1.07m Tangents and normals: gradient and equations

The curve \(C\) has equation \(y = \frac { 1 } { 8 } x ^ { 3 } - \frac { 24 } { \sqrt { x } } + 1\)

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in its simplest form.
   (3) The point \(P ( 4 , - 3 )\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

1. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , \quad x > 0\), passes through the point \(P ( 4,1 )\).

Given that \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }\)

1. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
   (4)
2. Find \(\mathrm { f } ( x )\).
   (5)
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1. A curve has equation \(y = \mathrm { f } ( x )\), where

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).

1. State the coordinates of the point \(P\).
2. Find \(f ^ { \prime } ( x )\).
3. Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found. The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
4. State the possible values of \(a\).
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1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 3 } + \frac { 4 } { \sqrt { x } } + \frac { 8 } { 3 x } - 5 \quad x > 0$$

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

6. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(C\) passes through the point \(P ( 8,2 )\)
• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )\)
  1. find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
     (3)
  2. Find, in simplest form, \(\mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
4. find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point \(P ( 4,12 )\) Give your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The curve with equation \(y = \mathrm { f } ( x )\) also passes through the point \(P ( 4,12 )\) Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
2. find \(\mathrm { f } ( x )\) giving the coefficients in simplest form.

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.

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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6. The curve \(C\) has equation \(y = \frac { 4 } { x } + k\), where \(k\) is a positive constant.

1. Sketch a graph of \(C\), stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the \(x\)-axis. The line with equation \(y = 10 - 2 x\) is a tangent to \(C\).
2. Find the possible values for \(k\). \(\_\_\_\_\) -

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

The point \(P \left( 4 , \frac { 32 } { 3 } \right)\) lies on the curve.
Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3\)
• \(\quad \mathrm { f } ^ { \prime } ( x ) = 5\) at \(P\) find
  1. the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found,
  2. \(\mathrm { f } ( x )\).
     

8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
2. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\) Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
3. find the value of \(\alpha\)

5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\) Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
2. find the area of triangle \(B P A\).

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

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

Given that

• \(f ^ { \prime } ( x ) = \frac { 4 x ^ { 2 } + 10 - 7 x ^ { \frac { 1 } { 2 } } } { 4 x ^ { \frac { 1 } { 2 } } }\)
• the point \(P ( 4 , - 1 )\) lies on \(C\)
  1. 1. find the value of the gradient of \(C\) at \(P\)
     2. Hence find the equation of 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 to be found.
  2. Find \(\mathrm { f } ( x )\).

13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
2. Hence find the coordinates of the turning point on the curve \(C\).
3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}

13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)

1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).

12. $$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C\), where \(A , B , C\) and \(k\) are constants to be determined.
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\).
3. Find an equation of the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 4\) 2. LIIIII

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,11 )\).
Line \(l\) is the tangent to \(C\) at the point \(P\).

1. Use calculus to show that \(l\) has equation \(y = 5 x - 9\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(x = 1\), the \(x\)-axis and the line \(l\).
2. Find, by using calculus, the area of \(R\), giving your answer to 2 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

9. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 9 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\). The point \(P ( - 2,7 )\) lies on \(C\).
3. Find an equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   

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

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\) The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
3. Use integration to find the exact area of the shaded region \(R\).

8.

1. Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$
2. show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\)
3. Hence find the possible values of \(b\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$ The point \(P ( 4,6 )\) lies on \(C\).

1. Use calculus to show that the normal to \(C\) at the point \(P\) has equation $$2 y + x = 16$$ The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
2. Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
3. Showing all your working, use calculus to find the exact area of \(R\).

11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).

1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.