P\) on the curve has coordinates \(( 2,3 )\).
   1. Show that the gradient of the curve at \(P\) is 5 .
   2. Hence find an equation of the normal to the curve at \(P\), expressing your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
3. Determine whether \(y\) is increasing or decreasing when \(x = 1\).

6 The curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the \(x\)-axis at the origin \(O\), and the point \(A ( 2 , - 6 )\) lies on the curve.

1. 1. Find the gradient of the curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) at the point \(A\).
   2. Hence find the equation of the normal to the curve at the point \(A\), giving your answer in the form \(x + p y + q = 0\), where \(p\) and \(q\) are integers.
2. 1. Find the value of \(\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

3. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   [0pt] [P1 June 2003 Question 8*]

1 A curve is defined for \(x > 0\) by the equation \(y = x + \frac { 2 } { x }\).

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that the gradient of the curve at the point \(P\) where \(x = 2\) is \(\frac { 1 } { 2 }\).
2. Find an equation of the normal to the curve at this point \(P\).

4 A curve has equation \(y = \frac { 1 } { x ^ { 2 } } + 4 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P ( - 1 , - 3 )\) lies on the curve. Find an equation of the normal to the curve at the point \(P\).
3. Find an equation of the tangent to the curve that is parallel to the line \(y = - 12 x\).
   [0pt] [5 marks]

3 The diagram shows a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(y\)-coordinate of the maximum point \(M\).
3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
   [0pt] [3 marks]

3 In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\).

The point \(P\) with \(x\) coordinate 3 lies on \(C\) \section*{Given}

• \(\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { 2 } + k x + 3\) where \(k\) is a constant
• the normal to \(C\) at \(P\) has equation \(y = - \frac { 1 } { 24 } x + 5\)
  1. show that \(k = - 5\)
  2. Hence find \(\mathrm { f } ( x )\).

The line \(3y + x = 25\) is a normal to the curve \(y = x^2 - 5x + k\). Find the value of the constant \(k\). [6]

The gradient of the curve \(C\) is given by $$\frac{dy}{dx} = (3x - 1)^2.$$ The point \(P(1, 4)\) lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\). [4]
2. Find an equation for the curve \(C\) in the form \(y = f(x)\). [5]
3. Using \(\frac{dy}{dx} = (3x - 1)^2\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2x\). [2]

For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),

1. find \(\frac{dy}{dx}\). [2]

The point \(A\), on the curve \(C\), has \(x\)-coordinate 1.

1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). [5]

The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,

1. find the coordinates of \(B\). [6]